The
diffinition
of
number
.
NVMBER
IS
AS
much
to
saye
,
as
a
multitude
compounde
of
many
vnities
,
as
two
is
compounde
of
twoo
vnities
,
three
is
compounde
of
three
vnities
,
foure
of
foure
vnities
,
fiue
of
fiue
vnities
ten
of
ten
,
fourtene
of
fourtene
,
fiftene
of
fiftene
,
twentie
of
twentie
vnities
,
&c.
And
therefore
an
vnitie
is
no
number
,
but
the
beginning
and
originall
of
number
,
as
if
you
do
multiplie
or
deuide
an
vnitie
by
it selfe
,
it
is
resolued
into
it selfe
withoute
any
increase
.
But
it
is
in
number
otherwise
,
for
there
can
be
no
number
,
howe
great
soeuer
it
bee
,
but
that
it
may
continuallye
be
encreased
by
adding
euermore
one
vnitie
vnto
the
same
.
¶
Numeration
.
The
first
Chapter
.
NVmeration
is
the
arte
whereby
to
expresse
and
declare
the
value
of
any
summe
or
number
proposed
,
and
is
of
twoo
kyndes
,
the
one
gathereth
the
value
of
a
summe
proposed
,
and
the
other
expresseth
any
summe
conceaued
by
due
figures
&
places
,
for
the
value
is
one
thing
,
&
the
figures
are
another
thing
:
&
that
commeth
partlye
by
the
diuersitie
of
figures
,
but
chiefly
of
the
places
wherin
they
be
orderly
set
.
And
first
marke
,
that
there
are
but
ten
figures
or
characters
which
are
vsed
in
Arithmetick
,
wherof
nine
of
them
are
called
signifying
figures
,
&
the
tenth
is
called
a
Ciphar
,
which
is
made
like
an
0
,
&
of
it self
signifieth
nothing
,
but
it
being
ioined
with
any
of
the
other
figures
,
encreseth
their
valu
,
&
these
be
thei
.
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
one
|
twoo
|
thrée
|
foure
|
fiue
|
sixe
|
seuen
|
eyght
|
nine
.
|
Also
you
shall
vnderstande
that
euerye
one
of
these
fygures
hath
twoo
values
:
One
is
alwaye
certaine
and
hath
hys
signification
of
hys
owne
forme
,
and
the
other
is
vncertaine
which
hee
taketh
of
his
place
.
A
place
is
called
the
seate
or
rome
that
a
figure
standeth
in
,
A
place
.
and
howe
many
figures
soeuer
are
written
in
one
summe
or
number
,
so
manye
places
hath
the
whole
value
therof
.
And
that
is
called
the
first
place
(
which
nexte
is
towarde
the
ryghte
hande
)
of
anye
summe
,
and
so
reckning
by
order
towarde
the
lefte
hand
,
so
that
that
place
is
last
which
is
nexte
the
lefte
hande
.
And
contrarywyse
,
when
you
expresse
the
value
of
the
fygures
in
any
summe
,
you
muste
begynne
at
the
lefte
hande
,
and
so
recken
toward
the
right
hand
.
Euerye
of
these
nyne
fygures
,
(
which
are
called
signifying
fygures
)
hath
his
own
simple
value
when
hee
is
founde
alone
,
or
in
the
first
place
of
any
summe
.
In
the
seconde
place
towarde
the
left
hande
,
he
betokeneth
his
own
value
ten
times
.
As
70.
is
,
seuen
times
ten
:
that
is
to
saye
seuentie
.
80.
is
eyght
tymes
ten
:
that
is
to
say
eyghtie
.
In
the
third
place
euery
figure
betokeneth
hys
owne
value
a
hundreth
tymes
.
As
700.
in
that
thirde
place
betokeneth
a
hundreth
times
7.
that
is
to
say
,
seuen
hundred
:
In
the
fourth
place
euerye
figure
betokeneth
hys
owne
value
a
thousande
times
.
As
7000.
is
seuen
thousande
,
and
8000.
is
eight
thousande
.
These
foure
first
places
must
be
had
perfectly
in
minde
,
yea
and
that
by
heart
,
for
by
the
knowledge
of
them
you
maye
expresse
all
kinde
of
numbers
how
great
so
euer
they
bee
.
In
the
fift
place
euery
figure
betokeneth
his
owne
value
ten
thousand
times
.
As
70000.
is
ten
times
seauen
thousande
,
that
is
to
saye
seuentie
thousande
:
In
the
sixt
place
euery
fygure
standeth
for
his
owne
value
,
a
hundreth
thousand
times
.
As
700000.
is
seuen
hundreth
thousande
.
The
seauenth
place
M.
M.
times
,
or
a
million
:
as
7000000.
is
seuen
M.
M.
or
seuen
millions
.
And
the
eyght
place
ten
M.
M.
times
,
or
ten
millions
,
so
that
euery
place
towarde
the
left
hande
,
excedeth
the
former
ten
tymes
.
But
nowe
for
the
easy
reading
,
and
redye
expressing
orderly
of
any
summe
proposed
,
you
shall
practise
thys
maner
,
folowing
.
And
for
example
I
propone
this
number
765432658.
in
the
which
are
.ix.
places
.
In
the
fyrste
place
is
.8
.
&
betokeneth
but
eyght
,
in
the
seconde
place
is
5.
and
betokeneth
ten
times
fiue
that
is
fiftye
,
in
the
third
place
is
6.
and
betokeneth
a
hundreth
times
sixe
,
that
is
vj.
C.
In
the
fourth
place
is
2.
and
that
is
twoo
M.
And
3.
in
the
fifte
place
is
ten
M.
times
3.
that
is
xxx
.
M.
So
.4
.
in
the
firste
place
is
C.
thousande
times
4.
foure
,
that
is
foure
.
C.M.
then
.5
.
in
the
seuenth
place
is
a
M.
M.
times
5.
that
is
fiue
M.M.
or
rather
fiue
millions
.
And
6.
in
the
eyght
place
is
six
times
ten
millions
,
that
is
lx.
millions
.
And
last
of
al
.vij.
in
the
ix
.
place
,
is
vij
.
C.
millions
.
Now
foloweth
the
practise
.
First
put
a
pricke
ouer
the
fourth
figure
,
and
so
ouer
the
seuenth
,
and
likewise
ouer
the
tenth
.
And
also
ouer
the
13.16
.
or
.19
.
if
you
had
so
many
,
and
so
still
leauing
twoo
figures
betweene
euerye
twoo
pricks
and
these
roomes
from
one
prick
to
an
other
are
called
ternaries
,
Ternaries
.
then
you
muste
pronounce
euery
three
figures
from
one
pricke
to
an
other
as
though
they
were
written
alone
from
the
rest
.
And
at
th
ende
of
their
value
,
adde
so
many
times
a
thousand
,
as
your
number
hath
pricks
(
that
is
to
say
,
if
there
be
but
1.
prick
,
it
is
but
1.
M.
if
2.
pricks
a
M.M.
or
else
a
million
,
if
3.
prickes
M.M.M.
or
a
M.
million
,
&
so
consequentli
of
al
other
figures
folowing
)
Then
come
likewise
to
the
next
iij.
fygures
,
and
sound
them
as
if
they
were
a
part
from
the
rest
,
and
adde
to
their
value
so
many
times
thousandes
as
there
are
pricks
betwene
them
&
the
first
place
of
your
whole
number
.
And
so
doe
by
the
next
iij.
figures
folowing
&
of
all
the
rest
likewise
as
in
example
.
451234678567.
The
first
prick
ouer
8.
in
the
fourth
place
,
which
is
the
place
of
a
M.
the
seconde
pricke
is
ouer
4.
in
the
seuenth
place
,
which
is
the
place
of
a
M.M.
or
one
million
,
the
thirde
prick
is
ouer
the
tenth
place
which
is
the
place
of
a
M.M.M.
or
of
a
M.
million
,
as
in
the
former
example
.
Then
for
the
expressing
of
thys
number
by
the
value
of
euery
figure
accordinge
to
the
place
wherein
they
stande
,
you
shall
fyrste
begynne
at
the
last
prick
ouer
1.
and
take
it
and
the
other
twoo
fygures
5.
and
4.
which
doe
folowe
hym
,
and
value
them
alone
and
they
are
foure
Clj.
MMM
.
or
else
CCCCli
.
M.
millions
.
Then
take
the
other
three
fygures
from
1.
to
the
next
pricke
,
and
value
them
as
if
they
were
a
parte
from
the
other
,
and
they
are
.234
.
which
are
CCxxxiiij
.
millions
,
or
234.
MM.
Then
come
to
the
thirde
prick
ouer
.8
.
and
take
the
other
two
figures
behinde
it
,
and
recken
them
likewise
as
if
they
were
alone
,
and
they
are
sixe
Clxxviij
.
M.
And
laste
of
all
come
to
the
other
thre
figures
which
remayne
,
that
is
.567
.
and
they
are
fiue
Clxvij
.
Thus
the
whole
sum
of
these
figures
,
is
four
Clj.
M.
two
Cxxxiiij
.
millions
,
six
Clxxviij
.
M.
fiue
Clxvij
,
as
before
.
Three
kindes
of
Number
.
Note
also
that
whole
number
is
deuided
into
thre
kindes
,
that
is
to
say
,
diget
number
,
article
and
mixt
or
compounde
number
.
The
diget
number
,
Diget
.
is
all
maner
of
numbers
vnder
ten
,
which
are
these
nine
fygures
,
1
2
3
4
5
6
7
8
9.
of
the
which
I
haue
spoken
before
.
Article
.
The
Article
number
is
any
kinde
which
beginneth
with
a
Cipher
as
this
0.
and
they
may
euer
be
deuided
iust
by
10.
without
anye
remaine
,
as
these
,
10.
20.
30.
40.
50.
100.
&
all
other
such
like
.
The
mixte
or
compounde
number
,
Mixt
or
compounde
.
conteineth
diuers
and
many
articles
,
or
at
the
least
one
article
,
and
a
diget
,
as
.11
.
12.
16.
19.
22.
38.
108.
1007.
and
so
forth
.
And
as
any
article
number
may
be
made
a
compounde
,
by
putting
therto
a
diget
,
euen
so
likewyse
euery
compounde
number
,
may
be
made
an
Article
number
by
adding
thervnto
a
0.
¶
And
here
foloweth
a
briefe
rehersall
of
the
order
and
Denominatours
of
the
places
.
And
this
shal
be
sufficient
for
Numeration
.
The
order
of
the
places
.
|
Tenth
.
|
Ninth
.
|
Eyght
.
|
Seuenth
.
|
Sixte
.
|
Fyfthe
.
|
Fourth
.
|
Thyrde
.
|
Seconde
.
|
Fyrst
.
|
|
|
4
|
3
|
2
|
1
|
0
|
1
|
8
|
3
|
4
|
5
|
|
|
M.
of
Millions
.
|
C.
of
Millions
.
|
X.
of
Millions
.
|
Millions
.
|
C.
of
Thousandes
.
|
X.
Thousandes
.
|
Thousandes
.
|
Hundrethes
.
|
Tenthes
.
|
Vnities
.
|
The
Denominatours
of
the
places
.
|
Addition
in
whole
number
.
Chap.
2.
example of numbers to be added, with a horizontal line below the bottom number
Nowe
beginne
alwayes
at
the
firste
places
towarde
your
right
hande
,
and
put
togither
the
three
first
fygures
of
these
thre
summes
,
and
looke
what
commeth
of
them
,
write
that
vnder
them
beneath
the
line
,
as
in
saying
3.4
.
and
1.
being
put
togither
doe
make
8.
wryte
8.
vnder
three
as
thus
.
And
then
go
to
the
seconde
fygures
and
do
likewyse
:
as
in
saying
.
2.3
.
&
3.
maketh
8.
write
8.
vnder
2.
as
here
you
see
.
And
likewyse
doe
wyth
the
fygures
that
be
in
the
thirde
place
,
in
saying
2.
3.
and
4.
are
9.
put
nine
vnder
them
,
&
so
wyll
your
whole
summe
appere
thus
:
whereby
you
maye
perceaue
that
those
thre
summes
being
added
togither
doe
make
988.
li.
And
this
is
the
art
of
addition
according
to
his
simplicitie
,
if
the
sum
of
any
place
do
not
exceede
a
diget
number
.
But
in
case
the
sum
of
ani
one
place
cannot
be
expressed
by
one
figure
,
but
by
twoo
,
you
shall
put
the
first
of
those
figures
vnder
the
line
,
and
keepe
the
other
in
your
minde
,
for
to
adde
it
vnto
the
first
figure
of
the
next
place
.
And
if
the
same
next
place
cannot
be
aualued
but
by
two
figures
,
you
must
in
lyke
maner
put
the
first
of
those
figures
vnder
the
lyne
,
and
reserue
the
seconde
for
the
other
place
next
after
,
and
thus
must
you
doe
from
one
place
to
another
vntil
you
haue
come
to
the
last
place
,
where
in
case
you
doe
finde
that
the
summe
be
of
twoo
figures
,
you
muste
set
them
both
downe
bycause
it
is
the
ende
of
that
worke
,
as
in
this
example
.
734682456
|
450932345
|
13467891
|
4672123
|
1203754815
|
Where
the
fyrst
figures
are
3.1.5.6
.
which
added
togither
maketh
15.
&
for
that
,
that
15.
is
of
twoo
figures
,
I
doe
put
the
fyrste
figure
5.
vnder
the
line
,
and
keepe
the
second
figure
(
which
is
1.
)
in
my
minde
,
the
which
.
I
must
adde
with
the
next
fygures
of
the
seconde
place
,
that
is
to
saye
wyth
2.9.4
.
and
5.
the
whych
togither
make
21.
I
write
1.
vnder
the
line
for
the
seconde
fygure
of
that
addition
,
that
is
to
saye
after
5.
and
I
kéepe
2.
to
be
added
vnto
the
third
place
the
which
wyth
the
other
fygures
1.8.3.4
.
doe
make
18.
therfore
I
put
8.
next
after
1.
in
the
thirde
place
vnder
the
line
,
and
keepe
1.
to
bee
added
vnto
the
fygures
of
the
fourth
place
,
which
is
wyth
2.7.2.2
.
the
which
with
the
1.
that
I
keepe
do
make
14.
I
set
downe
4.
for
the
fourth
fygure
(
vnder
the
line
)
that
is
to
say
,
after
8.
and
I
keepe
1.
to
bée
added
vnto
the
figures
of
the
fift
place
,
the
which
is
7.6.3.8
.
with
the
1.
that
I
kepe
maketh
25.
I
put
5.
in
the
fift
place
vnder
the
lyne
next
after
4.
and
keepe
2.
in
minde
to
be
added
with
the
fygures
of
the
sixt
place
,
that
is
wyth
6.4.9.6
.
and
that
2.
whych
I
keepe
,
maketh
27.
I
write
downe
7.
vnder
the
line
in
the
sixt
place
,
and
I
kepe
twoo
which
I
adde
with
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
do
put
0.
vnder
the
line
in
the
eyght
place
,
and
then
I
adde
1.
vnto
the
ninth
place
,
that
is
to
saye
with
4.7
.
and
they
make
12.
the
which
12.
I
write
at
length
vnder
the
lyne
bicause
it
is
the
ende
of
thys
addition
,
and
thys
is
to
be
done
of
all
such
like
.
And
for
the
easier
vnderstanding
of
that
which
wee
haue
spoken
of
addition
,
you
may
examine
these
twoo
other
examples
folowing
,
in
the
which
the
first
hath
these
numbers
.
3570.2763.579.28
.
which
beyng
added
togither
,
doe
make
thys
number
6940
,
and
in
the
seconde
example
doth
resulte
thys
number
51683.
by
adding
togither
of
these
numbers
,
47630
,
3756
,
272
,
25
,
as
here
vnder
written
.
The
numbers
to
be
added
.
|
3570
,
|
47630
|
|
2763
|
3756
|
|
579
|
272
|
|
28
|
25
|
The
line
put
between
.
|
|
|
The
summe
of
this
addition
.
|
6940
|
51683
|
Of
Substraction
in
whole
number
.
The.
3.
Chapter
.
SVbstraction
teacheth
howe
you
shal
abate
one
lesser
number
from
a
greater
,
&
what
ther
doth
remain
after
that
you
shal
haue
abated
the
same
,
I
speake
not
of
the
abating
of
one
egall
number
,
from
an
other
egall
vnto
it
,
for
the
facilitie
therof
requireth
no
rule
.
In
Substraction
are
founde
three
numbers
,
the
one
is
that
,
from
the
which
the
substraction
is
made
.
The
second
is
the
number
that
is
to
be
substracted
and
the
third
is
the
number
which
remaineth
after
the
substraction
is
ended
.
As
when
I
would
substract
.
25.
from
40.
The
40.
is
the
number
from
the
which
the
substraction
is
made
.
25.
is
the
number
to
be
substracted
,
&
15.
is
the
number
which
remaineth
after
you
haue
done
the
substraction
,
here
foloweth
the
practise
.
You
shall
put
the
lesser
number
vnder
the
greater
in
such
sorte
that
euerye
figure
of
the
one
number
may
aunswere
vnto
euerye
figure
of
the
other
orderly
,
and
then
draw
a
right
line
vnder
those
two
numbers
as
you
did
in
Addition
.
Then
must
you
beginne
at
the
right
hande
and
take
the
first
figure
of
the
vndermost
number
&
abate
that
from
the
firste
figure
of
the
vppermost
number
,
&
that
which
remaineth
you
muste
set
vnderneth
the
line
right
vnder
the
figure
which
you
haue
substracted
.
Then
afterwarde
take
likewise
the
second
figure
of
the
nethermost
number
,
and
abate
that
also
from
the
seconde
fygure
of
the
higher
number
.
The
thyrde
from
the
thirde
,
and
so
forth
of
all
the
rest
tyll
you
come
to
the
ende
,
putting
alwaies
the
remaine
of
euery
figure
vnder
the
line
in
hys
order
,
example
.
I
wyll
substract
.
2345.
from
9876.
after
that
I
haue
put
them
downe
according
to
the
maner
aforesaide
.
I
take
firste
5.
from
6.
and
there
resteth
1.
the
which
I
set
vnder
the
line
right
against
5.
Secondly
I
abate
4.
from
7.
and
there
resteth
3.
the
which
I
set
in
the
seconde
place
vnder
the
line
,
next
after
1.
Thirdlye
,
I
abate
3.
from
8.
and
there
resteth
5.
The
which
I
put
vnder
the
line
in
the
thirde
place
,
fynally
I
do
abate
2.
from
9.
and
there
resteth
7.
the
which
I
put
vnder
the
line
in
the
fourth
and
laste
place
,
and
thus
is
this
Substraction
ended
,
by
the
which
there
resteth
.
7531.
But
when
twoo
figures
of
one
likenesse
doe
chaunce
to
meete
,
so
that
the
one
must
be
abated
from
the
other
,
as
if
I
should
abate
7.
from
7.
there
remaineth
nothing
,
and
then
must
I
set
a
cipher
0.
vnder
the
line
.
But
when
the
fygure
which
is
to
be
abated
,
doth
exceede
the
figure
which
is
ouer
him
,
so
that
it
cannot
be
taken
out
of
the
same
figure
.
Then
muste
you
abate
the
nether
figure
from
10.
And
that
which
doth
remaine
you
shall
adde
vnto
the
same
figure
which
is
vppermost
.
And
the
sum
which
commeth
therof
shal
you
set
vnder
the
line
.
But
whensoeuer
you
doe
borrowe
any
such
10.
of
the
ouer
number
:
you
must
adde
1.
vnto
the
next
nether
figure
folowing
which
is
to
be
abated
.
And
there
is
nothing
else
to
be
done
in
substraction
.
Exaumple
,
I
wyll
substract
93576.
from
4037479.
after
that
I
haue
placed
my
twoo
numbers
,
as
I
ought
to
doe
,
I
doe
first
abate
.
6.
from
9.
and
there
resteth
3.
then
I
put
the
3.
vnder
the
line
ryght
against
6.
And
secondely
I
abate
7.
from
7.
And
there
resteth
nothing
.
I
doe
put
a
cipher
0.
vnder
the
line
right
against
7.
in
the
second
place
.
Then
I
come
to
the
thyrde
place
where
I
fynde
5.
which
I
cannot
bate
from
the
figure
ouer
him
,
which
is
but
4.
therfore
I
doe
abate
it
from
10.
as
before
I
taught
and
there
resteth
5.
the
which
I
doe
adde
wyth
the
4.
which
is
ouer
him
,
and
that
maketh
9.
I
put
9.
in
the
third
place
vnder
the
line
for
the
thirde
figure
.
Fourthly
,
for
the
10.
whych
I
borowed
I
adde
one
vnto
the
next
nether
fygure
which
is
three
,
and
they
make
4.
the
which
I
do
abate
from
the
ouer
fygure
7.
and
there
resteth
3.
I
put
3.
vnder
the
lyne
for
the
fourth
figure
.
And
then
I
come
to
the
fift
place
where
I
doe
fynde
9.
which
I
cannot
abate
from
the
figure
ouer
him
which
is
but
3.
but
I
abate
9.
from
10.
and
there
resteth
1.
the
which
I
doe
adde
wyth
3.
and
they
make
4.
I
put
4.
vnder
the
line
for
the
fift
figure
.
And
if
it
wer
not
,
for
that
I
did
last
borow
10.
the
substraction
should
haue
been
ended
.
But
for
bycause
that
I
must
(
for
euery
suche
ten
that
I
borowe
)
alwayes
adde
1.
vnto
the
next
nether
fygure
folowing
,
I
must
therefore
proceede
vnto
the
substraction
.
And
for
that
that
there
is
no
other
fygure
folowing
in
the
nether
number
,
it
shal
suffise
to
haue
kept
the
vnitie
,
and
to
abate
it
from
the
next
ouer
fygure
.
But
I
finde
there
0.
&
cannot
abate
1.
from
0.
therfore
I
abate
it
from
10.
and
there
resteth
9.
which
I
doe
put
vnder
the
line
in
the
sixte
place
,
finallye
for
the
ten
which
I
borowed
,
I
keepe
1.
in
minde
.
The
which
I
do
abate
from
4.
and
there
remaineth
3.
the
which
I
do
put
vnder
the
line
in
the
seuenth
place
after
9.
And
the
operation
is
thus
ended
.
An other
example
.
576084026
|
485675437
|
90408589
|
But
if
there
were
many
numbers
to
be
substracted
from
one
number
alone
,
then
must
you
fyrst
adde
those
numbers
togither
according
vnto
the
doctrine
of
the
Chapiter
going
before
,
and
then
to
make
your
substraction
as
aboue
saide
.
As
if
I
woulde
abate
these
three
summes
123.234.456
.
from
.
98925.
first
I
doe
adde
the
three
summes
into
one
,
&
they
are
.
813.
The
which
I
do
abate
from
98925.
and
there
resteth
.
98112.
Of
Multiplication
.
Chapter
.
4.
IN
multiplication
there
are
iij.
numbers
to
be
noted
,
that
is
to
say
the
numbre
which
is
to
be
multiplied
,
the
which
wee
will
call
the
Multiplicande
:
and
the
number
by
the
which
we
multiplie
,
wee
call
the
multiplyer
,
or
multiplicatour
.
And
the
thirde
number
is
that
which
commeth
of
the
multiplication
of
the
one
by
the
other
,
which
is
called
the
product
.
As
when
I
would
know
how
much
mounteth
10.
multiplied
by
9.
that
is
to
say
how
much
are
ten
times
nine
.
I
fynde
that
they
are
worth
.
90.
then
10.
is
the
multiplicand
,
9.
is
the
multiplier
.
And
90.
is
called
the
product
.
Then
for
to
multiplie
,
is
none
other
thing
,
but
to
finde
a
number
which
conteineth
the
multiplicande
so
many
times
,
as
the
multiplier
conteineth
vnities
,
As
10.
multiplied
by
9.
doe
make
90.
as
before
said
.
And
90.
conteineth
10.
so
many
times
,
as
9.
conteineth
vnities
,
that
is
to
saye
nine
tymes
.
In
multiplication
,
it
forceth
not
much
which
of
the
twoo
numbers
bee
the
multiplicande
,
nor
which
bee
the
multiplier
.
For
ten
multiplied
by
9.
maketh
as
many
as
9.
multiplied
by
10.
yet
neuerthelesse
it
shall
be
more
commodious
that
the
lesser
number
be
alwayes
the
multiplier
.
And
for
that
,
that
the
multyplication
of
figures
the
one
by
the
other
,
is
the
most
chief
&
necessariest
kind
wherby
to
know
how
to
work
in
the
multiplication
of
compound
numbers
,
and
that
euery
man
hath
not
at
the
fingers
ende
:
I
wil
therfore
giue
you
here
certeine
easye
wayes
of
multiplication
of
diget
numbers
.
When
you
would
multiplye
two
simple
figures
,
or
digets
the
one
by
the
other
,
abate
eche
of
those
dyget
numbers
from
10.
Then
multiplie
the
two
remaines
the
one
by
the
other
.
And
if
the
sum
do
exceede
10.
write
only
the
first
figure
,
&
kepe
the
other
to
be
added
to
the
next
operation
,
which
is
thus
.
Adde
your
two
simple
figures
togither
:
And
of
that
which
resulteth
of
the
addition
,
take
onely
the
first
figure
,
vnto
the
which
you
must
add
the
vnity
which
you
keepe
before
.
And
that
shall
be
the
second
figure
of
the
sum
which
you
do
seeke
.
Example
,
I
would
multipli
7.
by
6.
I
take
7.
from
10.
and
there
resteth
3.
likewise
I
abate
6.
from
10.
and
there
resteth
4.
then
I
say
thus
3.
times
4.
make
12.
I
write
2.
for
my
first
figure
,
&
I
keepe
1.
in
my
mind
,
them
I
adde
6.
with
7.
&
they
are
13.
of
the
which
I
caste
away
the
seconde
fygure
1.
and
I
take
only
the
fyrst
fygure
3.
vnto
the
which
I
adde
the
vnitie
which
I
kepte
,
&
they
make
4.
which
I
write
in
the
seconde
place
,
after
2.
And
thus
I
finde
42.
which
is
the
valure
of
7.
multiplied
by
6.
Otherwise
,
and
all
commeth
to
one
effect
,
set
down
your
two
diget
numbers
the
one
right
ouer
the
other
,
&
right
against
euery
of
them
towarde
the
right
hande
write
hys
owne
distaunce
from
.
10
:
Then
multiple
the
twoo
differences
togither
,
the
fygure
which
commeth
thereof
,
shall
you
set
downe
vnder
both
the
differences
.
But
if
there
be
two
fygures
,
set
downe
but
the
fyrst
,
and
keepe
the
other
in
your
minde
afterwardes
abate
(
from
one
of
the
two
diget
numbers
)
the
difference
of
the
other
diget
number
that
is
to
say
,
crossewise
.
And
vnto
the
remaine
adde
the
fygure
which
you
kepte
,
and
that
shal
be
the
seconde
number
,
and
thus
you
shall
haue
your
multiplication
.
Example
of
the
lyke
figures
that
is
to
say
of
7.
multiplied
by
6.
the
distaunce
of
7.
vnto
10.
is
3.
And
the
distaunce
of
6.
from
10.
is
4.
I
set
them
downe
croswayes
as
you
see
:
And
then
I
saye
three
times
foure
are
12.
I
set
downe
twoo
and
keepe
one
in
my
minde
,
then
I
abate
4.
from
7.
or
else
thre
from
6.
it
forceth
not
from
which
of
them
:
and
there
resteth
alwaies
3.
vnto
the
which
I
adde
the
vnitie
which
I
kepte
in
my
minde
,
and
they
are
foure
,
which
shal
be
the
seconde
fygure
of
the
multiplication
.
And
thus
I
finde
that
7.
multiplied
by
6.
maketh
42.
as
in
the
other
operation
.
Thys
practise
hath
no
place
where
the
12.
diget
numbers
(
doe
not
exceede
10.
)
by
adding
them
togither
,
and
then
is
multiplication
easye
ynough
without
any
rule
.
Another
way
to
knowe
the
multiplication
of
symple
numbers
,
is
by
thys
table
following
:
the
vse
wherof
is
thus
.
Fyrst
you
shall
vnderstande
that
the
numbers
from
1.
and
so
downewardes
to
9.
set
in
the
left
parte
or
hanging
margine
of
this
table
doe
betoke
the
multipliers
of
all
simple
numbers
.
And
the
elements
or
fygures
being
put
highest
in
euerye
square
roome
drawing
towarde
your
right
hand
right
against
euery
of
the
multipliers
,
do
signifie
the
multiplicands
,
vnto
the
multipliers
of
the
hanging
margin
.
And
the
lower
or
inferiour
numbers
in
euery
square
roome
,
do
betoken
the
product
of
that
multiplication
,
whych
is
made
in
multiplying
the
vpper
number
ouer
it
,
with
the
fygure
in
the
hanging
margin
,
answering
dyrectly
vnto
the
saide
square
:
as
by
example
,
The
Table
of
multiplication
by
all
the
Diget
numbers
.
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
|
49
|
56
|
63
|
8
|
8
|
9
|
64
|
72
|
9
|
9
|
81
|
First
bicause
1.
doth
not
multiplye
,
I
set
in
the
vpper
margin
the
figures
from
1.
to
9.
both
in
the
higher
and
also
in
the
inferiour
rowes
,
for
1.
in
the
hanging
margine
,
multiplied
by
1.
the
vpper
number
in
the
first
square
bringeth
but
1.
so
lykewyse
2.
beyng
the
higher
number
in
the
seconde
square
of
the
vpper
margine
multiplied
by
1.
in
the
hanging
margin
,
bringeth
two
for
the
lower
number
in
the
second
square
of
the
vpper
margine
,
for
one
times
1.
maketh
but
1.
and
1.
times
.
2.
maketh
2.
then
1.
times
3.
maketh
3.
and
1.
times
4.
maketh
4.
and
so
continuing
towarde
the
ryght
hande
vntill
you
come
to
the
figure
of
9.
which
is
1.
times
9.
maketh
9.
Then
after
multiple
twoo
of
the
hanging
margine
by
2.
the
vpper
number
of
the
square
nexte
towarde
the
right
hande
and
that
maketh
foure
which
is
the
product
of
2.
multiplied
by
2.
which
4.
is
set
vnder
the
2.
for
2.
times
.
2.
are
.
4.
and
2.
times
.
3.
maketh
6.
then
2.
times
4.
maketh
8.
and
two
times
5.
maketh
10.
and
so
continuing
vnto
2.
times
9.
which
maketh
18.
The
lyke
is
to
be
done
with
the
thirde
rowe
,
and
so
lykewyse
of
all
the
resydue
.
Exaumple
,
I
woulde
knowe
what
is
the
product
of
9.
multiplied
by
8.
I
seeke
in
the
hanging
mergin
the
multiplier
8.
and
amongest
the
squares
directlye
against
eyght
drawing
towarde
the
right
hande
,
I
seeke
the
multiplicand
9.
in
the
higher
rowe
,
and
I
fynde
the
product
right
vnder
9.
to
be
72.
Then
72.
is
the
number
which
commeth
of
the
multiplication
of
9.
by
8.
and
so
is
to
bee
vnderstande
of
all
the
reast
of
the
table
,
which
table
must
bee
of
all
men
learned
by
heart
,
or
as
they
saye
wythoute
booke
,
whych
being
learned
you
shall
the
better
attaine
to
the
rest
of
multiplication
.
To
come
nowe
vnto
the
practise
of
multiplication
,
when
you
woulde
multiplie
two
numbers
the
one
by
the
other
,
you
must
set
them
down
after
the
same
maner
as
you
did
in
addition
,
and
in
substraction
.
That
is
to
saye
,
the
fyrst
fygure
of
the
multiplier
vnder
the
fyrst
fygure
of
the
multiplicande
,
the
seconde
vnder
the
second
,
and
the
third
vnder
the
thirde
,
if
there
be
so
many
,
and
then
draw
a
right
line
vnder
them
,
as
in
the
other
operations
going
before
.
After
you
shall
multiply
all
the
figures
of
the
multiplicande
by
the
multiplier
,
and
set
downe
the
figures
(
comming
of
any
such
multiplication
)
vnder
the
line
euery
one
in
order
.
Exaumple
,
I
woulde
multiply
123.
by
3.
that
is
to
saye
,
I
woulde
knowe
how
much
amounteth
thre
times
one
hundreth
,
twentie
and
three
.
The
twoo
numbers
beyng
placed
in
suche
order
as
is
before
said
,
you
must
beginne
towardes
the
right
hande
:
and
say
thus
3.
tymes
3.
are
9.
write
down
9.
vnder
the
line
,
right
against
3.
for
the
first
fygure
:
secondly
by
the
same
3.
you
must
multiplie
the
second
figure
2.
&
they
make
6.
put
downe
6.
after
the
9.
vnder
the
lyne
:
Thirdly
by
the
same
3.
you
shall
multiplie
the
last
figure
1.
and
they
are
but
three
,
set
downe
3.
after
6.
for
the
thirde
&
last
figure
.
And
thus
is
the
worke
ended
:
wherby
you
shall
finde
,
that
123.
beyng
multiplyed
by
3.
maketh
369.
But
when
that
of
the
multiplication
of
one
figure
by
an
other
.
The
sum
which
commeth
therof
shal
be
of
twoo
fygures
,
as
it
happeneth
moste
often
,
then
shall
you
write
downe
the
first
fygure
,
and
keepe
the
other
fygure
to
be
added
vnto
the
multiplication
of
the
next
fygure
.
Exaumple
,
Syx
men
haue
gained
(
euery
one
of
them
)
345.
crownes
I
would
know
how
many
crowns
they
had
in
all
.
Firste
I
multiplie
6.
times
5.
are
30.
I
write
0.
vnder
the
line
,
&
keepe
3.
to
bee
added
to
the
next
multiplication
:
Secondely
I
saye
6.
times
4.
are
24.
vnto
the
which
I
adde
3
which
I
reserued
.
And
they
make
27.
I
write
7.
in
the
seconde
place
vnder
the
line
,
and
I
keepe
2.
to
be
added
to
the
next
multiplication
,
thirdely
I
saye
syxe
times
3.
are
18.
vnto
the
which
I
adde
the
2.
which
I
keepe
.
And
they
make
20.
the
which
I
write
al
downe
for
bicause
that
is
the
last
worke
.
And
so
I
finde
the
345.
being
multiplied
by
6.
do
make
2070.
When
the
multiplier
is
of
many
figures
you
must
multiplie
all
the
whole
multiplicand
by
euery
one
of
those
figures
,
and
write
the
productes
euery
one
vnder
hys
owne
fygure
.
Example
.
I
would
know
how
many
dayes
are
past
from
the
natiuitie
of
Iesus
Christe
vntyll
the
yeare
1560.
full
complete
.
I
haue
to
multiplie
1560.
by
365.
whych
are
the
dayes
of
one
whole
yeare
.
The
leape
yeares
not
being
reckened
,
which
haue
euery
one
of
them
366.
dayes
.
Fyrst
by
the
fygure
5.
I
multiplie
al
the
higher
figures
,
saying
thus
fyue
times
0.
maketh
0.
I
writ
0.
vnder
the
line
for
the
first
figure
,
and
bicause
I
keepe
nothing
for
the
next
place
,
I
proceede
&
say
5.
times
6.
are
30.
I
set
0.
vnder
the
line
for
the
second
figure
,
&
I
kepe
3.
to
be
added
to
the
next
multiplication
,
thirdlye
I
say
5.
times
5.
are
25.
The
which
with
3.
that
I
keepe
are
28.
I
set
downe
8.
and
keepe
2.
to
be
added
with
the
next
multiplication
.
Then
comming
vnto
the
fourth
and
last
fygure
,
I
saye
5.
times
1.
are
5.
the
which
with
2.
that
I
reserued
are
7
:
I
put
7.
for
the
last
figure
of
thys
fyrste
operation
by
the
fygure
5.
with
the
which
fygure
we
haue
no
more
to
doe
.
And
therefore
I
cancell
the
same
5.
with
a
little
strike
thorow
it
,
to
signifie
that
we
haue
finished
with
that
figure
.
And
forasmuch
that
in
multiplication
there
is
alwayes
as
many
simple
operations
,
as
the
multiplier
conteineth
figures
.
There
resteth
yet
2.
operatōis
to
be
made
.
I
come
then
vnto
the
seconde
operation
,
which
is
by
the
figure
6.
by
the
which
I
must
again
multiplie
al
the
figures
of
the
multiplicande
as
I
did
by
5
&
the
first
fygure
(
which
shal
be
produced
)
you
must
put
one
rank
more
lower
then
the
figures
of
the
operation
euen
now
made
by
5.
not
right
vnder
the
first
fygure
of
the
multiplier
5.
but
vnder
6.
that
is
to
say
:
one
place
more
forewarde
than
the
fift
toward
the
left
hande
:
&
one
ranck
more
lower
than
the
fyrst
operation
:
and
you
shall
put
afterwarde
euerye
of
the
other
figures
which
commeth
of
the
same
multiplication
in
their
order
:
thirdly
you
must
make
the
multiplication
by
the
thirde
figure
&
that
which
shal
come
thereof
you
must
set
in
his
ranck
,
as
here
vnder
you
se
.
And
now
we
neede
make
no
further
discourse
hereof
,
bicause
that
he
which
can
doe
the
fyrst
multiplication
by
5.
may
as
easely
doe
all
the
others
.
It
shall
therefore
suffice
to
set
here
vnder
the
examples
.
Now
,
if
you
wil
know
how
much
the
operations
thus
placed
do
amount
vnto
,
which
in
value
are
but
one
number
:
you
must
add
those
thre
numbers
togither
,
but
not
after
the
same
maner
as
we
haue
done
in
the
chapter
of
addition
,
the
first
figure
of
the
first
ranck
with
the
first
figure
of
the
seconde
ranck
,
&
of
the
thirde
:
but
you
muste
adde
them
in
the
same
sort
as
you
shal
finde
them
situated
or
placed
:
that
is
to
say
,
the
fyrst
figure
of
the
fyrst
rancke
alone
by
it selfe
:
the
second
of
that
ranck
with
the
fyrst
of
the
seconde
rancke
.
The
thirde
of
the
fyrst
rancke
wyth
the
seconde
figure
of
the
second
rancke
and
the
first
of
the
thirde
ranck
:
and
so
of
all
other
as
hereafter
doeth
appeare
.
And
thus
the
1560
1560
|
365
|
7800
|
9360
|
4680
|
569400
|
yeares
do
containe
fiue
hundreth
sixtye
&
nyne
thousande
foure
hundreth
dayes
not
counting
herein
the
daies
of
the
leape
yeares
,
which
are
here
in
number
390.
for
then
the
whole
sum
of
the
dayes
should
be
569790.
Another
example
.
34560
|
2456
|
207360
|
172800
|
138240
|
69120
|
84879360
|
The
summe
of
multiplication
,
when
you
woulde
multiplie
anye
number
by
10.
you
must
onely
adde
one
cipher
vnto
all
the
number
.
As
345.
multiplied
by
10.
maketh
3450.
if
you
wyll
multiplie
by
100.
Adde
vnto
the
whole
number
two
ciphers
.
00.
if
by
1000.
adde
000.
And
to
bee
briefe
.
when
the
laste
fygure
of
the
multiplier
is
1.
and
al
the
rest
bee
ciphers
,
add
so
many
ciphers
to
your
multiplicande
,
as
there
shal
be
found
in
your
multiplier
.
But
if
in
multiplying
,
the
last
figure
were
not
1.
but
that
there
were
onely
certaine
ciphers
in
the
beginning
:
&
that
the
other
were
signifying
fygures
,
and
likewise
those
of
the
multiplicand
,
then
shall
you
put
those
cyphers
apart
,
and
multiplie
the
signifying
figures
of
the
one
by
the
signifying
fygures
of
the
other
.
Then
adde
vnto
the
product
of
that
multiplication
,
al
the
ciphers
which
you
did
before
put
a
part
.
As
if
I
would
multiplie
.
46000.
by
3500.
I
put
a
part
the
three
ciphers
of
the
first
,
and
the
twoo
ciphers
of
the
seconde
numbers
.
And
then
I
multiplie
46.
by
35.
&
therof
commeth
1610
:
vnto
the
whych
I
adde
the
00000.
and
then
the
whole
product
will
be
.
161000000.
Of
Diuision
the
fift
Chap.
DIuision
or
partition
is
,
to
seeke
how
many
times
one
number
doth
conteine
an
other
for
in
this
operation
are
firste
required
twoo
numbers
for
the
fynding
out
of
the
thirde
.
The
firste
number
is
called
the
diuidende
or
number
which
is
to
be
diuided
&
that
muste
be
the
greater
number
,
the
other
number
is
called
the
diuisor
,
&
that
is
the
lesser
.
And
the
third
number
which
we
seke
is
called
the
quotient
.
As
if
I
would
diuide
36.
by
9.
the
diuidend
shall
be
.36
.
and
the
diuisour
is
9.
And
for
bicause
that
9.
is
conteined
in
36.
foure
times
,
that
is
to
say
that
4.
times
9.
do
make
36.
The
quotient
shall
be
4.
as
in
marking
how
many
times
9.
is
conteined
in
.36
.
¶
The
practise
.
Wryte
downe
fyrste
the
deuidende
in
the
higher
number
,
and
the
diuisour
vnderneth
,
in
suche
sorte
,
that
the
fyrst
fygure
of
the
diuisour
towarde
the
lefte
hande
be
vnder
the
fyrst
of
the
diuidend
and
euery
figure
of
the
same
diuisor
vnder
his
like
,
that
is
to
saye
,
the
fyrst
vnder
the
fyrst
,
the
seconde
vnder
the
seconde
,
the
thirde
vnder
the
third
,
and
so
consequentlye
of
the
other
,
if
there
be
any
more
,
which
is
contrarye
to
the
other
three
kindes
before
specifyed
,
but
you
muste
consider
if
all
the
lower
figures
of
the
diuisor
,
maye
be
taken
out
of
the
higher
fygures
of
the
diuidend
,
by
the
order
of
substraction
.
The
which
if
you
can
not
doe
,
then
muste
you
set
the
fyrst
fygure
of
the
Diuisor
(
toward
the
lefte
hande
)
vnder
the
seconde
fygure
of
the
diuidend
,
and
so
consequently
the
reast
,
if
any
be
to
be
set
down
euery
one
of
them
vnder
his
like
as
before
is
said
.
And
then
draw
a
lyne
betweene
the
diuidende
and
the
diuisor
.
And
at
the
ende
of
them
an
other
crooked
lyne
,
behinde
the
which
toward
the
right
hand
,
shall
be
set
your
quotient
.
As
by
this
example
folowing
where
the
diuisor
is
but
of
one
fygure
.
If
you
woulde
diuide
860.
by
4.
you
must
set
downe
4.
vnder
the
8.
with
a
line
betwene
them
as
here
vnder
you
may
see
.
The
diuidende
|
860
|
Diuisor
.
|
4
|
And
then
you
muste
seeke
howe
many
times
the
diuisor
is
conteined
in
the
higher
number
,
or
diuidend
aunswering
to
him
,
as
in
this
our
example
I
must
seke
how
many
times
4
is
conteined
in
8.
in
the
which
I
finde
2.
times
,
then
I
write
downe
2.
apart
behinde
the
crooked
line
,
as
you
se
,
which
shal
be
the
first
figure
of
the
quotient
to
come
,
secondly
by
this
fygure
(
being
thus
put
apart
)
I
must
multiple
the
diuisor
:
and
vnder
the
same
multiplication
.
I
must
set
that
number
which
commeth
of
the
same
multiplication
as
2.
times
4.
doe
make
8.
which
8.
I
do
set
vnder
the
4.
which
is
the
diuisor
.
Thirdly
,
I
do
substract
the
product
of
the
sayde
multiplication
(
of
the
quotient
by
the
diuisor
)
from
the
higher
number
correspondant
to
the
same
,
as
if
I
abate
8.
from
8.
there
remaineth
nothing
,
and
then
I
cansell
or
strike
out
that
whych
is
done
as
you
sée
.
In
these
three
operations
is
comprehended
the
arte
of
diuision
.
The
which
are
to
bee
obserued
from
point
to
point
for
ther
is
no
diuersitie
in
the
finishing
of
the
same
which
is
thus
.
I
must
remoue
my
diuisor
one
place
nerer
toward
my
right
hand
:
as
in
proceding
with
our
example
I
remoue
my
deuisor
4.
which
was
vnder
8.
and
I
set
it
vnder
6.
then
I
seeke
how
many
times
4.
is
conteined
in
6.
where
I
fynde
but
one
time
them
I
set
1.
behinde
the
crooked
line
behinde
2.
afterwarde
by
this
last
&
new
figure
1.
I
multiplie
the
diuisor
4.
&
that
maketh
but
4.
(
for
an
vnity
which
is
but
1.
encreaseth
nothing
)
I
abate
4.
from
the
higher
figure
6.
and
there
resteth
2.
the
which
2.
I
set
ouer
the
6.
&
I
cancell
the
6.
for
so
must
you
doe
when
there
resteth
anye
thing
after
you
haue
made
the
substraction
.
Thirdly
for
that
there
yet
remaineth
another
fygure
in
the
diuidend
,
I
remoue
again
the
diuisor
,
and
I
set
it
vnder
the
cipher
0.
Then
I
seeke
how
many
times
4.
is
in
the
higher
number
which
is
20.
where
I
fynde
5.
times
,
I
put
5.
behinde
the
crooked
line
for
the
thirde
and
laste
figure
of
the
quotient
.
Then
by
the
same
5.
I
multiply
the
diuisor
4.
and
that
maketh
20.
the
which
I
abate
from
the
higher
number
,
and
there
resteth
nothing
.
And
so
is
this
diuision
ended
:
&
I
haue
founde
yt.
860.
being
diuided
by
4.
bringeth
for
the
quotient
215.
that
is
to
say
,
that
4.
is
conteyned
in
860.
twoo
hundreth
&
fyftene
tymes
.
Thys
is
the
most
easiest
working
that
is
in
diuision
,
but
that
which
foloweth
,
apperteyneth
to
the
whole
and
perfect
vnderstanding
of
the
same
.
When
the
fyrste
fygure
of
your
diuisor
toward
your
left
hande
is
greater
than
the
fyrst
of
the
diuidende
,
you
must
not
place
the
fyrste
fygure
of
your
diuisour
right
vnderneth
the
fyrst
of
the
diuidende
,
but
vnder
the
seconde
fygure
of
the
same
diuidende
,
neerer
to
your
right
hande
,
as
before
is
said
.
When
the
diuisor
is
of
many
figures
,
and
that
you
haue
to
seeke
how
many
times
it
is
conteined
in
the
higher
number
(
for
the
more
easyer
working
)
you
muste
not
seeke
to
abate
the
diuisor
all
at
one
time
,
but
you
muste
see
and
marke
howe
many
times
the
fyrst
fygure
of
the
same
towarde
the
lefte
hande
is
conteyned
in
the
hygher
number
aunswering
to
the
said
number
,
&
then
to
worke
after
the
same
maner
as
is
before
taught
.
Example
,
I
haue
316215.
crownes
to
be
deuided
amonge
45.
men
for
to
make
my
diuision
I
muste
not
put
the
fyrste
fygure
of
the
diuisor
which
is
4.
vnder
the
first
of
the
diuidende
,
which
is
3.
bicause
that
4.
is
greater
number
than
3.
And
further
,
I
cannot
take
4.
out
of
three
,
wherefore
I
must
set
the
4.
vnder
the
seconde
fygure
of
the
hygher
number
which
is
1.
and
the
fygure
5.
of
the
diuisor
next
right
vnder
the
6.
as
you
may
see
.
I
must
first
seeke
,
how
many
times
45.
is
conteyned
in
316.
which
is
but
parte
of
the
diuidende
,
wherefore
for
the
more
easie
working
I
neede
but
to
seeke
howe
many
times
4.
is
conteined
in
31.
&
bicause
I
may
haue
it
seauen
tymes
I
put
7.
behinde
the
crooked
line
,
as
is
aforesaide
,
then
by
7.
I
multiply
all
the
diuisor
45.
and
they
are
315
:
the
which
I
set
vnder
the
same
diuisor
,
the
fyrst
fygure
vnder
the
fyrst
:
And
the
other
in
order
towarde
the
lefte
hande
.
Then
I
substract
.
315.
from
the
higher
number
316.
and
of
this
fyrst
working
there
remaineth
but
1.
the
which
I
set
ouer
the
6.
and
I
cancell
the.
315.
&
the
other
fygures
3
,
1
,
6
,
and
also
the
diuisor
:
and
then
it
wil
stande
thus
.
And
when
I
come
to
remoue
the
diuisor
,
and
that
I
must
seeke
how
many
times
it
is
conteined
in
the
higher
number
,
if
I
see
that
I
cannot
fynde
it
there
,
that
is
to
say
that
if
the
higher
number
be
lesser
than
the
diuisor
,
as
it
is
in
this
example
,
then
must
I
put
a
cipher
in
the
quotient
behinde
the
crooked
line
,
&
if
there
remain
any
figures
in
the
diuidende
whiche
are
not
finished
,
I
must
remoue
the
diuisor
agayne
nerer
towarde
my
right
hande
by
one
place
,
for
to
finde
a
newe
fygure
in
the
quotient
.
As
in
this
our
example
,
for
after
that
I
haue
remoued
the
diuisor
,
I
séeke
howe
many
times
.
45.
is
conteined
in
.
12.
and
because
I
cannot
haue
45.
in
12.
I
put
a
0.
behinde
the
croked
line
after
7.
then
without
multipliyng
or
abatyng
,
I
remoue
againe
the
diuisor
nerer
towarde
my
ryght
hande
,
and
I
seeke
howe
many
tymes
4.
(
whiche
is
the
first
figure
of
the
diuisor
)
is
in
the
hygher
numbre
,
that
is
to
say
,
in
12.
whereas
I
finde
it
3.
tymes
:
I
put
3.
behinde
the
crooked
lyne
,
for
the
thyrde
fygure
of
the
quotient
:
then
by
3.
I
multiplie
the
diuisor
.
45.
and
therof
commeth
.
135.
A
note
.
But
here
is
to
bee
noted
,
that
if
it
happen
that
the
fygure
beynge
laste
founde
whiche
is
put
in
the
quotient
,
doe
produce
or
bringe
foorthe
a
greater
noumbre
(
in
multipliyng
all
the
dyuisor
by
the
same
)
then
that
whiche
is
ouer
the
saied
diuisor
:
you
muste
then
make
the
same
figure
of
youre
quotient
(
whiche
you
doe
put
downe
)
lesser
by
one
:
and
after
that
you
haue
cancelled
the
firste
multiplication
,
you
must
make
a
newe
.
And
the
same
must
bee
so
doone
as
often
times
:
as
(
in
decreasing
the
same
)
it
produceth
a
lesser
noumbre
,
or
at
the
least
,
a
noumbre
egall
to
that
whiche
is
ouer
it
.
As
in
the
laste
woorke
:
for
because
that
the
diuisor
,
being
multiplied
by
3.
bringeth
foorthe
.
135.
whiche
amounteth
more
then
.
121.
the
same
producte
must
be
cancelled
.
And
likewyse
the
figure
.
3.
whiche
I
did
put
in
the
quotient
,
must
bee
chaunged
into
a
figure
of
2.
Then
by
the
saied
2.
I
must
multiplie
the
diuisor
.
45.
and
thereof
commeth
90
,
the
whiche
I
abate
from
.
121.
and
there
remaineth
.
31.
And
then
wil
the
somme
stand
thus
.
13
|
316215
|
45
2
|
135
(
703
|
90
|
And
here
is
also
to
bee
noted
that
the
somme
whiche
remaineth
must
be
alwayes
lesser
then
the
diuisor
.
A
note
.
Then
finally
I
remoue
the
diuisor
to
the
2.
last
figures
towarde
the
right
hande
,
and
I
seeke
howe
many
times
4.
is
in
31.
And
for
because
I
finde
it
7.
times
,
I
put
7.
in
the
quotient
:
by
the
whiche
I
multiplie
the
diuisor
,
and
thereof
commeth
315.
the
whiche
I
abate
from
the
hygher
noumbre
of
the
diuidend
,
and
there
remaineth
nothing
as
here
you
may
see
.
But
in
case
that
after
the
diuision
is
ended
,
there
doe
remaine
any
thing
in
the
diuidende
,
as
most
often
times
there
dothe
:
I
must
them
sette
that
remaine
aparte
behinde
the
croked
line
after
the
entier
quotient
,
and
the
diuisor
right
vnder
the
same
remayne
,
with
a
lyne
betwene
them
bothe
,
as
in
this
diuision
followyng
,
where
there
remayneth
3.
in
the
last
woorke
of
the
same
.
And
we
shal
sée
what
the
same
doth
signifie
,
when
we
shal
treate
of
fractions
or
broken
numbres
.
In
summe
,
all
the
whole
practise
of
diuision
may
bée
kepte
in
remembraunce
by
thrée
lettres
,
that
is
to
say
:
S.
M.
A.
whiche
thrée
letters
doe
signifie
to
seeke
,
to
multiplie
,
to
abate
.
First
,
I
must
séeke
howe
manye
times
the
diuisor
is
conteined
in
the
higher
numbre
:
then
,
by
the
quotient
(
whiche
I
finde
)
I
must
multiplie
the
diuisor
:
finally
,
I
must
abate
the
product
of
that
multiplication
,
from
the
higher
numbre
to
the
same
correspondent
,
that
is
to
saye
:
out
of
the
diuidende
,
aunswering
to
the
diuisor
.
And
further
,
besides
this
kynde
of
woorkyng
in
diuision
.
The
whiche
is
reguler
and
commune
:
I
wyll
here
put
an
other
manner
of
woorkynge
very
easye
.
The
whiche
shall
serue
for
soche
diuisions
as
are
difficyll
to
bée
wrought
.
That
is
to
wytte
,
when
the
number
to
bee
diuided
is
verye
great
,
and
the
diuisor
great
also
,
and
it
shall
serue
againe
for
to
auoyde
errour
in
supputacion
,
and
for
the
placing
of
fewer
figures
in
the
quotient
:
and
consequently
it
shall
saue
muche
labour
vnto
them
whiche
as
yet
haue
muche
studied
in
this
arte
.
The
practise
whereof
is
thus
,
as
foloweth
.
I
haue
to
deuide
7894658.
by
643.
In
the
firste
place
,
you
shall
vnderstande
,
that
althoughe
the
firste
figure
of
the
diuisor
towarde
your
lefte
hande
,
maye
bee
founde
many
times
in
the
hygher
noumbre
as
10.
times
12.
times
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
time
putte
any
noumbre
in
your
quotient
which
exceadeth
the
fygure
of
.
9.
that
is
to
saie
any
noumbre
being
greater
then
9.
for
to
come
then
vnto
our
practise
,
wryte
downe
your
dyuisor
one
time
:
and
behynde
it
towarde
your
ryghte
hande
,
drawe
a
lyne
downe
straighte
,
and
right
against
the
same
diuisor
behinde
the
lyne
put
this
fygure
1.
Then
double
your
saied
diuisor
,
and
righte
against
the
same
(
beyng
doubled
)
put
behinde
the
lyne
the
fygure
of
2.
After
,
adde
vnto
the
same
noumbre
(
whiche
you
doubled
)
your
saied
diuisor
and
right
against
the
same
product
,
behinde
the
line
put
the
fygure
of
.
3.
And
vnto
this
thyrde
producte
,
you
muste
adde
againe
your
diuisor
:
and
ryght
against
the
same
producte
behinde
the
lyne
sette
the
fygure
.
4.
And
this
muste
you
dooe
,
vntill
you
come
to
the
fygure
of
.
9
:
in
soche
sorte
that
euerye
of
the
productes
dooe
surmounte
so
muche
his
former
noumbre
,
as
all
the
diuysor
dothe
amounte
vnto
:
placing
at
the
right
syde
of
euerye
producte
behynde
the
lyne
,
the
noumbre
whiche
signifieth
howe
muche
he
is
in
order
.
That
is
to
saye
,
right
against
the
fifte
producte
,
you
must
put
.
5.
right
against
the
sixte
producte
,
you
muste
put
.
6
:
And
so
likewyse
of
all
the
other
.
Example
of
the
diuisor
proponed
,
643.
first
,
I
wryte
downe
643.
and
643
|
1
|
1286
|
2
|
1929
|
3
|
2572
|
4
|
3215
|
5
|
3858
|
6
|
4501
|
7
|
5144
|
8
|
5787
|
9
|
right
against
the
same
behind
the
lyne
I
put
.1
.
secondely
,
I
double
643.
and
thei
make
1286
&
right
againste
him
behinde
the
lyne
I
put
.2
.
Thirdlye
,
vnto
that
same
1286.
I
adde
the
diuisor
643.
and
they
are
1929.
and
right
against
the
same
I
sette
3.
Fourthely
,
vnto
the
sayed
1929.
I
adde
the
diuisor
643.
and
they
are
2572.
and
right
against
the
same
I
put
.4
.
And
thus
must
you
doe
alwayes
by
encreasing
so
muche
euery
product
,
as
the
diuisor
dothe
amounte
vnto
,
vntyll
you
haue
so
done
nyne
tymes
,
as
you
sée
in
this
present
table
.
This
being
done
,
you
must
sette
downe
your
diuisor
vnder
the
diuident
after
the
same
maner
as
is
before
declared
:
that
is
to
saye
,
643.
vnder
the
thrée
first
figures
of
the
diuidende
towarde
your
right
hande
,
which
are
.789
.
Then
must
you
seke
howe
many
times
.643
.
are
conteyned
in
.789
.
And
for
to
knowe
the
same
,
I
looke
in
my
foresaied
table
if
I
may
there
finde
the
same
numbres
,
789.
the
whiche
is
not
there
:
Therefore
I
must
take
a
lesser
noumbre
the
nerest
to
it
in
quantitie
that
I
can
finde
in
the
table
,
the
whiche
is
.643
.
whiche
noumbre
hathe
against
it
on
the
ryght
hande
of
the
lyne
this
diget
.
1.
Then
I
take
the
sayed
1.
and
I
put
it
behynde
the
croked
lyne
,
for
the
first
fygure
of
the
quotient
.
Then
I
dooe
abate
.643
.
from
789.
and
there
remayneth
.146
.
the
whiche
I
put
ouer
the
.789
.
and
I
cancell
the
789.
and
thus
is
the
fyrst
operation
ended
.
Then
I
sette
forwarde
the
deuisour
one
fygure
nerer
to
my
ryghte
hande
,
and
I
séeke
a
newe
quotient
as
I
soughte
this
,
where
I
finde
the
higher
noumber
ouer
my
diuisour
to
bee
1464.
The
whiche
I
doe
séeke
in
the
table
,
and
because
I
can
not
fynde
it
there
,
I
take
a
lesser
noumbre
,
the
nighest
to
it
that
I
can
finde
,
and
that
is
1286
:
whiche
noumbre
hathe
against
it
this
digette
.2
.
I
put
.2
.
for
the
seconde
figure
of
the
quotient
behinde
the
line
,
and
I
doe
abate
1286.
from
.1464
.
and
there
remaineth
.178
.
Thirdly
,
I
remoue
forewarde
the
diuisour
,
as
before
,
and
I
finde
the
higher
noumbre
to
bee
.1786
.
and
that
the
nexte
lesser
noumbre
to
it
in
my
table
,
is
againe
.1286
.
I
put
therefore
ones
againe
.2
.
in
the
quotient
for
the
third
figure
:
and
I
abate
.1286
.
from
,
1786.
and
there
remaineth
.500
.
Fourthly
,
I
set
forward
the
diuisour
,
and
the
higher
noumbre
ouer
it
,
is
5005.
and
the
next
lesser
noumbre
to
it
in
my
table
is
.4501
.
right
against
the
whiche
noumbre
is
.7
.
I
put
my
.7
in
the
quotient
,
for
the
fourth
figure
.
And
after
that
I
haue
abated
.4501
.
from
.5005
.
there
remayneth
.504
.
Finally
,
I
remoue
forwarde
my
diuisor
vnto
the
last
place
:
and
I
finde
the
higher
noumbre
to
bee
.5048
.
And
the
nexte
lesser
noumbre
to
it
in
my
table
,
is
.
4501.
I
set
.7
.
againe
in
the
quotient
,
for
the
fifte
and
last
fygure
.
Then
I
take
.4501
.
from
.5048
.
and
there
remaineth
.547
.
whiche
must
bee
put
at
the
ende
of
the
whole
quotient
with
the
diuisor
vnder
it
,
and
a
lyne
betwene
them
in
this
maner
folowing
.
¶
The
summe
of
diuision
.
WHen
you
would
diuide
any
numbre
by
.10
.
you
must
take
away
the
last
figure
next
towardes
your
right
hand
&
the
rest
shal
be
the
quotient
.
As
if
you
would
diuide
.46845
.
by
.10
.
take
away
the
.5
.
and
then
.4684
.
shal
bee
the
quotient
,
and
the
.5
,
shal
be
the
nombre
that
doth
remaine
.
Likewise
when
you
woulde
diuide
any
numbre
by
100.
take
awaye
the
twoo
last
figures
towardes
your
right
hande
,
and
if
you
woulde
diuide
by
.1000
.
take
away
thrée
figures
,
if
by
.10000
.
take
away
foure
figures
.
And
so
of
all
other
,
when
the
first
figure
of
the
dyuisor
towarde
the
lefte
hande
shal
be
onely
1.
and
the
rest
of
the
same
diuisor
being
but
cyphers
.
¶
Here
foloweth
the
proofes
of
addicion
,
substraction
,
multiplication
,
and
diuision
.
¶
The
profe
of
Addition
.
WHen
you
would
proue
whether
your
addicion
be
well
made
,
consider
the
figures
of
the
noumber
whiche
bée
added
,
euery
one
in
his
simple
value
:
not
hauinge
any
regarde
to
the
place
where
he
standeth
,
but
to
recken
him
as
though
he
were
alone
by
himselfe
and
then
recken
them
all
,
one
after
an
other
,
casting
away
from
them
the
noumber
of
9.
as
ofte
as
you
maye
.
And
after
your
discourse
made
,
kepe
in
minde
the
same
figure
which
remaineth
after
the
nynes
bee
taken
away
,
or
set
the
same
in
a
voyde
place
at
the
vpper
ende
of
a
line
.
For
if
your
addicion
be
well
made
,
the
like
figure
will
remaine
,
after
that
you
haue
taken
away
all
the
nines
,
out
of
the
totall
summe
of
the
same
addicion
,
as
of
ten
as
you
mayther
finde
any
:
as
in
this
addicion
whiche
here
you
se
.
Ther
remaineth
.2
.
for
echepart
.
¶
The
profe
of
substraction
.
ADde
the
noumbre
whiche
you
doe
substract
with
that
numbre
which
remaineth
after
the
substraction
,
is
made
:
and
if
the
totall
somme
of
that
addicion
,
be
like
vnto
the
nombre
from
the
whiche
the
substraction
was
made
you
haue
done
well
,
otherwyse
not
:
as
in
this
example
doth
appeare
,
where
you
sée
the
nombre
whiche
is
to
be
substracted
is
,
3584.
and
the
noumbre
whiche
doth
remayne
,
is
.1879
.
the
whiche
twoo
summes
being
added
together
,
dooe
make
.5463
.
whiche
is
like
to
the
higher
noumbre
,
out
of
the
whiche
the
substraction
was
made
,
as
before
is
saied
.
¶
The
profe
of
multiplication
.
THe
profe
of
multiplication
is
made
by
the
helpe
of
diuision
,
for
if
you
diuide
the
nombre
produced
of
the
multiplication
,
by
the
multiplier
:
you
shall
finde
the
higher
noumbre
,
whiche
is
the
multiplicande
.
¶
The
profe
of
diuision
.
TO
knowe
if
your
diuision
be
well
made
:
you
must
multiplie
all
the
quotient
by
your
diuisor
,
and
if
any
thinge
remained
after
your
diuision
was
made
.
Thesame
shall
you
adde
vnto
the
producte
whiche
commeth
of
the
multiplication
:
and
you
shal
finde
the
like
nombre
vnto
your
diuidend
if
you
haue
wel
diuided
:
otherwise
not
.
¶
Of
progression
the
vi
.
Chapiter
.
PRogression
arithmetical
,
Progression
arithmetical
.
is
a
briefe
&
spedy
assembling
or
adding
together
of
diuers
figures
or
nombres
,
euery
one
surmounting
the
other
continually
by
equall
difference
:
as
1.2.3.4.5
.
&c.
here
the
difference
,
from
the
first
to
the
second
is
but
of
1.
and
so
do
al
the
other
,
euery
one
excede
another
by
1.
still
to
th
end
.
Like
waies
.
Here
.2.4.6.8
&c.
do
procede
by
the
difference
of
2.
also
3.6.9
.
12.
&c.
doe
euery
one
differ
from
other
by
3.
and
so
may
these
noumbres
continue
.
Infinitelie
after
this
order
,
in
adding
vnto
the
thirde
noumbre
,
the
quantitie
wherein
the
seconde
dothe
differ
from
the
fyrste
:
Lyke
wayes
addinge
the
same
difference
vnto
the
fowerth
noumbre
,
also
to
the
fyfte
,
and
so
vnto
all
the
other
.
As
.1.4
.
the
difference
of
the
seconde
to
the
fyrst
is
3.
adde
3.
vnto
4
:
and
they
are
7.
for
the
thirde
noumbre
:
Then
adde
3.
vnto
7
:
and
thei
make
10.
for
the
fowrth
noumbre
,
and
so
of
all
other
.
Then
if
you
will
adde
quickely
the
noumbre
of
any
progession
,
you
shall
dooe
thus
,
first
tel
howe
many
noumbres
there
are
,
and
wryte
their
somme
downe
by
it selfe
,
as
in
this
example
,
2.
5.
8.
11.
14.
where
the
noumbres
are
5.
as
you
maye
sée
,
therefore
you
must
sette
downe
5.
in
a
place
alone
,
as
I
haue
done
here
in
the
margent
.
Then
shall
you
adde
the
first
noumbre
and
the
last
together
,
whiche
in
this
exaumple
are
.14
.
and
2.
and
they
make
.16
.
take
halfe
thereof
whiche
is
.8
.
and
multiplie
it
by
the
5.
whiche
I
nooted
in
the
margente
for
the
noumbre
of
the
places
,
and
the
somme
whiche
amounteth
of
that
multiplicacion
,
is
the
iust
somme
of
al
those
figures
added
together
,
as
in
this
example
:
8.
multiplied
by
.5
.
doe
make
.40
.
and
that
is
the
somme
of
all
the
figures
.
An other
example
of
parcels
that
are
euen
,
as
thus
.1.2.3.4.5.6
.
in
this
example
you
must
likewaies
note
doun
the
nomber
of
the
places
,
as
before
is
taught
,
and
than
adde
together
the
last
nomber
and
the
first
.
And
the
somme
,
whiche
cometh
of
that
addicion
,
shall
you
multiplie
by
halfe
the
nomber
of
the
places
,
whiche
before
are
noted
,
and
that
,
whiche
resulteth
of
the
same
multiplicacion
,
is
the
wholesomme
of
all
those
figures
,
as
in
this
former
example
,
where
the
nomber
of
the
places
is
.6
.
I
note
the
.6
.
a
part
,
and
then
I
adde
.6
.
and
.1
.
together
,
whiche
are
the
laste
and
firste
nombers
,
and
thei
make
.7
.
the
whiche
I
multiplie
by
.3
.
whiche
is
halfe
the
nomber
of
places
,
and
thei
make
21
and
so
moche
amounteth
all
those
figures
,
added
together
.
Progression
Geometricall
is
,
Progession
,
geometricall
.
when
the
second
nomber
containeth
the
first
in
any
proporcion
:
2.3
.
or
.4
.
times
and
so
forthe
.
And
in
like
proporcion
shall
the
thirde
nomber
contain
the
second
,
and
the
fowerth
,
the
third
,
and
the
fift
the
fowerth
.
&c.
As
.2.4.8.16.32
,
64
:
here
the
proporcion
is
double
.
Likewaies
.3.9.27.81.243
.
are
in
triple
proporcion
.
And
.2.8.32.128.512
.
are
in
proporcion
quadruple
.
That
is
to
saie
,
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
Arithmeticalle
,
differeth
from
Progression
Geometricalle
for
that
,
that
in
the
Arithmeticalle
.
The
excesse
is
onelie
in
quantitie
,
but
in
the
Geometricalle
,
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
multiplicacion
,
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
quocient
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
shall
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
:
which
is
the
total
somme
of
the
nombers
of
that
progression
.
Like
wise
.4
.
16.64
256.1024
.
whiche
are
in
proporcion
quadruple
:
therfore
multiplie
1024.
by
.4
.
and
thereof
cometh
4096
from
the
whiche
abate
the
firste
nomber
.4
.
and
there
resteth
.4092
:
The
whiche
you
must
diuide
by
.3
.
and
you
shall
finde
in
your
quotiente
.1364
.
whiche
is
the
total
somme
of
that
progression
,
and
this
shal
be
sufficient
for
progression
.
¶
The
.vii.
chapiter
treateth
of
the
Rule
of
.3
.
called
the
golden
Rule
.
THe
rule
of
three
is
the
chifest
the
moste
profitable
,
and
the
moste
excellent
rule
of
al
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
iudgement
,
it
is
called
the
rule
of
proporcions
of
noumbers
.
But
now
in
these
daies
,
by
vs
it
is
called
the
rule
of
thre
,
because
it
requireth
three
nombers
in
his
operacion
.
Of
the
whiche
three
nombers
,
the
twoo
first
are
set
in
a
certain
proporcion
.
And
in
soche
proporcion
as
their
bée
stablished
,
this
rule
serueth
to
finde
out
vnto
the
third
nomber
,
the
fourth
nomber
to
hym
proporcioned
,
in
soche
sort
as
the
second
is
proporcioned
vnto
the
firste
.
Not
for
that
,
that
the
fower
noumbers
,
nor
yet
the
three
,
are
or
bee
proporcionall
,
or
set
in
one
proporcion
,
but
soche
proporcion
,
as
is
from
the
firste
to
the
seconde
,
ought
to
bee
from
the
thirde
vnto
the
fowerth
,
that
is
to
saie
,
if
the
seconde
noumber
dooe
conteine
the
firste
twoo
tymes
or
more
,
so
many
tymes
shal
the
fowerth
nomber
conteine
the
thirde
.
And
note
well
that
the
firste
nomber
,
and
the
thirde
in
euery
rule
of
three
,
oughte
and
must
bee
alwaies
semblable
,
and
of
one
condicion
.
And
the
second
nomber
,
and
the
fowerth
muste
likewise
bee
of
one
sembleaunce
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
multiplicacions
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
dissembleaunte
by
the
other
:
that
is
to
saie
,
the
fowerth
nomber
by
the
second
(
which
are
dissembleaunte
to
the
other
twoo
nombers
)
your
twoo
quocientes
will
bee
egall
.
Regul.
The
stile
of
this
rule
is
thus
,
you
muste
sette
doune
your
three
noumbers
in
a
certaine
order
,
as
by
example
here
vnder
shall
appere
.
And
then
multiplie
the
thirde
noumber
,
by
the
seconde
.
And
the
producte
thereof
you
muste
diuide
by
the
firste
noumber
,
or
otherwise
,
diuide
the
firste
noumber
by
the
seconde
.
And
the
quocient
therof
shal
bee
diuisor
vnto
the
third
nomber
,
that
is
to
saie
,
the
thirde
noumber
shall
bee
diuided
by
the
quotient
of
the
foresaied
diuision
,
that
is
of
the
firste
noumber
diuided
by
the
seconde
.
Or
otherwise
diuide
the
seconde
noumber
by
the
firste
.
And
that
whiche
cometh
into
your
quotiente
,
you
shall
multiplie
it
by
the
thirde
nomber
.
And
thus
shall
you
haue
the
fowerth
noumber
,
whiche
you
seke
for
.
¶
Example
.
IF
.8
.
bee
worthe
.12
.
what
are
14.
worthe
after
the
rate
,
or
els
if
.8
.
require
.12
.
for
his
proporcionall
,
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
multiplicacion
)
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
moche
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
8/12.
wherfore
abreuie
8/12
and
thei
are
1/9
for
your
quotient
,
then
diuide
the
thirde
nomber
.14
.
by
the
saied
⅔
,
and
you
shall
haue
.21
.
as
before
.
Or
els
diuide
the
seconde
nomber
.12
.
by
the
firste
nomber
.8
.
thereof
commeth
.1.1/2.
the
which
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
noumbers
of
this
rule
maie
be
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
dooe
none
otherwise
,
then
you
did
in
the
last
example
.
But
in
case
thai
be
broken
noumbers
,
or
broken
and
whole
noumbers
together
,
the
maner
and
waie
to
dooe
theim
,
receiueth
a
certaine
variacion
,
and
difficultee
,
accordyng
to
the
varietie
of
the
noumbers
,
the
whiche
operacion
easely
to
doe
,
and
vnuariablie
,
this
rule
teacheth
.
The
three
noumbers
beyng
sette
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
fraccion
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
,
the
same
broken
must
remain
in
their
estate
.
¶
A
Rule
.
This
beyng
doen
,
you
shall
multiplie
the
denominatour
of
the
firste
nomber
,
by
the
numeratour
of
the
seconde
,
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
cometh
of
this
multiplicacion
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
the
same
price
,
that
the
twoo
clothes
did
cost
,
sette
doune
your
three
nombers
thus
.
Lib.
Clothes
.
Lib.
15.
|
2.
|
300.
|
1
|
|
|
|
2
|
600
|
|
|
|
600
|
155
|
(
40
|
|
|
|
1
|
|
And
than
as
you
sée
,
multiplie
the
third
nomber
,
whiche
is
.300
.
li.
by
.2
.
which
is
the
second
nomber
,
and
therof
cometh
600.
the
whiche
.600
.
you
must
diuide
by
the
first
nomber
.15
.
and
you
shal
find
in
your
quotient
40.
whiche
is
.40
.
clothes
,
and
so
many
clothes
shall
I
buye
for
.300
.
li.
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
likewise
the
seconde
&
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
will
.48
.
poundes
gaine
me
:
Here
you
se
that
the
denominacion
of
the
first
nomber
is
nobles
,
and
the
denominacion
of
the
thirde
,
is
poundes
,
wherefore
,
before
you
dooe
procede
to
worke
by
the
rule
of
three
,
you
muste
firste
turne
the
poundes
into
nobles
in
multiplying
.48
.
poundes
by
three
nobles
and
thei
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
shall
you
sette
doune
your
.3
.
nombers
in
order
thus
.
If
.12
.
nobles
doe
gaine
me
.6
.
nobles
,
what
shall
.144
.
nobles
gaine
?
the
whiche
.
144.
are
the
nobles
which
are
in
.48
.
li.
Then
multiple
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
muse
,
to
see
all
the
three
nombers
in
this
rule
of
three
,
to
bee
of
one
denominacion
,
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
denominacion
of
the
thirde
,
that
is
to
saie
the
.12
.
nobles
into
poundes
,
then
shall
the
firste
and
the
thirde
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
.
Poundes
.
|
Nobles
.
|
Poundes
.
|
|
|
4.
|
6.
|
48.
|
|
|
|
|
6.
|
|
Nobles
.
|
|
|
288.
|
288
|
(
72
|
|
|
|
44
|
|
There
is
yet
a
more
exacte
waie
to
woorke
in
this
rule
of
three
,
whiche
is
thus
.
You
must
marke
if
the
third
and
first
nombers
in
the
rule
of
thrée
,
maie
bee
bothe
diuided
by
one
like
diuisor
:
the
whiche
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
.
Crounes
doe
buye
me
.44
.
yardes
of
clothe
,
howe
many
yardes
shall
I
haue
for
120
:
here
you
maie
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
.
Crounes
.
|
Yardes
.
|
Crounes
.
|
5.
|
44.
|
12.
|
|
12.
|
|
|
88.
|
3
|
|
44
|
528
(
105
⅕
|
|
528
|
355
|
Multiplie
.44
.
by
.12
.
and
thereof
commeth
.528
.
diuide
thesame
.528
.
by
5.
and
you
shall
finde
in
your
quocient
105.
⅗
.
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
woorkyng
by
the
rule
of
three
,
but
for
that
thei
require
the
knowledge
of
fraccions
,
and
because
thei
are
not
so
easie
as
this
first
waie
,
whiche
is
common
,
therefore
content
your selues
with
this
same
,
vntill
you
haue
tasted
the
fraccions
,
the
whiche
by
gods
helpe
I
intende
to
set
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
,
whereof
wee
haue
nowe
treated
.
For
in
the
directe
rule
of
thrée
the
greater
the
thirde
noumber
is
,
so
moche
the
greater
will
the
fowerth
be
But
here
in
this
backer
rule
it
is
contrariwise
,
for
the
greater
the
third
nomber
is
,
so
moche
lesser
will
the
fowerth
bee
.
Then
,
where
as
in
the
rule
of
.3
.
directe
,
the
thirde
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
thesame
by
the
thirde
,
and
the
nomber
whiche
commeth
in
the
quociente
,
answereth
to
the
question
.
For
soche
practise
commeth
often
tymes
in
vse
:
In
soche
sorte
,
that
if
you
woorke
thesame
by
the
rule
of
three
directe
(
not
hauyng
a
regarde
vnto
the
Proposicion
of
the
question
)
you
should
then
commit
an
euident
and
open
errour
.
¶
Example
.
If
15.
shillinges
worth
of
wyne
wyll
serue
for
the
ordinarye
of
46.
men
,
when
the
Tonne
of
wyne
is
worth
12.
pounds
:
for
how
many
men
will
the
same
15.
shillings
suffise
when
the
tonne
of
wyne
is
worth
but
eight
poundes
?
It
is
certaine
,
that
the
lower
the
price
is
that
the
tonne
of
wine
doth
cost
,
and
so
many
more
persons
will
the
said
15.
shillings
in
wine
suffise
.
Therefore
set
downe
your
numbers
thus
,
if
12.
poundes
suffise
46.
men
,
how
many
will
8.
poundes
suffise
,
you
must
multiply
46.
by
12.
and
thereof
commeth
552.
the
which
552.
you
shal
diuide
by
8
,
and
therof
commeth
69.
and
vnto
69.
men
wyll
the
saide
15.
shillings
worth
in
wine
suffyse
,
when
the
tonne
of
wyne
is
worth
but
eyght
poundes
,
as
hereafter
doth
appere
by
practise
.
Lib.
|
Men.
|
Lib.
|
|
|
12.
|
46.
|
8.
|
|
|
|
12.
|
|
7
|
|
|
92.
|
|
552
|
(
69.
|
|
46.
|
|
88
|
|
|
552.
|
|
|
|
Likewise
,
a
messenger
maketh
a
iourney
in
24
dayes
,
when
the
day
is
but
12.
houres
long
:
how
many
daies
shall
hee
be
vppon
the
same
iourney
when
the
day
is
16.
houres
in
length
?
Here
you
must
perceaue
,
that
the
more
houres
are
in
a
day
,
the
fewer
dayes
wyll
the
messenger
bee
in
going
hys
iourney
.
Therefore
wryte
downe
your
numbers
thus
as
here
you
mai
se
.
Houres
.
|
Daies
.
|
Houres
.
|
|
|
12.
|
24.
|
16.
|
4
|
|
|
12.
|
|
12
|
|
|
48.
|
|
288
|
|
|
24
|
|
166
|
|
|
288.
|
|
1
|
(
18
|
And
then
multiplie
24.
dayes
by
12.
houres
,
and
thereof
commeth
288
:
diuide
the
same
288.
by
the
thyrde
number
16.
and
you
shall
fynde
18.
the
which
is
18.
dayes
,
and
in
so
many
dayes
wyll
the
messenger
make
hys
iourney
when
the
day
is
16.
houres
long
.
Likewyse
,
when
the
bushell
of
wheate
doth
cost
3.
shillings
,
the
penny
loafe
of
breade
waieth
4.
lib.
I
demaunde
what
the
same
pennye
loafe
shall
waye
,
when
the
bushell
of
wheate
is
worth
but
twoo
shillings
:
here
is
to
bee
considered
that
the
better
cheape
the
wheate
is
,
the
heauier
shall
the
peny
loafe
waie
,
and
therefore
write
down
your
3.
numbers
thus
.
Shill.
|
Lib.
|
Shill.
|
|
|
3.
|
4.
|
2.
|
|
|
|
3.
|
|
12
|
|
|
12.
|
|
2
|
(
6.
|
Then
multiplie
4.
lib.
which
is
the
seconde
number
,
by
the
fyrst
number
3.
and
they
make
12.
the
which
12.
you
shall
diuide
by
the
thirde
number
2.
and
thereof
commeth
6.
li.
&
so
much
must
the
peny
loafe
of
breade
waye
,
when
the
bushel
of
wheate
is
worth
but
two
shillings
as
may
appeare
.
And
nowe
,
according
to
my
former
promise
,
shall
follow
the
seconde
parte
of
Arithmeticke
,
whiche
teacheth
the
working
by
Fractions
.
¶
Here
endeth
the
first
parte
of
Arithmetick
.
The
seconde
part
of
Arithmetick
,
which
treateth
of
Fractions
or
broken
numbers
.
¶
The
first
Chapter
treateth
of
Fractions
,
or
broken
numbers
,
and
the
difference
thereof
.
BRoken
number
is
as
much
as
a
parte
or
many
parts
of
1.
wherof
there
are
two
numbers
with
a
line
betwene
them
both
:
that
is
to
say
,
the
one
which
is
aboue
the
line
is
called
the
numerator
.
And
the
other
vnderneth
the
line
is
called
the
denominator
,
as
by
example
,
thre
quarters
,
which
must
be
set
downe
thus
,
¾
:
whereof
3.
which
is
the
higher
number
aboue
the
line
is
called
the
numerator
,
and
4.
which
is
vnder
the
line
is
called
the
denominator
.
And
it
is
alwaies
conuenient
that
the
numerator
be
lesse
in
number
,
than
the
denominator
.
For
if
the
numerator
,
and
the
denominator
were
egall
in
value
:
then
shoulde
they
represent
a
whole
number
thus
,
as
1/1
,
2/2
,
3/3
,
which
are
whole
numbers
:
by
reason
that
the
numerators
of
these
,
and
all
such
like
,
may
be
diuided
by
their
denominators
,
and
theire
quotientes
will
alwayes
be
but
1.
But
in
case
that
the
numerator
doe
exceede
hys
denominator
,
then
it
is
more
than
one
whole
:
as
20/18
,
is
more
than
a
whole
number
by
2/18
,
other
diffinition
doth
not
hereunto
appertaine
.
Furthermore
it
is
to
be
vnderstande
that
the
middest
of
all
broken
numbers
is
the
iust
half
of
1.
whole
,
as
6/12
,
7/14
,
8/16
,
9/18
,
and
other
lyke
,
are
the
halfes
of
one
whole
number
,
wherof
doth
growe
,
and
come
forth
2.
progressions
naturall
:
the
one
progreding
by
augmenting
,
or
encreasing
,
as
these
.
½
⅔
¾
⅘
⅚
6/7
⅞
8/9
9.
/
10.
&c.
And
they
doe
proceede
infinitely
and
wyll
neuer
reache
to
make
a
whole
number
thus
1/1.
And
the
other
progression
,
doth
progrede
by
diminishing
or
decreasyng
,
as
thus
.
½
⅓
¼
⅕
⅙
1/7
⅛
1/9
1/10.
&c.
And
these
do
proceede
infinitely
,
and
shall
neuer
come
to
make
a
0.
which
signifieth
nothing
,
but
shall
euer
retaine
some
certaine
number
whatsoeuer
,
whereby
it
doth
appeare
that
broken
numbers
are
infinite
.
¶
The
seconde
Chapter
treateth
of
the
reducing
or
bringing
togither
,
of
2.
numbers
,
or
many
broken
dissembling
,
vnto
one
broken
sembling
.
REduction
,
is
as
much
as
to
bring
togither
,
or
to
put
in
semblaunce
2.
or
manye
numbers
dissembling
one
from
the
other
,
in
reducynge
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
.
Multiplie
the
Denominators
the
one
by
the
other
,
and
so
you
shal
haue
a
new
denominator
common
to
all
,
the
which
denominator
diuide
by
the
perticuler
denominators
,
and
multiplye
euery
quotient
by
his
numerator
and
so
you
shall
haue
newe
numerators
,
for
the
numbers
which
you
woulde
reduce
,
as
appeareth
by
thys
example
followyng
.
¶
Reduction
in
common
denomination
.
IF
you
wyll
reduce
⅔
and
⅘
togither
,
you
must
fyrst
multiplie
the
two
denominators
the
one
by
the
other
,
that
is
to
say
3.
by
5.
maketh
15.
which
is
your
common
denominator
,
that
set
vnder
the
crosse
,
then
diuide
15.
by
the
denominator
3.
&
you
shall
haue
5.
which
multiply
by
the
numerator
2.
and
you
shall
finde
10.
set
that
ouer
the
⅔
and
they
are
10/11
,
for
the
⅔
.
Afterwardes
diuide
15.
by
the
denominator
5.
and
therof
commeth
3.
the
which
multiply
by
the
numerator
4.
and
you
shall
finde
12.
which
set
ouer
the
head
of
the
⅘
and
they
make
12/15
for
the
⅘
:
as
appeareth
more
plainer
in
the
margent
.
2.
If
you
will
reduce
½
,
⅔
,
¾
,
⅚
,
togither
,
you
must
multiplie
all
the
denominators
the
one
by
the
other
,
that
is
to
say
,
2.
by
3.
maketh
6
,
then
6.
by
4.
and
mounteth
24.
Last
of
all
24.
by
6.
and
thereof
commeth
144.
for
the
common
denominator
.
Then
,
for
the
fyrst
diuide
144.
by
the
denominator
2.
and
thereof
commeth
72.
the
which
multiplye
by
the
numerator
1.
and
it
is
still
72.
set
that
ouer
the
½
and
it
is
72/144
,
for
the
½
:
Then
diuide
144.
by
the
seconde
denominator
3.
&
therof
commeth
48
:
the
which
multiplie
by
the
seconde
numerator
2.
and
they
are
96.
which
set
ouer
the
⅔
and
they
make
96/144
,
for
the
⅔
:
Then
diuide
144.
by
the
thirde
denominator
4.
&
therof
commeth
36.
the
which
multiplye
by
the
thyrde
numerator
3.
&
they
make
108.
which
set
ouer
the
¾
and
they
are
108/144
for
the
¾
.
Finally
diuide
144.
by
the
last
denominator
6
,
&
thereof
commeth
24
:
The
which
multiply
by
the
last
numerator
5.
&
therof
commeth
120.
Which
set
ouer
the
⅚
and
they
are
120/144
,
for
the
⅚
,
as
appeareth
here
by
practise
.
¶
Reduction
of
broken
numbers
of
broken
.
IF
you
wyl
reduce
the
broken
of
broken
togither
,
as
thus
,
the
⅔
of
7/4
of
⅘
,
you
must
multiplye
the
numerators
the
one
by
the
other
to
make
one
broken
number
of
the
thre
broken
numbers
,
that
is
to
saye
2.
by
1.
maketh
2.
and
then
2.
by
4.
maketh
8.
which
is
your
numerator
.
Then
multiplye
the
Denominators
the
one
by
the
other
,
that
is
to
saye
3.
by
4.
maketh
12.
and
then
12.
by
5.
maketh
60.
for
your
denominator
,
set
8.
ouer
60.
with
a
line
betwene
them
,
and
they
be
1/60
which
being
abbreuied
are
2/15
and
so
much
are
the
⅔
of
⅙
of
⅘
as
appereth
in
the
margent
.
¶
Another
example
of
the
same
reduction
and
of
the
seconde
reduction
.
IF
you
will
reduce
⅔
of
,
¼
,
of
⅘
,
the
¾
,
of
5/7
:
And
the
½
,
of
the
½
,
of
the
⅔
of
⅓
.
First
it
behoueth
you
of
euery
partie
of
the
broken
numbers
to
make
of
eche
of
them
one
broken
,
as
by
the
third
reduction
is
taught
:
that
is
to
say
,
in
multiplying
the
numerators
by
numerators
&
denominators
by
denominators
:
first
,
for
the
fyrst
part
which
is
⅔
of
¼
of
⅘
,
you
must
as
is
said
before
,
multiplie
2.
by
1.
and
then
by
4.
&
you
shall
haue
8.
for
the
numerator
lykewise
multiplie
3.
by
4.
and
the
product
by
5.
and
you
shall
haue
60.
for
the
denominator
,
so
they
make
,
1/60
which
being
abreuied
are
2/15
for
the
first
part
,
that
is
to
say
,
for
the
⅔
of
¼
of
⅘
,
secondely
for
the
¾
of
5/7
multiplye
lykewise
the
numerator
3.
by
5.
maketh
15.
for
the
numerator
,
&
multiplie
4.
by
7.
maketh
28.
for
the
denominator
,
and
then
they
be
15/29
for
the
seconde
part
that
is
to
say
for
the
¾
of
5/7.
Thirdely
for
the
½
of
½
of
⅔
of
⅓
multiply
the
numerators
the
one
by
the
other
,
that
is
to
say
,
1.
by
1.
and
then
by
2.
and
last
by
1.
and
al
maketh
but
2.
for
the
numerator
,
likewyse
multiplie
2.
by
2.
maketh
4.
and
4.
by
3.
maketh
12.
and
then
12.
by
3.
maketh
36.
for
denominator
,
and
they
are
2/36
,
which
beyng
abreuied
maketh
1/18
,
for
the
thyrde
part
,
the
is
to
say
for
½
of
the
½
of
⅔
of
⅓
.
Last
of
all
take
the
2/15
the
15/28
and
the
1/18
&
reduce
them
according
to
the
order
of
the
seconde
reduction
,
and
you
shall
finde
1008/7560
for
the
2/15.
And
4050/7500
for
the
15/28.
And
420/7560
for
the
1/18
:
and
thus
are
broken
numbers
of
broken
,
reduced
as
appereth
by
practise
.
8
|
15
|
2
|
|
|
|
⅔¼⅘
|
¾
5/7
|
½½⅔⅓
|
|
|
|
1008
|
4050
|
420
|
60
|
28
|
36
|
2/15
|
15/28
|
1/18
|
|
|
|
|
760
|
|
¶
Reduction
of
broken
numbers
,
and
the
partes
of
broken
togither
.
IF
you
wyll
reduce
⅓
and
the
½
of
⅓
togither
to
bring
them
into
one
broken
number
,
you
must
first
set
down
the
⅓
and
½
as
appereth
in
the
marget
with
a
crosse
betweene
them
,
&
then
multiplie
the
twoo
denominators
the
one
by
the
other
,
that
is
to
say
,
2.
by
3.
maketh
6.
set
that
vnder
the
crosse
,
then
multiplie
the
fyrst
Numerator
,
one
by
the
laste
denominator
twoo
,
and
that
maketh
2.
vnto
the
whych
adde
the
laste
numerator
one
,
and
they
be
three
,
whych
set
aboue
your
crosse
,
so
you
shall
finde
that
the
⅓
and
the
½
of
⅓
doe
make
⅓
which
being
abbreuied
dothe
make
⅓
,
which
is
as
much
as
the
⅓
and
the
½
of
⅓
.
Likewise
if
you
wyll
reduce
the
⅔
and
the
¼
of
⅓
,
you
must
do
as
before
,
set
downe
the
⅔
and
¼
with
a
crosse
betweene
them
,
multiplie
the
two
denominators
the
one
by
the
other
,
that
is
to
say
,
3.
by
4.
maketh
12.
which
set
vnder
the
crosse
as
you
see
in
the
margent
and
then
multiply
the
first
numerator
2.
by
the
last
denominator
4.
and
thereof
commeth
8.
whereunto
adde
the
last
numerator
1.
and
that
maketh
9.
which
set
ouer
the
Crosse
,
so
shall
you
finde
that
the
⅖
and
the
¼
of
⅓
are
worth
9/12
,
which
abbreuied
doe
make
¼
,
as
appeareth
by
example
in
the
margent
.
Reduction
of
whole
numbers
and
broken
togither
into
a
Fraction
.
IF
you
will
reduce
whole
number
with
broken
,
you
muste
bring
the
whole
number
into
broken
,
as
by
this
example
may
appere
:
reduce
17.
⅝
into
a
broken
number
,
first
you
must
multiplye
the
whole
number
17.
by
the
denominator
of
the
broken
,
which
is
eight
in
saying
eight
times
17.
doe
make
136.
vnto
the
which
you
must
adde
the
numerator
of
⅝
which
is
5.
and
all
amounteth
to
141.
which
set
ouer
8.
with
a
line
betweene
them
,
&
thei
wil
be
141/8
so
much
is
17.
⅛
worth
in
a
Fraction
as
appeareth
here
by
practise
.
17.
|
|
141.
|
|
|
8.
|
|
5.
|
|
|
136.
|
17.
|
8.
|
maketh
|
141.
/
8.
|
5.
|
|
|
|
|
141.
|
|
|
|
|
In
case
you
haue
whole
number
and
broken
to
bee
reduced
,
with
broken
you
muste
bring
the
whole
number
into
his
broken
,
in
multiplying
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
10
⅔
al
into
thirds
,
as
by
the
syxt
reduction
,
and
you
shal
finde
32/3
,
then
reduce
the
32/3
and
4/7
togither
,
by
the
fyrste
reduction
,
and
you
shall
finde
224/21
for
the
32/3
:
and
12/21
for
4/7
,
as
appereth
here
by
practise
.
Also
in
case
you
haue
in
both
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
with
14.
⅔
to
bring
them
into
one
denomination
,
fyrst
bring
the
12.
¼
all
into
fourthes
,
and
you
shall
fynde
49/4
:
then
likewise
reduce
14.
⅔
all
into
thirdes
,
and
you
shall
haue
44/3
,
for
the
14.
⅔
,
then
reduce
49/4
and
44/3
togither
,
by
the
order
of
the
fyrst
Reduction
,
and
you
shall
fynde
147/12
for
the
49/4.
And
170/12
for
the
14.
⅔
as
here
by
practise
doth
plainly
appere
.
¶
The
thirde
Chapter
treateth
of
abbreuiation
of
one
great
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
which
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
may
fynde
in
the
same
broken
number
,
&
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
abreuiat
54/81
,
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
cannot
take
a
whole
number
out
of
the
denominator
,
81.
but
that
there
will
bee
either
more
or
lesse
than
a
whole
number
,
therefore
if
you
diuide
54.
54.
/
81.
|
|
|
1
|
|
|
27
|
|
|
27
|
(
2
|
|
|
|
⅔
|
2
|
|
|
81
|
|
|
27
|
(
3
|
|
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
shal
find
⅔
and
thus
by
this
rule
the
54/81
are
abreuied
vnto
⅔
,
as
appeareth
in
the
margent
,
and
so
is
to
be
vnderstande
all
other
.
¶
The
forme
&
maner
how
to
finde
out
the
greater
number
,
by
the
which
you
mai
wholy
diuide
the
numerator
&
denominator
(
to
th
ende
that
you
may
abreuiat
them
)
is
thus
.
Fyrst
,
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
vntyll
you
haue
so
diuided
that
there
may
nothing
remaine
,
then
is
it
to
be
vnderstande
,
that
your
last
diuisor
(
wherat
you
did
ende
,
and
that
0.
did
remaine
after
your
last
diuision
)
is
the
greatest
number
,
by
the
which
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
be
abreuied
.
Example
,
of
54/11
diuide
81.
(
which
is
the
denominator
)
by
54.
which
is
his
numerator
,
and
there
resteth
27.
then
diuide
54.
by
27.
and
there
remaineth
nothing
,
wherefore
your
last
diuisor
27.
is
the
number
,
by
the
which
you
must
abreuiat
54/81
as
in
the
last
example
is
specifyed
.
¶
Another
stile
of
abbreuiation
.
2.
Mediate
the
numerator
,
and
also
the
denominator
of
your
fraction
in
case
the
numbers
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
it
your
numerator
,
also
of
½
the
denominator
,
make
your
denominator
,
and
so
continue
as
often
as
you
may
in
takyng
alwayes
the
½
of
the
numerator
,
and
semblablie
of
the
denominator
,
or
else
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
bee
noted
,
that
with
whatsoeuer
number
of
these
,
you
doe
abbreuiate
the
Numerator
of
your
Fraction
,
by
the
same
you
must
abbreuiate
lykewyse
the
Denominator
,
so
continuyng
vntill
they
can
no
more
bee
abbreuied
.
And
it
is
to
bee
vnderstande
that
if
the
Numerator
and
the
Denominator
be
euen
numbers
,
as
you
may
knowe
when
the
fyrst
fygure
is
an
euen
number
,
or
a
0
,
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
thre
.
And
if
thei
be
odde
numbers
,
then
must
you
consider
if
they
maye
bee
abbreuied
by
9.
by
7.
by
5.
or
by
3
:
but
when
the
fyrst
number
,
as
well
of
the
Numerator
,
as
of
the
Denominator
are
euen
numbers
,
then
may
you
wel
knowe
that
such
numbers
maye
bee
abbreuied
by
2.
as
is
aforesaide
.
And
if
you
adde
the
fygures
of
the
Numerator
togither
,
in
such
manner
as
you
doe
in
makyng
the
proofe
by
nine
in
whole
numbers
:
that
is
,
if
you
fynde
9.
it
appeareth
that
you
maye
abbreuye
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
number
that
you
doe
abreuiate
the
numerator
,
by
the
same
must
you
abreuiate
likewyse
the
denominator
,
and
if
the
fyrst
fygures
of
the
same
number
be
.5
.
or
0.
you
may
abreuiate
them
by
5.
but
if
the
fyrste
fygures
be
both
0.
they
may
be
abreuied
by
10.
in
cutting
awaye
the
twoo
Cyphers
thus
,
as
which
maketh
⅔
,
as
sometimes
by
100.
thus
,
as
in
cutting
away
the
foure
ciphers
after
this
sort
,
and
then
the
100/200
doe
make
½
,
and
after
this
maner
haue
I
set
here
diuers
examples
,
although
that
all
numbers
cannot
be
abreuied
by
this
rule
,
that
is
to
saye
,
all
those
which
may
bee
well
abreuyed
by
the
fyrste
rule
aforesaide
.
Abreuiated
.
by
|
10.
|
3840/7080
|
by
|
9.
|
1890/4725
|
by
|
8.
|
384/768
|
by
|
7.
|
210/525
|
by
|
6.
|
48/69
|
by
|
5.
|
30/75
|
by
|
4.
|
8/16
|
by
|
3.
|
6/15
|
by
|
2.
|
2/4
|
|
|
2/6
|
|
|
½
|
|
|
|
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
may
then
take
one
of
them
of
the
numerator
,
and
also
one
of
them
of
the
denominator
,
and
it
shall
bee
abreuyed
as
555/888
,
being
abreuiated
after
this
maner
commeth
to
⅝
.
And
yet
it
happeneth
sometimes
,
that
two
,
or
many
fygures
of
the
numerator
are
proportioned
vnto
two
,
or
many
fygures
of
their
denominators
and
the
other
fygures
of
the
same
number
doe
beholde
the
one
the
other
in
thys
proportion
?
Then
may
you
take
two
or
many
figures
,
as
well
of
the
numerator
as
of
the
denominator
,
and
by
this
maner
the
same
number
shal
be
abbreuied
,
as
4747/5959
which
being
abbreuied
by
this
rule
,
do
come
to
47/59.
4.
Also
it
happeneth
sometimes
that
you
woulde
abbreuiate
one
number
vnto
the
semblaunce
or
likenesse
of
another
.
And
for
to
know
if
the
same
may
be
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
quotients
be
egall
,
then
is
one
of
them
the
number
by
the
which
the
saide
fraction
must
be
abbreuied
,
as
by
example
of
115/207.
I
would
know
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
both
the
quotients
23.
by
the
whych
it
appeareth
that
this
number
may
be
abbreuied
by
23.
5/9.
|
115/207
|
115
|
●
|
|
|
55
(
23
|
207
|
|
|
|
99
(
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
the
one
to
the
other
you
must
reduce
than
into
a
common
denomination
,
which
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
folowing
.
1.
If
you
wyll
adde
⅔
with
¾
,
you
must
fyrst
reduce
the
twoo
fractions
both
into
one
denomination
,
according
to
the
introduction
of
the
fyrste
reduction
,
that
is
to
saye
,
in
multiplying
the
denominator
of
the
firste
fraction
which
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
the
fyrst
numerator
2.
by
the
last
denominator
4.
and
thereof
commeth
8.
which
set
ouer
the
⅔
,
and
then
multiply
the
last
numerator
3.
by
the
fyrst
denominator
3.
and
therof
commeth
9.
which
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
which
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
5/12
,
and
so
much
are
the
⅔
added
with
¾
as
doth
appere
.
¶
Addition
in
broken
numbers
.
2.
Also
if
you
wyll
adde
½
,
⅔
,
¾
,
⅘
,
togither
,
you
muste
fyrste
adde
the
½
and
⅔
togither
,
according
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
31/20.
Then
finally
adde
the
7/6
(
which
came
of
the
½
and
⅔
added
togither
)
with
13/20
,
&
you
shall
fynde
by
the
foresaide
addition
that
they
amount
vnto
326/120.
Wherefore
diuide
326.
by
120.
&
therof
commeth
2.
and
86.
remaineth
,
which
is
86/120
of
one
whole
,
&
thei
being
abreuied
do
make
43/60
:
&
thus
the
½
,
⅔
,
¾
,
⅘
,
added
togither
doe
amount
to
2.43/00
,
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
⅝
:
first
you
must
reduce
the
numbers
according
to
the
order
of
the
fourth
reduction
,
in
multiplying
the
numerator
of
the
fyrst
3.
fractions
,
the
one
by
the
other
,
and
of
the
product
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
fyrst
thre
broken
numbers
,
which
being
abbreuied
do
make
⅖
,
them
reduce
the
other
3.
fractions
,
by
the
saide
fourth
reduction
,
in
multiplying
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
that
⅖
which
came
of
the
fyrst
3.
broken
numbers
,
&
25/96
which
are
of
the
last
3.
fractions
,
both
togither
,
by
the
instruction
of
the
first
addition
&
you
shal
find
317/480
which
cannot
be
abbreuied
,
but
is
the
product
of
the
addition
:
so
much
are
the
⅔
of
¾
of
⅘
added
with
the
⅚
of
½
of
⅝
as
hereafter
by
practise
doth
euidently
appere
.
¶
Addition
of
broken
number
with
the
partes
of
broken
togither
.
4.
Likewise
if
you
will
adde
the
⅓
,
and
the
½
of
⅓
with
the
⅘
and
¼
of
⅕
,
you
must
reduce
the
⅔
½
by
the
fyft
reduction
and
therof
commeth
⅚
for
the
⅔
&
½
,
of
one
of
the
saide
thirdes
,
them
reduce
the
⅘
and
¼
by
the
saide
fift
reduction
,
and
thereof
commeth
17/20.
Last
of
all
adde
the
⅚
and
17/20
togither
according
to
the
fyrst
rule
of
addition
,
and
you
shall
fynde
202/120
which
beyng
diuided
bringeth
1.
&
82/120
part
remaining
,
which
abreuied
maketh
41/60
and
thus
you
doe
perceaue
that
the
⅔
&
½
added
with
the
⅘
and
¼
doe
amount
vnto
1.41/60
as
hereafter
by
practise
doth
plainely
appere
.
¶
Addition
of
whole
number
and
broken
togither
.
5.
Also
if
you
will
adde
12.
⅘
with
20.
⅚
,
you
may
(
if
you
will
)
adde
12.
&
20.
togither
,
and
they
make
32.
&
then
adde
the
twoo
broken
numbers
togither
,
that
is
to
saye
⅘
and
⅚
by
the
order
of
the
fyrst
addition
&
they
make
49/30
:
therefore
diuide
49.
by
30.
and
thereof
commeth
1.
and
19/30
partes
remaine
,
which
1.
you
must
adde
vnto
the
32.
&
the
whole
addition
wil
be
33.
19/30
,
or
otherwise
,
you
may
reduce
12.
⅘
into
the
likenesse
of
a
Fraction
by
the
syxt
reduction
and
they
will
be
64/5
,
&
likewise
by
the
same
reduction
,
reduce
20.
⅚
and
they
bee
125/6
,
then
adde
84/5
with
the
125/6
,
by
the
fyrst
addition
and
you
shall
fynde
1009/30.
Therefore
diuide
1009.
by
30.
and
therof
commeth
33.19/30
as
before
,
&
as
by
practise
of
the
same
both
the
wayes
,
doth
here
vnder
appere
.
¶
The
fift
Chapter
treateth
of
Substraction
in
broken
numbers
.
IF
you
wyll
substract
⅔
from
¾
you
must
fyrst
reduce
bothe
the
fractions
into
a
common
denomination
by
the
fyrst
reduction
,
and
you
shall
fynde
8/12
for
the
⅔
,
and
9/12
for
the
¾
.
Therefore
abate
the
numerator
8.
from
the
numerator
9.
&
there
remaineth
1/12
as
maye
appere
here
by
practise
.
2.
But
if
you
haue
a
broken
number
to
be
substracted
from
a
whole
number
,
you
must
borowe
one
of
the
whole
number
,
&
resolue
it
into
a
fraction
of
like
denomination
as
is
that
fraction
,
which
you
woulde
abate
from
the
same
whole
number
,
&
then
abate
the
saide
fraction
therefrom
,
&
you
shall
finde
what
doth
remaine
,
as
by
thys
example
.
If
you
abate
⅘
from
8.
you
must
borow
one
of
the
said
8.
and
resolue
it
into
fiftes
like
vnto
the
fraction
,
bicause
it
is
4.
fifts
,
and
that
1.
wil
be
5.
fiftes
thus
5/5
,
therfore
abate
⅘
from
5/5
&
there
will
remain
⅕
,
and
substract
the
1.
which
you
borowed
from
8.
and
there
doth
remaine
7.
and
also
the
⅕
.
Thus
the
⅘
being
substracted
from
8.
doth
leaue
7.
⅓
as
by
practise
dothe
plainely
appeare
.
3.
If
you
will
substract
broken
number
from
whole
number
and
broken
being
togither
:
thus
,
as
if
you
would
substract
¾
from
6.
⅚
,
you
may
by
the
fyrst
substraction
,
abate
¾
from
⅚
,
and
there
wil
remaine
1/12
,
and
the
6.
doth
still
remaine
whole
,
bicause
the
¾
may
bee
abated
from
the
⅚
,
thus
¾
being
abated
from
6
⅚
leaueth
6.1/12
as
appereth
by
practise
.
Lykewise
if
you
wyll
abate
⅔
,
from
14.
⅖
,
you
must
fyrst
reduce
14.
⅖
all
into
fiftes
by
the
6.
reduction
,
and
they
be
72/5
,
then
reduce
⅔
into
a
common
denomination
,
by
the
fyrste
reduction
,
and
you
shall
fynde
10/15
for
the
⅔
:
and
216/15
for
the
72/5
:
them
substract
the
numerator
10.
of
the
fyrst
fraction
,
from
the
numerator
216.
of
the
seconde
fraction
,
&
there
remaineth
206/15.
Therfore
diuide
206.
by
15.
and
therof
commeth
13.11/15
,
and
so
much
remaine
of
this
substraction
,
as
may
appere
.
4.
If
you
will
substract
whole
number
and
broken
from
whole
&
broken
,
as
thus
,
if
you
will
substract
9.
¼
from
20.
½
you
muste
reduce
9.
¼
into
fourthes
,
&
likewise
the
20.
½
into
halfes
by
the
syxt
reduction
:
&
you
shal
find
37/4
for
the
9.
¼
And
41/2.
for
the
20.
½
.
Then
reduce
37/4.
and
41/2
into
one
denomination
,
according
vnto
the
fyrste
reduction
and
you
shall
fynde
74/8
for
the
37/4
,
and
164/8
for
the
41/2
them
abate
the
numerator
of
74/8
which
is
74.
from
the
numerator
of
164/8
and
there
remaineth
90/8
then
diuide
90.
by
8.
and
thereof
commeth
11.
¼
which
is
the
remaine
of
thys
substraction
.
¶
Substraction
of
broken
numbers
of
broken
.
5.
If
you
will
substract
,
the
2/1
of
⅔
of
⅗
from
the
⅚
of
¾
of
⅞
you
must
first
bring
the
½
of
⅔
of
⅗
into
one
fraction
by
the
3.
reduction
,
&
the
3/6
of
¾
of
⅞
likewise
into
one
fraction
by
the
same
reduction
,
&
you
shall
fynde
6/30
for
the
fyrst
3.
broken
numbers
,
which
being
abbreuied
do
make
⅕
:
&
for
the
other
3.
broken
numbers
,
you
shal
finde
105/192
:
which
being
likewise
abbreuied
do
make
35/64
,
then
you
shal
substract
⅕
from
35/64
by
the
instruction
of
the
first
substraction
,
in
reducing
both
the
fractions
into
a
common
denomination
,
as
before
is
done
,
&
you
shall
find
remaining
111/320
as
may
appere
by
example
.
¶
The
sixt
Chapter
is
of
multiplication
in
broken
numbers
.
FIrst
for
to
multiplie
in
broken
number
,
there
is
a
rule
which
is
thus
,
multiply
the
numerator
of
the
one
fraction
by
the
numerator
of
the
other
.
And
then
diuide
the
fraction
if
you
may
,
or
else
abbreuiate
it
,
and
you
haue
done
,
but
if
there
be
whole
number
&
broken
togither
,
you
muste
reduce
the
whole
numbers
into
broken
,
&
adde
thereunto
the
numerator
of
his
broken
,
and
then
multiply
as
is
before
saide
,
as
also
hereafter
by
examples
shall
more
plainely
appere
.
1.
If
you
wyll
multiply
⅔
by
¾
,
you
must
multiplie
the
numerator
2.
by
the
numerator
3.
&
therof
commeth
6.
for
the
numerator
:
Likewise
multiplye
the
denominators
the
one
by
the
other
,
that
is
to
say
3.
by
4.
&
thereof
commeth
12.
fpr
the
denominator
,
so
that
thys
multiplication
commeth
to
6/12
,
which
being
abbreuiated
doe
make
½
and
so
much
amounteth
the
multiplication
of
the
⅔
by
¾
as
by
practise
.
2.
Likewise
if
you
will
multiplye
a
broken
number
by
whole
number
,
or
whole
number
by
broken
,
which
is
all
one
,
as
⅘
by
18
,
or
else
18.
by
⅘
,
you
must
set
1.
vnder
18.
thus
8/1
:
and
then
multiply
18.
by
the
numerator
4.
and
thereof
commeth
72.
the
which
diuide
by
the
denominator
5.
&
thereof
commeth
14.
⅖
for
the
whole
multiplication
,
or
otherwise
abate
from
18.
his
⅕
part
which
is
3.
⅗
,
and
there
remaineth
14.
⅖
as
aboue
.
|
72
|
|
22
|
|
⅘
|
|
18/1
|
72
|
|
|
5
|
|
55
|
(
14
2/5
|
|
18
|
|
3
|
|
|
1●/1
|
|
⅕
|
18
|
18
|
|
|
5
|
|
5
(
3.
|
3
|
⅗
|
|
|
|
|
14
|
⅖
|
3.
Also
if
you
wil
multiply
a
whole
number
,
by
whole
number
and
broken
,
or
else
whole
number
&
broken
by
a
whole
number
,
which
is
al
one
.
As
by
example
,
if
you
multiplye
15.
by
16.
¾
or
else
16.
¾
by
15.
First
reduce
16
¾
all
into
fourthes
,
in
multiplying
16.
by
the
denominator
of
¾
whych
is
4.
and
therof
commeth
64.
wherunto
adde
the
numerator
3.
and
it
maketh
67/4
which
multiplie
by
15/1
according
vnto
the
instruction
of
the
laste
example
,
&
you
shall
finde
the
product
of
this
multiplication
to
be
251.
¼
as
by
practise
doth
here
appere
.
|
|
|
|
|
67
|
|
67
|
|
1005
|
|
15
|
16
|
¾
|
1●/●
|
|
●7/4
|
335
|
|
|
|
4
|
|
67
|
|
|
|
|
|
1005
|
4.
And
if
you
wyll
multiplye
a
broken
number
,
by
whole
number
and
broken
,
or
else
whole
number
&
broken
by
a
broken
.
As
by
example
,
if
you
wil
multiplie
¼
by
18.
⅔
,
or
else
18.
⅔
by
¼
,
which
is
all
one
:
you
must
reduce
the
whole
number
into
hys
broken
by
the
syxt
reduction
.
And
you
shall
fynde
56/3
,
which
you
shal
multiplie
by
the
¼
,
after
the
doctrine
of
the
first
multiplication
,
that
is
to
say
:
in
multiplying
the
Numerator
56.
by
the
Numerator
of
¼
,
which
is
1.
And
it
is
still
56.
bicause
1.
doth
neither
multiplie
nor
diuide
.
And
likewise
you
must
multiplye
the
Denominator
3.
by
the
Denominator
4.
and
it
maketh
12.
then
diuide
56.
by
12.
and
thereof
commeth
4.
⅔
.
And
so
muche
amounteth
the
multiplication
of
the
18.
⅔
multiplyed
by
¼
as
by
example
.
|
56
|
|
56
|
|
18
|
1/8
|
⅔
|
¼
|
|
●6/3
|
56
|
|
|
|
12
|
|
12
(
4
2/3
|
5.
If
you
wyll
multiplye
whole
number
and
broken
,
with
whole
and
broken
,
you
muste
fyrste
put
eyther
whole
number
into
hys
broken
,
according
to
the
instruction
of
the
syxte
reduction
,
and
then
multiply
the
one
numerator
by
the
other
,
and
of
the
product
make
your
numerator
.
And
likewise
multiply
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
thys
multiplication
.
Example
,
If
you
woulde
multiplye
12.
⅘
by
6.
¾
:
fyrst
by
the
syxt
reduction
the
12.
⅘
wil
make
64/3
,
and
the
6.
¾
wyll
make
27/4
,
then
multiplye
the
numerator
64.
by
the
numerator
27.
and
therof
commeth
1728.
for
the
numerator
.
And
then
you
must
multiply
the
denominator
25.
by
the
denominator
4.
and
they
doe
make
20.
then
diuide
1728.
by
20.
and
thereof
commeth
86.
⅖
for
the
whole
multiplication
,
as
by
example
.
|
|
1728
|
|
64
|
|
64
|
|
27
|
27
|
12
|
⅘
|
|
6
¾
|
448
|
|
|
20
|
|
128
|
|
|
|
|
1728
|
|
|
1
|
|
|
|
|
1728
|
|
|
|
200
(
86.
|
|
|
|
2
|
|
|
6.
If
you
wyll
multiplie
one
broken
number
by
many
broken
numbers
,
thus
:
As
to
multiply
⅔
by
5/7
&
by
4/9
:
you
must
multiply
the
numerators
of
al
the
fractions
,
the
one
by
the
other
,
&
of
the
product
make
the
numerator
that
is
to
saye
:
2.
by
5.
and
they
be
10.
then
10.
by
4.
and
they
be
40.
for
the
Numerator
.
Likewise
you
muste
multiply
the
denominators
the
one
by
the
other
,
that
is
to
say
3.
by
7.
maketh
21.
then
21.
by
9.
maketh
189.
for
the
denominator
:
then
set
40.
ouer
the
189
wyth
a
line
betweene
them
,
and
they
make
40/189.
And
so
much
amounteth
the
whole
multiplication
of
the
⅔
multiplied
by
5/7
and
4/9
as
by
example
folowing
.
And
thus
is
to
be
vnderstand
of
all
such
lyke
.
|
|
|
2
|
3
|
|
|
|
|
5
|
7
|
|
|
40
|
|
10
|
21
|
|
⅔
|
5/7
|
4/9
|
4
|
9
|
40/189
|
|
189
|
|
40
|
189
|
|
¶
The
7.
Chapter
treateth
of
Diuision
in
broken
numbers
.
NOte
that
in
Diuision
of
broken
numbers
,
you
must
sette
your
diuisor
downe
fyrst
,
next
vnto
the
lefte
hande
,
and
the
diuidende
or
number
which
is
to
bee
diuided
alwaies
towarde
the
right
hande
.
And
then
multiplie
crosse wise
,
that
is
to
saye
the
numerator
of
your
diuisor
by
the
denominator
of
the
diuidende
,
and
the
product
shal
be
the
denominator
,
which
afterwarde
shall
bee
your
Diuisor
.
And
lykewise
you
must
multiplie
the
Denominator
of
your
fyrste
number
,
that
is
to
saye
of
your
Diuisor
:
by
the
Numerator
of
the
Diuidende
,
which
afterwarde
shall
be
the
Diuidend
,
and
that
must
be
set
ouer
the
Crosse
,
and
the
Denominator
vnder
the
Crosse
,
then
shall
you
diuide
the
Numerator
by
the
Denominator
if
it
maye
bee
diuided
,
if
not
,
you
must
abbreuiate
them
,
as
hereafter
by
exaumples
shall
more
plainelye
appere
.
1.
If
you
will
diuide
¾
by
⅖
,
you
muste
sette
the
Diuisor
(
which
is
2/3
)
next
to
the
lefte
hande
,
and
the
diuidende
¾
toward
your
right
hand
,
with
a
crosse
betwene
them
:
as
may
appere
by
this
example
in
the
margent
.
Then
you
shal
multiply
the
numerator
of
the
⅔
,
which
is
2.
by
the
denominator
of
the
¾
which
is
4.
and
and
thereof
commeth
8.
which
shal
be
your
new
diuisor
:
set
that
8.
vnder
the
crosse
,
as
the
denominator
,
then
multiply
the
numerator
of
the
diuidende
,
that
is
to
saye
of
the
¾
which
is
3.
by
the
denominator
of
the
diuisor
,
that
is
to
wit
of
the
⅔
which
is
3.
set
that
ouer
the
crosse
,
and
it
is
9.
for
the
numerator
,
which
shal
be
now
the
diuidende
,
or
number
to
be
diuided
.
Then
finally
you
shall
diuide
9.
by
8.
and
thereof
commeth
into
the
quotient
1.
⅛
,
and
so
oftentimes
is
⅔
conteined
in
¾
.
as
doth
appere
before
in
the
margent
.
But
in
case
you
woulde
diuide
⅔
by
¾
,
you
muste
lykewise
set
your
diuisor
¾
next
to
your
left
hande
,
as
is
before
saide
.
And
then
proceede
,
as
is
aboue
declared
,
&
you
shal
finde
that
⅔
diuided
by
¾
bringeth
into
the
quotient
8/9
,
which
cannot
be
diuided
nor
abbreuied
,
wherfore
it
appereth
that
⅔
diuided
by
¾
bringeth
but
8/9
of
one
vnitie
into
the
quotient
as
doth
appere
.
2.
Likewise
if
you
wil
diuide
a
broken
number
by
a
whole
number
or
else
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
,
&
thereof
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
them
set
the
¾
next
your
left
hand
and
put
one
vnder
13.
as
in
the
last
example
,
&
it
is
13/1
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
52/3
,
then
diuide
52.
by
3.
and
therof
commeth
17.
⅓
,
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
else
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
35/6
for
your
diuisor
,
then
put
1.
vnder
20.
And
it
wyll
bee
20/1
,
then
shal
you
multiply
35.
by
1.
and
20.
by
6.
as
is
taught
in
the
other
diuisions
,
and
you
shall
finde
120/35
:
then
diuide
120.
by
35.
and
you
shall
fynde
in
your
quotient
3.
3/7
&
so
many
tymes
is
5.
⅚
conteined
in
20.
as
in
the
margent
doth
appere
.
But
if
you
wyll
diuide
5.
9/6
by
20.
you
must
diuide
35.
by
120
,
which
you
can
not
,
wherefore
you
shall
abbreuiate
35/126
,
and
thereof
commeth
.
7/24.
4.
If
you
will
diuide
a
broken
number
,
by
whole
number
and
broken
,
or
else
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
9/164.
But
if
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
set
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.
Likewyse
multiply
3.
by
3.
for
your
Numerator
,
and
it
amounteth
to
93.
the
whych
diuision
wyll
bee
thus
93/164
as
before
doth
appere
.
But
if
you
will
diuide
13.
⅔
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
,
must
be
diuided
in
such
maner
as
broken
numbers
are
,
&
there
is
no
difference
,
sauing
only
that
of
many
broken
numbers
you
must
make
but
two
broken
numbers
,
that
is
to
say
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
49/40
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
189/320
for
this
diuision
as
doth
appere
.
But
if
you
woulde
diuide
8/21
by
9/40.
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
shal
diuide
the
same
by
½
:
likewise
if
you
will
triple
any
fraction
you
must
diuide
it
by
1/3.
And
for
to
quaduple
any
broken
number
,
you
shall
diuide
it
by
¼
,
and
so
is
to
be
vnderstande
of
all
other
.
Example
of
Duplation
.
IF
you
wil
double
⅜
you
shal
diuide
⅜
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
⅜
,
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
number
,
that
is
to
say
,
not
euen
,
them
you
may
multiply
the
numerator
by
2.
or
else
double
the
numerator
,
which
is
al
one
thing
,
and
that
fraction
shal
bee
doubled
.
Example
,
if
you
will
double
⅗
you
must
only
multiply
the
numerator
3.
by
2.
&
they
be
6.
which
maketh
that
fraction
to
be
6/5
,
the
which
6.
being
diuided
by
5.
bringeth
1.
⅕
and
so
much
is
the
double
of
⅗
.
Example
of
Triplation
.
If
you
will
triple
⅗
you
must
diuide
⅗
by
⅕
and
thereof
commeth
9/5
whiche
beinge
diuided
bryngeth
1
⅘
,
or
otherwise
,
bicause
the
denominator
is
an
odde
number
you
may
multiplie
the
numerator
3.
by
3
,
and
thereof
commeth
9.
which
maketh
9/3
as
before
.
Example
of
quadruplacion
.
If
you
will
quadruple
⅘
,
you
shall
diuide
⅘
by
¼
and
thereof
commeth
16/5
which
16.
being
diuided
by
5.
bringeth
3
⅕
,
or
otherwise
,
bicause
the
denominator
of
the
fraction
is
an
odde
number
,
you
shall
multiplie
the
numerator
of
the
⅘
that
is
to
say
4.
by
4.
and
therof
commeth
16.
the
which
diuide
by
5.
and
you
shall
finde
3.
⅕
as
before
,
and
this
sufficeth
for
duplacion
,
triplacion
and
quadruplacion
.
¶
The
9.
Chapter
treateth
of
the
prooues
of
broken
numbers
.
And
first
of
Reduction
.
IF
you
doe
abbreuiate
the
broken
numbers
which
bee
reduced
,
you
shall
retourne
them
into
their
first
estate
:
as
by
example
,
if
you
reduce
⅖
with
⅘
you
shall
fynde
10/15
and
12/15
,
then
abbreuiate
10/15
and
you
shall
finde
⅔
,
abbreuiate
likewyse
12/15
and
therof
commeth
⅘
as
before
.
The
proofe
of
Abbreuiation
.
IF
you
doe
multiplye
that
number
which
you
haue
abbreuied
by
that
or
those
numbers
,
by
the
which
you
haue
abbreuied
them
,
you
shall
returne
them
againe
into
theire
fyrst
estate
.
Example
,
if
you
will
abbreuiate
32/48
by
16.
in
taking
the
1/16
part
both
of
the
numerator
,
and
also
of
the
denominator
,
you
shall
finde
⅔
,
the
proofe
is
thus
,
you
must
multiplye
both
the
numerator
&
denominator
of
⅔
by
16.
that
is
to
say
,
three
by
16.
maketh
48.
for
the
denominator
,
&
2.
by
16.
maketh
32.
for
the
numerator
,
them
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
⅓
with
¼
you
shall
fynde
7/12.
The
proofe
is
,
if
you
substract
⅓
from
7/12
you
shall
fynde
remaining
the
other
number
which
is
¼
,
or
else
if
you
doe
substract
¼
from
7/12
there
wil
remaine
the
other
number
,
which
is
⅓
.
The
proofe
of
Substraction
.
If
you
do
adde
that
number
which
remaineth
,
with
the
number
whych
you
did
substract
,
you
shall
fynde
the
totall
summe
,
oute
of
the
which
you
made
the
abatement
:
or
otherwise
,
if
you
adde
the
two
lesser
numbers
togither
,
you
shal
finde
the
greater
.
Example
:
if
you
doe
abate
or
substracte
¼
from
⅓
there
wyll
remaine
1/12.
The
proofe
is
thus
:
you
must
adde
1/12
&
¼
togither
,
and
you
shall
fynde
⅓
,
which
is
the
greatest
number
.
The
proofe
of
Multiplication
.
If
you
diuide
the
producte
of
the
whole
multiplication
,
by
the
multiplicator
,
you
shall
fynde
in
your
quotient
,
the
multiplicande
or
number
by
the
which
you
haue
multiplyed
:
or
else
if
you
diuide
the
total
sum
which
is
come
of
the
multiplication
,
by
the
multiplicande
:
you
shall
finde
in
the
quotient
the
multiplicator
.
Example
,
if
you
multiply
⅔
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
shal
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
wil
bee
8/9
the
proofe
is
thus
,
you
must
multiply
8/9
by
¾
,
and
therof
commeth
24/36
which
being
abbreuiated
are
⅔
whiche
is
your
diuidende
,
&
by
this
maner
all
whole
numbers
haue
theire
proofes
as
well
as
broken
numbers
.
¶
The
tenth
Chapter
treateth
of
certaine
questions
done
by
broken
numbers
.
And
first
by
Reduction
.
FInde
two
numbers
,
wherof
the
2/7
of
the
one
number
may
bee
egal
vnto
the
⅜
of
the
other
.
Aunswere
:
you
shal
reduce
2/7
&
⅜
crosse-wise
,
and
you
shall
fynde
16.
ouer
the
2/7
and
21.
ouer
the
⅜
,
which
are
the
two
numbers
that
you
seeke
:
for
the
⅜
of
16.
are
6.
and
so
are
the
2/7
of
21.
lykewise
6.
wherefore
you
may
perceaue
that
the
⅜
of
16.
which
are
6.
are
egall
vnto
the
2/7
of
21.
which
is
also
6.
2.
Finde
two
numbers
,
wherof
the
⅔
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
twoo
numbers
that
you
seeke
.
For
the
⅔
of
3.
which
is
2.
is
double
vnto
the
¼
of
4.
which
is
but
1.
3.
Finde
two
numbers
wherof
the
⅓
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
⅕
togither
,
&
they
are
9/20
,
then
reduce
7/17
&
9/20
crosse-wise
,
&
you
shall
haue
140.
ouer
the
7/12
&
108.
ouer
the
9/20
,
which
are
the
two
numbers
that
you
seeke
.
For
63.
which
are
the
7/12
of
108.
are
also
the
9/20
of
140.
4.
Finde
two
numbers
,
wherof
the
½
the
⅓
and
the
¼
of
the
one
of
them
,
may
be
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
107/210.
Then
reduce
13/12
and
107/210
crossewise
,
as
by
the
fyrst
question
of
reduction
,
and
you
shall
finde
2730.
ouer
the
13/12
and
1284.
ouer
the
107/210
,
which
are
the
two
numbers
that
you
seeke
:
for
1391
which
is
the
½
the
⅓
and
¼
of
1284.
is
lyke
to
the
11/56
&
1/7
of
2730
,
which
is
also
1391.
5.
Finde
three
numbers
,
wherof
the
⅖
of
the
first
,
the
3/7
of
the
seconde
,
&
the
4/9
of
the
thirde
,
may
be
egall
the
one
to
the
other
.
Aunswere
:
set
downe
the
23/57
and
4/9
,
and
then
multiplie
the
Denominator
of
the
⅖
that
is
to
say
5.
by
the
Numerators
of
the
other
two
Fractions
,
that
is
to
say
,
by
the
Numerator
of
3/7
,
and
by
the
Numerator
of
4/9
,
which
is
3.
and
4.
And
thereof
commeth
60.
for
your
fyrst
number
,
then
shall
you
multiplye
the
Denominator
of
the
3/7
which
is
7.
by
the
Numerators
of
⅕
and
4/9
,
that
to
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
3/7
that
is
by
2.
and
by
3.
and
therof
commeth
54.
for
the
thirde
number
.
And
thus
the
⅖
of
60.
which
is
24.
is
likewise
the
3/7
of
56.
which
is
the
second
number
:
and
the
4/9
of
54.
which
is
the
thyrde
number
.
6.
Finde
three
numbers
,
of
which
the
fyrste
and
the
seconde
may
bee
in
such
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
⅕
.
Then
say
by
the
Rule
of
three
,
if
5.
do
gyue
mee
4.
what
shall
two
gyue
mee
,
which
is
the
seconde
proportional
,
multiply
the
seconde
number
4.
by
the
thyrde
number
two
,
and
therof
commeth
eyght
,
the
which
diuide
by
the
first
number
5.
and
therof
commeth
1.
⅗
for
the
thirde
proportionall
,
and
you
shal
fynde
that
3.2.1
.
⅗
are
the
three
numbers
proportionall
that
I
demaunde
,
or
else
15.10
.
&
8.
in
whole
numbers
.
1.
What
number
is
that
,
vnto
the
which
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
.
Aunswere
:
:
abate
⅖
from
⅚
,
and
there
will
remaine
13/30
,
which
is
the
number
that
you
desyre
.
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
wil
be
4.7/12
which
is
the
number
that
you
desire
to
know
.
4.
What
number
is
that
,
whereunto
if
you
adde
the
¾
of
it selfe
,
that
is
to
say
,
of
the
number
that
you
seeke
,
the
whole
addition
may
be
⅚
.
Aunswere
:
Here
foloweth
a
generall
rule
for
all
such
like
questions
.
Fyrst
,
of
3.
which
is
the
numerator
of
¾
make
still
the
numerator
and
likewise
of
3.
and
4.
togither
,
which
is
both
the
numerator
,
and
the
denominator
of
the
¾
make
your
denominator
,
so
you
shal
finde
3/7
,
then
take
the
3/7
of
⅚
which
is
15/42
or
5/14
,
and
substract
them
from
⅚
,
and
there
wyll
remaine
10/21
which
is
the
number
that
you
seeke
.
5.
What
number
is
that
,
vnto
the
which
if
you
adde
his
owne
⅔
that
is
to
say
⅔
of
it selfe
,
the
whole
addition
shall
bee
20.
Aunswere
:
doe
as
in
the
laste
question
:
of
the
numerator
of
⅔
,
that
is
to
saye
,
of
2.
make
still
your
numerator
.
And
likewise
of
the
numerator
.
2.
and
the
denominator
3.
of
the
⅔
,
make
of
them
both
,
your
denominator
,
and
you
shall
finde
⅖
,
them
take
the
⅖
of
20.
which
are
8.
And
abate
them
from
20.
and
there
will
remaine
12.
which
is
the
number
that
you
desyre
and
so
is
to
bee
done
of
all
suche
lyke
reasons
.
1.
What
number
is
that
,
from
the
which
if
you
do
abate
17.
the
rest
may
be
19.
Aunswere
:
adde
17.
and
19.
togither
and
you
shall
finde
36.
which
is
the
number
that
you
seeke
.
2.
What
number
is
that
,
from
the
which
if
you
abate
⅗
the
rest
maye
be
⅛
.
Aunswere
:
adde
⅗
and
⅛
togither
:
and
you
shall
finde
29/40
whiche
is
the
number
that
you
demaunde
.
3.
What
number
is
that
,
from
the
which
if
you
deduct
13.
½
the
rest
maye
be
5.5/7.
Aunswere
:
adde
13.
½
and
5.5/7
together
,
and
thereof
commeth
19.3/14
,
which
is
the
number
that
you
seeke
.
4.
What
number
is
that
,
from
the
which
if
you
substracte
hys
⅖
the
rest
may
be
.
12.
Aunswere
:
&
a
rule
for
such
like
reasons
,
that
is
to
saye
from
the
denominator
of
⅖
which
is
5.
abate
2.
which
is
his
numerator
,
and
there
resteth
3.
for
the
denominator
,
and
thus
of
⅖
you
haue
now
made
⅔
,
then
take
the
⅔
of
12.
which
are
8
:
and
adde
them
vnto
12.
and
thereof
commeth
20.
for
the
number
which
you
desyre
.
5.
What
number
is
that
,
from
the
which
if
you
doe
abate
hys
¾
,
the
rest
maye
be
8/9.
Aunswere
:
from
the
Denominator
of
¾
which
is
4.
substracte
hys
Numerator
3.
and
there
resteth
1.
Thus
of
¾
you
haue
made
3/1.
Then
multiplye
3/1
by
8/9
,
and
thereof
commeth
2
⅔
,
the
which
adde
vnto
8/9
,
and
you
shal
haue
3.5/9
,
which
is
the
number
that
you
seeke
.
6.
What
number
is
that
,
from
the
which
if
you
abate
his
⅘
,
the
rest
maye
be
12.
⅔
.
Aunswere
:
Doe
as
you
dyd
in
the
laste
question
,
and
you
shal
fynde
that
the
⅘
wil
bee
4/1.
And
therfore
multiplie
12.
⅔
by
4/1
,
and
thereof
commeth
50.
⅔
,
the
which
adde
vnto
12.
⅔
,
and
you
shal
fynde
93.
⅓
,
for
the
number
that
you
demaunde
.
And
thus
of
all
such
lyke
questions
.
1.
What
number
is
that
,
which
being
multiplied
by
13.
the
whole
.
Multiplication
shall
mount
to
221.
Aunswere
:
diuide
221.
by
13.
and
thereof
commeth
17.
which
is
the
number
that
you
seeke
.
2.
What
number
is
that
which
being
multipled
by
15.
the
whole
multiplication
wil
amount
to
¾
.
Aunswere
:
diuide
¼
by
15/1
and
therof
commeth
1/20
which
is
the
number
,
that
you
seeke
.
3.
What
number
is
that
,
which
being
multiplyed
by
21.
the
whole
multiplication
wil
be
16
⅘
.
Aunswere
:
diuide
16.
⅘
by
21/1
,
and
you
shall
finde
⅘
whych
is
the
number
that
you
demaunde
.
4.
What
number
is
that
,
which
being
multiplyed
by
¾
the
multiplication
will
amount
to
18.
Aunswere
:
diuide
18/1
by
¾
,
and
thereof
commeth
24.
which
is
the
number
that
you
desyre
to
knowe
.
5.
What
number
is
that
,
which
if
it
be
multiplied
by
⅔
the
whole
multiplication
wyll
bee
¼
.
Aunswere
:
diuide
¼
by
⅔
and
the
quotient
will
bee
⅜
which
is
the
number
that
you
require
to
know
.
6.
What
number
is
that
,
whych
beyng
multiplied
by
⅝
,
the
product
of
the
multiplication
wil
be
16.
⅔
.
Aunswere
:
diuide
16.2/3
by
⅝
and
thereof
commeth
26.
⅔
which
is
the
number
that
you
seeke
.
¶
Here
ensueth
other
necessarie
Questions
,
which
are
wrought
by
Multiplication
in
broken
numbers
.
I
Demaunde
how
much
the
⅝
of
20.
shillings
are
worth
,
or
what
are
the
⅝
of
20.
shillings
.
Answere
:
you
must
multiplye
⅝
by
20/1
and
the
product
wil
be
100/8
,
therefore
diuide
100.
by
8.
and
therof
commeth
12.
½
,
which
is
to
saye
,
12
.
s
.
6.d
.
and
so
much
are
the
⅝
of
20.
shillings
worth
.
2.
I
demaunde
what
the
¾
of
⅚
of
a
pounde
of
money
are
worth
,
that
is
to
say
of
20.
s.
Auns
.
multiplye
¾
by
⅚
.
And
thereof
commeth
⅝
.
Then
take
the
⅝
of
20.
shil
.
as
in
the
last
question
going
before
,
and
you
shall
finde
12
.
s
6.
pence
,
and
so
much
are
the
¾
of
⅚
of
20.
s.
worthe
.
3.
I
demaunde
what
the
⅔
of
8.
d.
½
are
worth
.
Aunswere
:
multiply
8
.
½
by
⅔
or
else
⅔
by
8.
½
,
which
is
all
one
,
and
you
shall
fynde
34/6.
Then
diuide
34.
by
6.
and
your
quotient
wyll
be
fyue
pence
⅔
,
and
so
muche
are
the
⅔
of
8.
pence
½
worth
.
4.
What
are
the
¾
of
14.
pence
⅗
.
Aunswere
:
multiplie
14
⅗
by
¾
,
and
thereof
commeth
219/20.
Therefore
diuide
219.
by
20.
and
your
quotient
wil
be
ten
pence
19/20
,
and
so
much
are
the
¾
of
14.
⅗
.
5.
How
many
quarters
or
fourthe
partes
are
contayned
in
7.
⅔
.
Auns
.
multiply
7
.
⅔
by
4/1
(
bicause
one
whole
contayneth
4.
quarters
)
and
thereof
commeth
30
.
⅔
,
and
so
many
quarters
are
in
the
7
.
⅔
,
that
is
to
say
30.
quarters
,
and
⅔
of
a
quarter
.
6.
Howe
many
thirdes
are
in
¾
and
½
,
that
is
to
saye
in
3.
quarters
,
and
½
of
one
quarter
,
which
are
⅞
by
the
fifte
reduction
.
Aunswere
:
multiplie
⅞
by
3/1
(
for
bicause
that
in
1
whole
are
conteined
3.
thirdes
)
and
thereof
will
come
⅔
and
⅝
of
a
thirde
,
and
so
many
thirdes
are
in
¾
and
½
or
in
⅞
,
which
is
all
one
.
¶
Questions
done
by
diuision
in
broken
number
.
1.
What
number
is
that
,
which
being
diuided
by
17.
the
quotient
wil
be
13.
Aunswere
:
multiplye
17.
by
13.
And
thereof
commeth
221.
whiche
is
the
number
that
you
seeke
.
2.
What
number
is
that
,
which
being
diuided
by
¾
the
quotient
wil
be
21.
Aunswere
,
multiply
21/1
by
¾
&
thereof
commeth
63/4.
Then
diuide
63.
by
4.
and
thereof
commeth
15.
¾
,
which
is
the
number
that
you
seeke
.
3.
What
number
is
that
,
whych
being
diuided
by
⅛
,
the
quotient
wil
be
⅔
.
Aunswere
:
multiplye
⅔
by
⅓
&
thereof
commeth
2/24
which
being
abbreuiated
are
1/12
for
the
number
which
you
require
.
4.
What
number
is
that
,
whiche
being
diuided
by
⅘
the
quotient
will
be
16.
⅔
?
Aunswere
:
multiply
16.
⅔
,
by
⅘
,
and
thereof
commeth
200/15.
Therefore
diuide
200.
by
15.
and
thereof
commeth
13.
⅓
which
is
the
number
that
you
desire
to
finde
.
5.
What
number
is
that
,
which
being
diuided
by
13
⅓
,
the
quotient
will
be
20.
Aunswere
:
multiplie
20/1
by
13
⅓
,
and
therof
commeth
800/3
,
then
diuide
800.
by
3.
and
thereof
commeth
266.
⅔
,
for
the
number
,
which
you
seke
.
6.
What
number
is
that
,
which
if
it
be
diuided
,
by
12.
½
the
quotient
will
be
⅞
Aunswere
:
multiplie
by
12
½
and
therof
commeth
175/16
,
then
diuide
175
by
16.
and
thereof
commeth
10.15/16
,
for
the
number
which
you
desyre
.
¶
Other
necessarye
questions
done
by
Diuision
in
broken
number
.
I
Demaunde
what
part
30.
is
of
70.
Aunswere
:
diuide
30.
by
70.
which
you
can
not
,
for
they
are
30/70
,
but
abbreuiate
them
&
they
are
3/7.
Thus
30.
are
the
3/7
of
70.
2.
I
demaunde
what
parte
10.
is
of
16
⅔
.
Aunswere
:
diuide
10/1
by
16.
⅔
,
and
thereof
commeth
30/50
which
being
abbreuiated
are
⅗
.
And
thus
10.
is
found
to
be
⅗
of
16
⅔
.
3.
More
,
what
part
is
25.
of
⅝
.
Auns
.
diuide
⅝
by
25/1
,
and
thereof
commeth
5/200
,
which
being
abbreuiated
is
,
1/40.
And
thus
⅝
is
but
the
1/40
of
25.
4.
More
,
⅚
what
part
are
they
of
⅞
.
Aunswere
:
diuide
⅚
by
⅞
,
and
you
shal
finde
40/42
which
abbreuiated
are
20/21.
5.
More
,
⅘
what
parte
are
they
of
13.
1
/
3.
Aunswere
:
diuide
⅘
by
13.
⅓
and
you
shall
fynde
12/200
,
which
being
all
breuiated
are
3/80.
And
thus
4/1
are
the
5/50
of
13.
⅓
.
6.
More
12.
½
what
part
are
they
of
30.
Aunswere
,
diuide
12.
½
by
30/1
,
&
you
shal
find
25/60
which
being
abbreuiated
are
5/12
and
thus
12.
½
,
are
the
5/12
of
30.
7.
More
,
16
⅔
what
parte
are
they
of
of
57
1/7.
Aunswere
,
diuide
16.
⅔
by
57
1/7
and
thereof
commeth
350/1200
which
being
abbreuiated
are
7/24
,
and
thus
16.
⅔
are
the
7/24
of
57.1/7.
8.
More
,
¾
and
⅔
of
¼
,
or
3.
quarters
&
⅔
of
one
quarter
,
what
parte
are
they
of
1.
Aunswere
,
reduce
5/4
and
the
⅔
of
¼
into
one
broken
by
the
first
reduction
,
and
you
shall
finde
11/12.
And
thus
the
¾
and
⅔
of
¾
are
the
11/12
of
one
whole
.
9.
More
,
of
what
number
are
9.
the
2/3.
Aunswere
,
diuide
9.
by
⅔
,
&
thereof
commeth
13.
½
,
which
is
the
number
wherof
9.
are
the
⅔
.
10.
More
of
what
number
are
⅖
the
¾
.
Aunswere
,
diuide
⅖
by
¾
,
and
thereof
commeth
8/15
which
is
the
number
wherof
⅔
are
the
¾
of
the
same
number
.
11.
More
,
of
what
number
are
5.
¾
the
3/7.
Aunswere
:
diuide
5.
¾
by
3/7
,
and
you
shall
fynde
13.5/12
which
is
the
number
whereof
5.
¾
are
the
3/7.
12.
More
,
9
⅔
what
part
are
they
of
33.
½
.
Aunswere
:
diuide
9
⅔
by
33.
½
.
And
thereof
commeth
58/201
:
and
thus
9.
⅔
are
the
58/201
of
33.
½
as
appeareth
.
The
thyrde
parte
treateth
of
certaine
briefe
rules
,
called
rules
of
practise
,
with
diuers
necessarie
questions
profitable
for
Marchaunts
.
The
first
Chapter
.
SOme
ther
be
,
which
doe
call
these
rules
of
practise
briefe
rules
,
for
that
by
them
many
questions
may
be
done
wyth
quicker
expedition
,
than
by
the
rule
of
three
.
There
be
others
which
call
them
the
small
multiplication
,
for
bicause
that
the
product
,
is
alwaies
lesse
in
quantitie
,
than
the
number
whiche
is
to
bee
multiplied
.
This
practise
commeth
not
in
vse
,
but
onely
amonge
small
kindes
of
numbers
,
whiche
haue
ouer
them
,
other
numbers
that
are
greater
.
And
thys
beyng
well
considered
,
is
no
other
thyng
but
to
conuert
lesser
and
particuler
kindes
of
number
,
into
greater
,
the
which
maye
be
done
by
the
meanes
of
diuision
,
in
taking
the
halfe
,
the
thirde
,
the
fourth
,
the
fift
,
or
suche
other
partes
of
the
summe
,
which
is
to
be
multiplied
,
as
the
multiplier
is
part
of
hys
greater
kinde
,
and
that
which
commeth
thereof
is
worth
as
muche
(
not
in
quantitie
,
but
in
his
owne
forme
)
as
if
you
did
multiplye
simplie
the
two
summes
,
the
one
by
the
other
:
And
for
the
better
vnderstanding
of
suche
conuersions
,
you
must
haue
respect
to
one
of
these
two
considerations
.
The
first
is
,
when
one
woulde
demaunde
this
question
.
At
6.d
.
the
yarde
of
Cotton
,
what
are
18.
yardes
worth
by
the
price
?
It
is
manifest
that
they
are
worth
18.
peeces
of
6.
pence
the
peece
,
or
18.
half
shillings
,
which
must
be
turned
into
shillings
,
in
taking
the
halfe
of
18
.
s
.
&
they
make
9
.
s
.
Or
otherwise
you
must
consider
,
that
at
1
.
s
.
the
yarde
,
the
18.
yardes
are
worth
18
.
s
.
wherefore
at
6.d
.
they
shall
be
but
halfe
so
much
,
for
6.d
.
is
but
the
½
of
1
.
s
.
Therefore
you
must
take
the
½
of
18.
and
they
make
9
.
s
.
which
are
worthe
as
much
as
108.d
.
that
is
to
say
,
as
18.
times
6.
pence
.
2.
First
,
if
you
will
multiplie
anye
number
after
thys
maner
by
pence
whereof
the
number
of
the
same
pence
,
doe
not
extende
vnto
12.
and
therof
to
bring
shillings
into
the
product
:
you
must
know
the
certaine
partes
of
12.
which
are
these
:
that
is
to
say
,
6.4.3
.
2.
and
1.
For
6.
is
the
½
of
12.
and
4.
is
the
⅓
of
12
:
3.
is
the
¼
:
2.
is
the
⅙
:
and
1
,
is
the
1/12.
Then
for
6.d
.
which
is
the
halfe
of
1.
shilling
,
you
must
take
the
½
of
al
the
number
which
is
to
be
multiplyed
.
And
that
which
commeth
thereof
,
shal
be
shillings
,
if
there
doe
remaine
1.
it
is
6.
pence
.
For
foure
pence
you
muste
take
the
½
of
all
the
number
that
is
to
bee
multiplyed
:
and
if
anye
vnities
doe
remaine
,
they
shall
bee
thyrdes
of
a
shilling
,
euery
one
being
in
value
4.
pence
.
For
3.
pence
you
must
take
the
¼
of
al
the
summe
:
if
anye
vnities
doe
remaine
,
they
shall
bee
fourthes
of
a
shilling
,
euery
one
being
worth
thre
pence
.
For
2.
pence
you
must
take
the
⅙
of
all
the
summe
,
and
if
any
vnities
doe
remaine
,
they
shall
be
sixte
partes
of
a
shilling
,
being
euerye
one
of
them
worth
two
pence
.
For
d.
take
the
1/12
of
the
whole
summe
,
if
anye
vnities
remaine
,
they
are
12.
partes
of
a
shilling
,
eche
of
them
being
in
value
1.d
.
as
by
these
examples
following
doth
plainely
appere
.
j.
At
6.
Pence
the
yarde
.
|
What
59.
yardes
.
|
29.
shill.
6.
Pence
.
|
ij
.
At
4.
Pence
.
|
What
82.
|
27.
shill.
4.
Pence
.
|
iij.
At
3.
Pence
.
|
What
927.
|
24.
shill.
3.
Pence
.
|
iiij
.
At
2.
Pence
.
|
What
346.
|
57.
shil
.
8.
Pence
.
|
v.
At
1.
Pence
.
|
What
343.
|
28.
shil
.
7.
Pence
.
|
Here
you
may
see
in
the
fyrst
example
that
59.
yardes
,
at
6.
pence
the
yarde
is
worth
.29
.
shil.
6.d
.
in
taking
the
½
of
59.
And
in
the
seconde
example
,
the
82.
yardes
at
4.
pence
the
yarde
,
is
worth
,
27
.
s
.
4.d
.
in
taking
the
⅓
of
82.
Likewise
,
in
the
thyrde
example
,
97.
yardes
,
at
three
pence
the
yarde
,
bringeth
24.
shil.
3.
pence
,
in
taking
the
¼
of
97.
Also
in
the
fourth
example
,
346.
yardes
,
at
2.
pence
the
yarde
,
maketh
57.
shillings
eyght
pence
in
taking
the
⅙
of
346.
And
fynally
in
the
fift
example
.343
.
yardes
,
at
1.d
.
the
yarde
,
amount
to
28.
shil.
7.d
.
in
taking
the
1/12
of
343.
And
so
is
to
bee
done
of
all
such
lyke
,
when
the
number
of
the
pence
,
is
any
of
the
certaine
partes
of
12.
But
if
the
number
of
the
pence
be
not
a
certayne
parte
of
12.
you
muste
reduce
them
into
some
certayne
partes
of
12.
and
after
the
foresayd
maner
you
shal
make
two
or
three
productes
as
neede
shal
require
,
and
adde
them
togyther
into
one
summe
as
5.d
.
may
be
reduced
into
4.
&
1.
or
els
into
3.
&
2
:
wherfore
if
you
wil
work
by
4.
&
by
1
:
you
must
for
4.d
.
take
fyrst
the
⅓
.
of
the
number
,
that
is
to
be
multiplied
,
and
for
1.d
.
take
the
2/12
,
or
rather
for
1.d
.
ye
may
take
the
¼
of
the
producte
which
did
come
of
the
4.d
.
bycause
that
1.d
.
is
the
¼
of
4.d
.
But
if
you
wyl
worke
by
3
,
and
2
,
you
shal
take
for
3.d
.
the
¼
.
of
the
number
which
is
to
bee
multiplied
:
and
likewyse
for
2.d
.
the
⅙
of
the
same
number
,
adding
togyther
both
the
productes
.
The
totall
summe
of
those
two
numbers
shall
be
the
solution
to
the
question
.
And
in
like
maner
is
to
be
done
of
all
other
.
As
by
these
formes
folowing
may
appeare
.
j.
At
5.
d.
the
yarde
.
|
What
49.
yardes
?
|
16.
shil.
4.
d.
|
4.
shil.
1.
|
20.
shil.
5.
d.
|
ij
.
At
7.
d.
|
What
5●
?
|
18.
shil
.
0
|
13.
shil
.
6
|
31.
shil.
6.
d.
|
iij.
At
8.
d.
|
What
40
?
|
13.
shil
.
4
|
13.
shil
.
4
|
26.
shil.
8.
d.
|
iiij
.
At
9.
d.
|
What
73
?
|
36.
shil
.
6
|
18.
shil
.
3
|
54.
shil.
9.
d.
|
v.
At
10.
d.
|
What
32
?
|
16.
shil
.
0
|
10.
shil
.
8
|
26.
shil.
8.
d.
|
vj.
At
11.
d.
|
What
27
?
|
9.
shil
.
0
|
9.
shil
.
0
|
6.
shil
.
9
|
24.
shil.
9.
d.
|
Here
in
this
same
first
exaumple
where
it
is
demaunded
(
at
5.
pence
the
yard
)
how
much
are
nine
and
fourty
yardes
worth
?
Fyrst
for
foure
pence
,
I
take
the
⅓
of
49
.
s
.
and
thereof
commeth
16
.
s
.
4.d
.
them
for
1.d
.
I
take
the
¼
of
the
same
product
,
that
is
to
say
,
of
16
.
s
.
4.
d.
and
that
bringeth
.4
.
shil.
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
yard
.
For
7.d
.
take
the
⅓
and
the
¼
of
the
whole
summe
which
is
to
be
multiplied
,
and
adde
them
togither
,
that
is
to
say
,
for
4.d
.
the
⅓
and
for
3.d
.
the
¼
:
bycause
4.d
.
is
the
⅓
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
yard
what
are
54.
yardes
worth
?
Fyrst
for
4.d
.
I
take
the
⅓
of
54
:
and
they
make
18
.
s
.
Likewyse
for
3.d
.
I
take
the
¼
of
54.
and
they
are
13
.
s
.
6.d
.
Then
I
adde
18
.
s
.
and
13
.
s
.
6.
d.
togither
,
so
both
amount
to
31
.
s
.
6.d
and
so
much
are
the
54.
yardes
worth
at
5.d
.
the
yarde
.
Otherwyse
for
7.d
.
take
first
the
½
,
of
the
whole
summe
for
6.d
.
Then
for
1.d
.
take
the
⅙
of
thesame
product
,
and
adde
them
togither
,
so
shall
you
haue
the
lyke
summe
as
before
.
For
eight
pence
you
must
first
take
⅓
of
the
whole
summe
for
4.
pence
,
and
another
⅓
for
other
4.d
.
and
adde
than
togyther
as
in
the
example
doth
euidently
appeare
.
Where
the
question
is
thus
,
at
8.d
.
the
yarde
,
what
are
40
yardes
worth
?
Fyrste
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
whiche
is
also
13.
shillings
&
4.
pence
.
These
two
summes
being
added
togither
,
do
make
26.
shillings
8.
pence
,
and
so
much
are
the
40.
yards
worth
at
8.
pence
the
yard
as
in
the
third
example
abouesayd
doth
appeare
.
Otherwayes
,
for
eyght
pence
you
mai
take
first
the
½
of
the
whole
summe
for
6.d
.
Then
for
2.d
.
you
shal
take
the
⅓
of
the
product
,
which
did
come
of
the
sayd
½
,
and
adde
them
togither
,
so
shal
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
⅓
of
the
saide
product
,
that
is
to
say
of
20
.
s
.
which
bringeth
6
.
s
.
8.d
.
these
two
summes
(
20
.
s
and
6
.
s
.
8.d
.
)
I
adde
togither
&
they
make
26
.
s
.
8.d
.
as
before
.
For
9.d
.
you
must
take
the
½
&
the
¼
of
the
whole
summe
,
and
adde
them
togither
:
or
else
for
6.d
.
take
fyrst
½
of
the
whole
summe
,
then
for
3.d
.
take
the
½
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
two
summes
doe
I
adde
togither
,
&
they
make
54.
shil.
9.d
.
as
in
the
saide
fourth
example
is
euident
.
For
10.d
.
take
first
the
½
,
then
the
⅓
of
the
whole
summe
,
&
adde
them
together
For
11.d
.
take
fyrst
½
for
4.
pence
,
secondely
,
another
⅓
for
other
4.d
.
and
thirdely
¼
for
3.d
.
of
all
the
whole
summe
:
and
adde
them
togither
.
Or
else
for
11.d
.
take
first
the
½
then
the
⅓
of
the
whole
summe
,
and
finallye
the
¼
of
the
laste
product
,
adding
them
togither
.
3.
Lykewise
by
the
same
reason
,
when
you
wil
multiply
(
by
shillings
)
anye
number
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
⅕
:
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
½
of
the
number
,
which
is
to
bee
multiplied
,
and
you
shall
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
number
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
.
s
.
you
must
take
the
⅕
of
the
number
which
is
to
bee
multiplied
.
And
if
there
do
remaine
any
vnities
,
they
shal
be
fift
parts
of
a
pound
euery
one
being
worth
four
shillings
.
At
10.
shillings
the
Peece
.
|
What
75.
Peeces
?
|
37.
li.
10.
shil
.
|
At
5.
shil
.
|
What
89.
|
22.
li.
5.
shil
.
|
At
4.
shil
.
|
What
93.
|
18.
li.
12.
shil
.
|
For
2.
shillings
you
must
take
the
1/10
of
the
number
that
is
to
be
multiplied
.
Wherefore
,
if
you
wyll
take
the
1/10
of
any
number
:
you
muste
seperate
the
last
figure
of
the
same
number
which
is
nerest
your
ryghte
hande
,
from
all
the
other
fygures
.
For
all
the
other
figures
which
doe
remayne
towarde
your
lefte
hande
,
from
the
same
fygure
,
which
is
seperated
,
shall
bee
the
sayde
1/10
of
a
pounde
:
and
that
seperated
fygure
,
towarde
your
right
hand
shall
be
so
many
peeces
of
2.
shillings
the
peece
:
the
which
fygure
muste
be
doubled
,
to
make
therof
shillings
,
as
by
these
examples
appeareth
.
At
2.
shil
.
|
What
9/8.
|
9.
li
16.
shil
.
|
At
2.
shil
.
|
What
40/3.
|
40.
li.
6.
shil
.
|
Herevpon
dependeth
another
exact
way
for
to
multiply
by
shillings
(
if
the
number
of
shillings
be
euen
)
which
is
thus
:
you
shal
take
½
the
number
of
the
same
shillings
,
and
conuert
them
into
peeces
of
2.
shillings
.
Then
by
the
number
of
this
halfe
,
you
must
firste
multiply
the
last
figure
toward
your
right
hande
,
of
the
number
which
is
to
be
multiplied
:
And
if
ther
be
any
tennes
in
the
same
product
,
those
must
ye
reserue
in
your
minde
:
But
if
(
wyth
the
same
or
els
without
the
same
)
you
doe
finde
any
diget
number
,
the
same
diget
number
shall
you
double
,
&
put
it
in
the
place
of
shillings
:
Then
muste
you
proceede
to
the
multiplication
of
the
other
figures
,
adding
vnto
the
product
the
tennes
which
you
before
reserued
:
and
therof
shal
come
pounds
.
Now
,
for
your
better
vnderstanding
of
this
which
hath
bene
said
and
by
the
way
of
example
,
I
wil
propone
vnto
you
this
question
.
At
8.
shillings
the
grosse
,
what
are
97.
grosse
worth
after
the
rate
?
Firste
in
this
example
I
take
halfe
the
number
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
apereth
by
diuers
examples
.
At
8.
shil
.
the
Grosse
.
|
4
)
What
9/1
|
38.
li.
16.
shil
.
|
At
6.
shil
.
|
3
)
What
9/9
|
29.
li.
14.
shil
.
|
At
12.
shil
.
|
9
)
What
34/5
|
207.
li.
0
shil
.
|
At
14.
shil
.
|
7
)
What
21/0
|
127.
li.
0.
shil
.
|
Now
in
the
first
example
,
where
it
is
demaunded
,
at
8
.
s
.
the
grosse
,
what
are
97
grosse
?
First
the
½
of
8
.
s
.
which
is
4
.
s
.
being
set
apart
behind
the
croked
line
,
as
before
is
sayd
:
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
,
which
before
I
dyd
reserue
,
and
they
make
38.
Therefore
I
put
38
,
vnder
the
lyne
in
the
place
of
poundes
,
and
the
whole
summe
wil
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
multiplying
by
6.
Likewise
of
6.
shillings
if
you
multiply
by
3
,
also
of
14.
if
you
multiply
by
7.
And
so
of
all
euen
numbers
after
the
same
manner
.
For
1.
Shilling
you
must
take
the
½
of
the
1/10
parte
of
any
number
that
is
to
be
multiplied
.
At
1.
shil
.
|
What
35/0
|
17.
li.
10.
shil
.
|
And
if
any
thyng
do
remayne
,
they
are
shil
.
Thus
by
this
manner
shil
.
are
conuerted
into
poundes
:
for
it
is
euen
lyke
,
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
seperate
the
laste
fygure
of
350.
nexte
to
my
ryght
hand
,
which
is
the
0
wyth
a
line
betweene
it
and
the
figure
5.
Then
I
make
a
line
vnder
the
35/0
,
and
I
take
the
½
of
35
,
after
this
maner
:
saying
the
½
of
3.
is
1.
and
1.
remayneth
,
which
remayn
signifieth
10.
in
that
second
place
.
Then
I
put
1.
vnder
the
line
agaynst
3
,
&
I
proceede
to
the
rest
,
saying
:
the
halfe
of
15
,
is
7.
(
which
15.
came
of
the
1.
that
remayned
,
and
of
the
5.
in
the
first
place
)
I
put
7.
vnder
the
line
right
agaynst
5
,
and
they
make
17.
li.
The
1
,
which
did
last
remayne
,
is
10
.
s
.
Therfore
I
put
10
.
s
aparte
vnder
the
line
,
and
the
whole
summe
is
17.
li.
10
.
s
.
so
muche
are
350.
worth
at
1
.
s
.
the
peece
.
But
when
the
number
of
shillings
is
not
some
certayne
parte
of
20.
shil
.
you
muste
then
conuert
the
same
number
of
shillings
,
into
the
certayn
partes
of
20.
and
make
twoo
or
three
products
,
as
nede
shal
require
,
the
which
muste
bee
added
togyther
after
thys
maner
following
.
For
3.
shillings
you
must
firste
take
for
2.
shil
.
the
1/10
of
the
number
that
is
to
be
multiplied
,
then
for
1.
shilling
you
must
take
the
½
of
the
producte
which
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
.
Firste
,
for
2
shillings
I
take
the
1/10
of
684
,
which
is
68
:
At
3.
shil
.
|
What
68/4
?
|
68.
li.
0.
sh
|
34.
4.
|
102.
li.
12.
sh
|
in
seperating
the
laste
figur
4
,
which
I
must
double
,
and
they
be
8.
I
set
eyghte
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
.
which
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
saye
,
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.
shil
.
take
3/10
of
the
number
which
is
to
be
multiplied
:
that
is
to
say
,
first
1/10
,
then
double
the
product
of
the
same
1/10
and
adde
them
togither
.
Or
otherwise
for
4
.
s
.
take
fyrst
the
⅕
of
the
number
that
is
to
be
multiplied
,
them
take
the
½
of
the
product
which
is
for
two
.
s.
and
adde
them
togither
.
Or
else
take
for
5.
shil
.
the
¼
of
the
whole
summe
,
then
for
1.
shil
.
the
⅕
of
the
product
and
adde
them
togither
.
Likewyse
for
7.
shil
.
take
fyrst
for
5.
shil
.
the
¼
then
for
●
.
shillings
take
the
1/10
of
the
number
which
is
to
be
multiplied
,
and
adde
them
togither
.
For
eyght
shillings
take
the
⅖
at
two
sundry
tymes
,
that
is
to
say
,
first
⅕
for
4.
shil
.
and
then
as
much
more
for
other
4.
shil
.
and
adde
them
togyther
.
For
9.
shil
.
take
first
the
¼
and
lykewise
the
⅕
of
the
number
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
⅕
part
of
the
product
,
and
adde
them
togither
.
For
13.
shil
.
take
the
¼
then
the
⅕
,
&
agayne
another
⅕
of
the
number
which
is
to
be
multiplied
.
And
adde
the
productes
togither
,
that
is
to
say
:
fyrste
for
5.
shil
.
take
the
¼
,
then
for
4.
shil
.
take
the
⅕
.
And
agayne
,
another
⅕
for
the
other
4.
shil
.
and
assemble
the
three
productes
,
the
like
is
to
be
done
in
al
others
,
when
the
price
of
the
thing
which
is
valued
,
is
onely
of
shillings
.
And
as
by
these
examples
following
doth
playnly
appeare
.
At
6.
shil
.
|
Whas
67.
|
13.
.8
|
6.
.14
|
20.
li.
2.
shil
.
|
At
7.
shil
.
|
What
347.
|
86.
.15
|
34.
.14
|
121.
li.
9.
shil
.
|
At
8.
shil
.
|
What
540.
|
108.
.0
|
108.
.0
|
216.
li.
0.
shil
.
|
At
9.
shil
.
|
What
230.
|
57.
.10
|
46.
.00
|
103.
li.
10.
shil
.
|
At
11.
shil
.
|
What
159.
|
79.
.10
|
7.
.19
|
87.
li
9.
shil
.
|
At
12.
shil
.
|
What
349.
|
174.
.10
|
34.
.18
|
209.
li
8.
shil
.
|
At
13.
shil
.
|
What
267.
|
66.
.15
|
53.
.8
|
53.
.8
|
173.
li.
11.
shil
.
|
4.
Likewise
in
multiplying
by
pence
you
shal
haue
(
at
the
first
instant
)
poundes
in
the
product
,
in
case
you
knowe
the
certayne
partes
of
the
1/10
of
a
pound
or
of
24.
pence
,
which
are
these
12
,
8
,
6
,
4
,
3
,
and
2.
For
12
,
is
the
½
of
24
:
8.
is
the
⅓
:
6
is
the
¼
:
4
is
the
⅙
:
3
is
the
⅛
:
and
2.
the
1/12
:
but
for
12.d
.
which
is
1.
shil
.
we
haue
before
made
mencion
thereof
.
For
8.d
.
you
muste
take
the
⅓
of
the
1/10
and
the
rest
which
are
the
peeces
of
8.d
.
must
be
doubled
to
make
of
them
peeces
of
4.d
.
And
of
the
same
number
being
doubled
,
you
must
take
the
⅓
which
wil
be
shillinges
,
&
if
there
do
yet
remayn
any
thing
,
they
are
thirds
of
a
shilling
beeing
in
value
4.
pence
the
peece
.
For
6.d
.
take
the
¼
of
the
1/10
,
and
of
that
which
remayneth
you
must
take
the
½
which
shall
be
shillings
,
if
there
do
yet
remayne
1
,
it
shall
bee
in
value
6.
pence
.
For
4.d
.
you
must
take
the
⅙
of
the
1/10
and
of
that
which
resteth
,
take
the
1/●
to
make
therof
shillings
,
if
any
thing
do
yet
remayne
,
they
are
thirdes
of
a
shilling
,
being
in
value
4.d
.
the
pece
.
For
3.d
.
take
the
⅛
of
the
1/10
,
and
of
that
which
remayneth
,
take
the
¼
,
to
make
of
them
shillings
:
if
any
thing
do
yet
remaine
,
they
are
fourthes
of
a
shilling
,
euery
one
of
them
beeing
worth
3.d
.
For
2.d
.
take
the
1/12
of
the
1/10
:
and
of
that
which
resteth
take
the
⅙
the
which
are
shillings
,
if
there
do
still
remayne
any
thing
,
they
shall
be
sixt
parts
of
a
shilling
,
euery
one
being
in
value
2.d
For
1.
d.
it
is
not
possible
with
ease
,
to
bring
of
pence
,
pounds
(
into
the
product
)
vpon
the
total
summe
:
But
first
you
must
bring
them
into
shillings
by
the
order
of
the
second
rule
of
this
chapter
,
and
then
afterward
you
shal
conuert
them
into
pounds
,
if
nede
so
require
.
As
by
this
example
following
may
apeare
.
At
|
8.
d.
|
|
|
What
|
59/5.
|
|
|
|
19.
li.
17.
shil.
4.
d.
|
At
|
6.
d.
|
|
|
What
|
67/8.
|
|
|
|
16.
li.
19.
shil
.
|
At
|
4.
d.
|
|
|
What
|
93/4.
|
|
|
|
15.
li.
11.
shil.
4.
d.
|
At
|
3.
d.
|
|
|
What
|
57/1.
|
|
|
|
7.
li.
2.
shil.
9.
d.
|
At
|
2.
d
|
|
|
What
|
36/4.
|
|
|
|
3.
li.
0.
shil.
8.
d.
|
At
|
1.
d.
|
|
|
What
|
66/5.
|
|
|
|
5●5
.
|
00.
shil
.
|
4.
d.
|
|
2.
li.
|
16.
shillings
.
|
4.
d.
|
But
if
the
number
of
pence
,
be
not
a
certayne
parte
of
24.
pence
.
Then
must
you
bring
them
into
the
certain
partes
of
24.
and
make
therof
diuers
productes
,
which
must
be
added
togither
,
as
shall
hereafter
appeare
.
For
5.
pence
you
shall
firste
take
for
3.
pence
,
then
for
2.
pence
,
and
adde
them
togither
,
according
to
the
instruction
of
the
last
rule
.
Or
else
firste
take
for
4.
pence
,
and
then
for
1.
d.
For
7.d
.
firste
take
for
4.
d.
then
for
3.
d.
and
adde
them
togither
:
For
9.
d.
fyrste
take
for
6.d
.
then
for
3.
d.
adding
them
togither
.
For
10.
d.
fyrste
take
for
6.
d.
then
for
4.
d.
and
adde
them
togither
.
For
11.
d.
take
first
for
8.
d.
then
for
3.
d.
&
adde
them
togither
:
as
by
these
examples
following
doth
appeare
.
At
|
5.
d
|
|
|
What
|
92/7.
|
|
|
|
11.
|
.11
.
|
.9
|
|
7.
|
.14
.
|
.6
|
|
19.
li.
|
6.
shil
.
|
3.
d.
|
At
|
7.
d.
|
|
|
What
|
51/2.
|
|
|
|
8.
|
.10
.
|
.8
|
|
6.
|
.8
.
|
.0
|
|
14.
li.
|
18.
shil
.
|
8.
d.
|
At
|
9.
d.
|
|
|
What
|
54/6.
|
|
|
|
13.
|
.13
.
|
.0
|
|
6.
|
.16
.
|
.6
|
|
20.
li.
|
9.
shil
.
|
6.
d.
|
At
|
10.
d.
|
|
|
What
|
27/3.
|
|
|
|
6.
|
.16
.
|
.6
|
|
4.
|
.11
.
|
.0
|
|
11.
li.
|
7.
shil
.
|
6.
d.
|
At
|
11.
d.
|
|
|
What
|
26/4.
|
|
|
|
8.
|
.16
.
|
.0
|
|
3.
|
.6
.
|
.0
|
|
12.
li.
|
2.
shil
.
|
0.
d.
|
5.
If
you
wil
multiply
any
number
by
shil
.
and
pence
,
being
both
togither
you
must
take
first
for
the
shil
.
according
to
the
instruction
of
the
rule
of
this
first
chapter
,
then
take
for
the
pence
after
the
order
of
the
fourth
rule
before
mencioned
:
but
if
ther
be
any
certayn
partes
of
1.
li.
contayning
both
shil
.
and
pence
,
them
for
such
parts
you
shal
take
the
like
part
of
the
number
that
is
to
be
multiplied
,
as
the
number
is
part
of
1.
li.
the
which
certain
parts
are
these
,
6.
s.
8.
d
:
3.
s.
4.
d
:
2.
s.
6.
d
:
&
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
:
&
1.
s.
8.
d.
is
the
1/12.
then
for
6.
s.
8.
d.
you
must
take
the
⅓
of
the
number
that
is
to
bee
multiplied
:
&
if
ani
thing
do
remain
,
they
are
thirds
of
a
li.
euery
one
being
worth
6.
s.
8.
d
For
3.
s.
4.
d.
you
must
take
the
⅙
if
ani
thing
do
remain
,
they
are
sixt
parts
of
a
li.
euery
one
being
in
value
3.
s.
4.
d.
For
2.
s.
6.
d.
you
must
take
the
⅛
:
if
any
thing
be
remaining
they
ar
eight
parts
of
a
li.
ech
one
being
worth
2
.
s
.
6.
For
1.
shil.
8.
d.
you
shal
take
the
1/12
if
there
do
any
thing
remaine
,
they
are
twelfth
partes
of
a
pounde
euery
one
being
valued
at
1.
shil.
8.d
.
At
6.
shil.
8.
d
|
What
647.
|
215.
li.
13.
shil.
4.
d.
|
At
3.
shil.
4.
d
|
What
220.
|
36.
li.
13.
shil.
4.
d
|
At
2.
shil.
6.
d
|
What
47.
|
5.
li.
17.
shil.
6.
d
|
At
1.
shil.
8.
d
|
What
400.
|
33.
li.
6.
shil.
8.
d
|
6.
Heere
shall
you
accustome
youre selfe
,
to
multiply
by
all
sortes
of
summes
,
being
compound
of
shillings
,
and
pence
,
which
mai
come
to
practise
.
As
thus
,
for
1.
s.
1.
d.
for
1.
s.
2.
d.
1.
s.
3.
d.
for
1.
s.
4d
.
Likewise
for
2
.
s
.
1.d
.
2
.
s
.
2.d
.
2
.
s
.
3.
d.
2.
s.
4.
d.
And
so
of
all
other
:
considering
moreouer
,
many
subtile
abbreuiations
,
which
happen
oftentimes
that
are
easy
to
be
conceyued
.
As
thus
at
11.
s.
3.
d.
after
that
I
haue
taken
first
the
½
for
10.
s.
Then
for
1.
s.
3.
d.
I
take
the
⅛
of
the
product
,
bycause
1.
s.
3.
d.
is
the
⅛
of
10.
s.
in
taking
the
sayd
⅛
of
the
product
.
And
by
this
meanes
,
when
ye
haue
taken
one
product
,
ye
may
oftentimes
vpon
the
same
,
take
another
more
briefly
than
vpon
the
summe
that
is
to
be
multiplied
,
which
thing
you
must
foresee
.
At
11.
shil.
3.
d.
|
What
53.
|
26.
.10
.
.0
|
3.
.6
.
.3
|
29.
li.
16.
shil.
3.
d.
|
At
6.
shil.
3.
d.
|
What
58.
|
14.
10.
.
|
3.
12.
6.
|
18.
li.
2.
6.
d.
|
At
12.
8.
d.
|
What
64.
|
32.
0.
.
|
6.
8.
.
|
2.
2.
8.
|
40.
li.
10.
shil.
8.
d.
|
7.
But
if
you
wil
multiply
,
by
pounds
,
shillings
and
pence
being
altogither
.
Firste
you
muste
wholy
multiply
by
poundes
.
Then
take
for
the
shillings
and
pence
,
as
in
the
fifte
rule
of
thys
chapter
is
plainly
declared
.
And
as
by
these
examples
folowing
may
apere
.
At
|
3.
li.
|
6.
shil
.
|
8.
d.
|
What
|
49.
|
|
|
147.
|
.0
|
.
|
|
16.
|
.6
|
.8
|
|
163.
li.
|
6.
shil
.
|
8.
d.
|
At
|
5.
li.
|
18.
shil
.
|
4.
d.
|
What
|
543.
|
|
|
|
2715.
|
.0
.
|
.
|
|
271.
|
.10
.
|
.
|
|
135.
|
.15
.
|
.
|
|
90.
|
.10
.
|
.0
|
|
3212.
li.
|
15.
shil
.
|
0.
d.
|
At
|
2.
li.
|
7.
shil
.
|
4.
d.
|
What
|
927.
|
|
|
|
1854.
|
.0
.
|
.
|
|
185.
|
.8
.
|
.
|
|
154.
|
.10
.
|
.
|
|
2193.
li
|
18.
shil
.
|
0.
d.
|
8.
So
these
rules
do
serue
both
to
bye
and
sel
,
at
such
a
price
the
elle
,
the
yard
,
the
pece
,
the
pound
waight
,
or
any
other
thing
:
how
much
such
a
thing
Lykewyse
they
are
very
necessary
to
conuert
al
peces
of
gold
and
syluer
into
poundes
:
for
I
may
as
wel
say
,
at
4.
shil.
8.
d.
the
French
crowne
,
what
are
135.
crownes
worth
?
9.
When
anye
one
of
the
summes
(
which
is
to
be
multiplied
)
is
compound
of
many
denominations
:
&
the
other
is
of
one
figure
alone
:
than
shal
ye
multiplye
all
the
Denominations
of
the
other
summe
,
by
the
same
one
figure
beginning
first
wyth
that
summe
which
is
least
in
value
towardes
your
right
hande
,
and
bring
the
product
of
those
pence
into
shillings
,
and
the
producte
of
the
shillings
into
pounds
,
as
by
this
example
doth
appeare
At
3.
li.
9.
shil.
8.
d.
|
What
7.
|
24.
li.
7.
shil.
8.
d.
|
10.
But
(
if
in
any
of
the
numbers
which
are
to
be
multiplied
)
ther
be
with
it
a
broken
number
,
you
must
(
according
to
his
denominator
)
take
one
or
many
partes
of
the
other
number
,
as
neede
doth
require
:
and
set
the
number
which
commeth
thereof
,
vnder
the
productes
,
adding
thesame
togither
.
As
thus
:
At
5.
li.
7.
s.
8.
d.
the
grosse
,
what
shall
.
34.
grosse
½
cost
?
First
At
5.
li.
7.
sh.
8.
d
|
What
34
1/7.
|
170.
.0
|
11.
.6
.
.8
.
|
1.
.14
.
.0
|
2.
.13
.
.10
|
185.
li.
14.
shil.
6.
d
|
you
shall
multiply
5.
li.
7.
sh.
8.
d.
by
34.
grosse
,
saying
5.
tymes
34.
doe
make
170.
li.
then
for
6.
sh
8.
d.
take
the
⅓
of
34
which
is
11.
li.
6.
shil
8.
d.
Thirdely
,
for
1
sh.
take
34.
shillings
,
which
is
1.
li.
14.
shillings
0.
Lastly
,
for
the
½
grosse
,
you
must
take
½
of
the
5.
li.
7.
s.
8.
d.
which
is
2.
li.
13.
s.
10
And
then
adde
them
all
togither
,
so
you
shall
finde
that
the
34.
grosse
½
at
5.
pound
7.
shillings
8.
pence
is
worth
185.
pound
,
14.
shillings
6.
pence
,
as
apeareth
in
the
margent
.
And
as
in
this
last
exaumple
,
you
did
take
the
half
of
the
money
,
(
which
one
grosse
was
worth
)
for
the
½
grosse
Bycause
that
1.
grosse
beeing
worth
5.
pound
7.
shillings
8.
pence
,
the
½
grosse
muste
bee
worth
halfe
so
muche
.
So
likewise
,
if
you
haue
⅓
of
a
grosse
,
or
of
any
other
thing
,
you
muste
take
the
⅓
of
the
price
,
that
one
grosse
is
worthe
.
Semblably
,
for
the
¼
of
any
thing
you
shal
take
the
¼
of
the
price
,
also
if
you
haue
⅔
,
take
the
⅔
of
the
price
that
one
is
worth
,
and
of
all
other
fractions
,
as
by
these
examples
folowing
doth
appeare
.
At
4.
li.
6.
shil.
8.
d.
|
What
46.
½
.
|
184.
.0
.
.0
|
15.
.6
.
.8
|
2.
.3
.
.4
|
201.
li.
10.
0.
d.
|
At
8.
li.
0.
shil.
9.
d.
|
What
54.
⅓
|
432.
.0
.
.0
|
1.
.7
.
.0
|
0.
.13
.
.6
|
2.
.13
.
.7
|
43.
li.
14.
shil.
1.
d.
|
At
3.
li.
16.
shil.
8.
d.
|
What
17.
¾
|
51.
.0
.
.0
|
8.
.10
.
.0
|
5.
.13
.
.4
|
.
.19
.
.2
|
68.
li.
.
00.
shil.
10.
d.
|
11.
If
you
will
make
the
proofe
of
these
rules
aforesayd
,
you
must
first
abate
the
summe
of
money
(
which
the
fraction
of
the
multiplication
doth
import
)
from
the
totall
summe
.
And
diuide
the
rest
of
the
pounds
of
the
sayd
total
summe
,
by
the
whole
multiplicand
,
the
fraction
only
accepted
.
And
if
any
thing
do
remain
after
the
diuision
is
made
,
that
remaine
shal
be
multiplied
by
20.
and
vnto
the
product
of
that
multiplication
,
you
shall
adde
the
shillings
which
remained
of
the
rest
of
the
total
summe
Agayn
,
if
any
thing
do
remaine
after
the
same
diuision
,
you
must
multiply
thesame
by
12
,
&
vnto
the
product
adde
the
pence
of
the
total
summe
that
remayned
,
if
any
be
left
.
And
thus
if
ye
haue
truly
wrought
,
you
shal
find
again
the
higher
summe
of
your
question
that
is
to
say
,
the
price
that
one
grosse
or
any
other
thing
is
worth
,
whereof
you
demaund
.
Or
otherwise
reduce
the
remayne
of
the
totall
summe
(
the
value
of
the
moneye
that
the
fraction
is
worthe
,
beeing
fyrste
deducted
)
all
into
pence
,
in
multiplying
the
pounds
by
20
,
and
the
shillings
by
12.
adding
thereunto
,
the
shillings
and
pence
,
which
are
ioyned
wyth
the
remayne
of
the
sayd
totall
summe
,
if
any
such
be
,
then
diuide
those
pence
by
the
foresayde
number
that
is
to
be
multiplied
,
the
fraction
of
the
same
number
beeing
also
abated
.
So
shall
you
finde
the
price
that
one
peece
,
one
Grosse
,
or
any
other
thing
is
valued
at
.
As
in
the
firste
example
going
before
,
where
the
totall
summe
is
201.
pound
10.
shillings
,
from
the
which
I
doe
first
abate
the
price
of
the
half
grosse
,
which
is
2.
li.
3.
s.
4.
d
,
the
rest
is
199.
li.
6.
s.
8.
d.
which
beeing
reduced
into
pens
bringeth
47840.
d.
I
diuide
the
same
by
46.
and
thereof
commeth
1040.
pence
.
Then
I
diuide
that
1040.
pence
by
12.
and
they
bring
86.
shillings
8.
pence
,
that
is
to
say
,
4.
li.
6.
shillings
eyght
pence
,
whiche
is
the
price
that
one
grosse
,
or
any
other
thing
did
cost
,
as
in
that
fyrste
example
doth
appeare
.
12.
The
lyke
is
to
be
done
of
any
manner
of
thinge
that
is
solde
by
the
hundred
or
by
the
Kyntall
.
As
thus
:
at
12.
pound
7.
shillings
.
6.
d
the
100.
pounde
wayghte
:
what
shall
374.
pound
wayght
cost
?
You
shall
fyrst
multiply
twelue
pounde
,
seuen
shillings
,
sixe
pence
,
by
three
:
that
is
to
saye
,
by
three
hundreth
.
Then
for
50.
li.
waight
,
you
At.
12.
li.
7.
sh.
6.
d
|
What
3/74.
|
37.
.2
.
.6
|
6.
.3
.
.9
|
2.
.9
.
.6
|
0.
.9
.
.10
.
⅘
|
46.
li.
5.
sh.
7.
d.
⅘
|
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.
li.
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
,
which
will
make
the
summe
of
64.
li.
5
.
s
.
7.
d.
⅘
,
as
by
this
example
aboue
written
doth
appeare
.
The
proofe
is
made
by
reducyng
the
totall
summe
into
pence
.
And
to
diuide
the
product
by
the
number
that
is
to
be
multiplyed
,
that
to
to
saye
by
374.
likewise
diuide
the
quotient
produced
of
that
first
diuisyon
by
12.
so
shal
you
finde
againe
the
higher
summe
12.
li.
7.
shil.
6.
d.
which
is
the
price
of
100.
li.
waight
,
as
before
.
13.
Also
the
like
maye
be
done
of
our
vsuall
waight
here
in
Englande
(
which
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
,
which
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
rate
?
Fyrst
,
you
shall
multiply
24.
hundreth
by
3.
which
is
the
3.
li.
&
thereof
commeth
72.
li.
then
for
6.
s.
8.
d.
which
is
the
⅓
of
20.
s.
you
shal
take
the
⅓
of
24
which
is
8.
li.
for
At
3.
li.
6.
sh.
8
d.
|
What
26.
.3
.
21.
li.
|
72.
.0
.
.0
|
8.
.0
.
.0
|
1.
.13
.
.4
|
.
.16
.
.8
|
.
.8
.
.4
|
.
.4
.
.2
|
83.
li.
2.
sh.
6.
d
|
24.
nobles
maketh
8.
li.
afterwarde
,
for
the
3.
quarters
of
the
C.
you
shal
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.
(
which
is
the
quar.
of
a
C.
)
you
shall
take
the
¼
of
3.
li.
6.
s.
8.
d.
or
else
the
½
of
the
product
,
which
came
of
56.
li.
which
is
16.
s.
8.
d.
likewise
for
14.
li.
take
the
⅛
of
3.
li.
6.
s.
8.
d.
which
is
8.
s.
4.
d.
or
else
the
½
of
the
product
of
28.
li.
which
is
al
one
:
lastly
for
7.
li
take
the
1/16
of
3.
li.
6.
s.
8.
d.
or
else
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.
li.
waight
worth
after
3.
li.
6.
s.
8.
d.
the
C.
as
appereth
in
the
margent
.
The
proofe
hereof
is
made
,
lyke
to
the
other
proofes
aforesaide
,
sauing
that
wher
in
those
proofes
,
you
abate
the
price
of
the
money
,
that
the
fraction
was
worthe
,
from
the
totall
summe
:
here
in
thys
example
(
and
in
such
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
,
diuiding
the
product
of
the
multiplication
by
the
iust
number
of
the
hundrethes
,
so
shall
you
finde
the
pence
the
one
hundreth
is
worthe
,
which
you
shall
bring
into
poundes
by
the
order
of
diuision
,
&
so
all
other
.
¶
The
seconde
Chapter
treateth
of
the
rule
of
three
compounde
,
which
are
foure
in
number
.
THere
belongeth
to
the
fyrste
&
seconde
partes
of
the
rule
of
thre
compounde
alwaies
fyue
numbers
:
wherof
(
in
the
fyrst
part
of
the
rule
of
three
compounde
the
seconde
number
and
the
fift
,
are
alwaies
of
one
semblaunce
,
and
like
denomination
:
whose
rule
is
thus
,
multiply
the
first
number
by
the
seconde
,
&
that
shal
be
your
diuisor
:
then
multiplye
the
other
three
numbers
the
one
by
the
other
to
be
your
diuidende
.
Example
,
of
this
first
part
:
if
100.
crowns
in
12.
monthes
,
do
gaine
16.
li.
what
will
60.
crownes
gaine
in
8.
monthes
?
Aunswere
,
first
multiplie
100.
crownes
by
12.
monthes
,
&
therof
commeth
1200.
for
your
diuisor
:
then
multiply
15.
li.
by
60.
crownes
,
&
by
8.
monthes
,
&
you
shall
haue
7200.
diuide
7200.
by
1200.
&
therof
commeth
6.
li.
so
many
li.
wyll
6.
crownes
gaine
in
8.
monthes
:
thys
question
may
be
done
by
the
double
rule
of
3.
that
is
to
say
bi
the
rule
of
3.
at
2
times
:
yet
this
rule
of
3
compound
is
more
brief
Crownes
.
|
monthes
.
|
pounds
.
|
crownes
.
|
monthes
.
|
100.
|
.12
.
|
.15
.
|
.60
.
|
.8
.
|
|
1
|
|
|
|
|
7
2
|
00
|
|
|
|
|
1
2
|
00
|
(
6.
li.
|
|
|
2.
In
the
seconde
part
of
the
rule
of
thre
compound
,
the
3.
number
is
like
vnto
the
fift
,
wherof
the
rule
is
thus
:
multiplie
the
3.
number
by
the
4
,
the
product
shal
be
your
diuisor
:
them
multiply
the
first
number
by
the
seconde
,
&
the
product
therof
by
the
fift
,
the
which
number
shal
be
your
diuidend
,
or
number
that
is
to
be
diuided
:
as
by
example
.
When
60.
crownes
in
8.
monthes
do
gaine
6.
li.
in
how
many
monthes
wil
100.
crownes
gain
.
15.
li.
Aunswere
:
Multiply
the
thirde
number
6.
by
the
fourth
number
100
:
&
therof
commeth
600
then
multiplye
the
firste
number
60.
by
the
second
number
8.
&
by
the
fift
number
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
lykewise
be
done
by
the
double
rule
of
3.
Crownes
|
monthes
.
|
pounds
.
|
crownes
.
|
pounds
.
|
60.
|
.8
.
|
|
.6
.
|
.100
.
|
.15
.
|
|
1
|
|
|
|
|
|
72
|
00
|
monthes
.
|
|
|
|
66
|
00
|
(
12
|
|
|
3.
In
the
thirde
part
of
the
rule
of
3.
compound
,
there
may
be
5.
numbers
or
more
:
&
in
this
rule
the
first
number
&
the
last
are
alwayes
dissemblaunt
the
one
to
thother
:
&
the
question
is
from
the
last
number
vnto
the
first
,
wherof
the
rule
is
thus
:
multiply
that
number
which
you
would
know
by
those
numbers
which
do
giue
the
value
,
&
diuide
the
product
of
the
same
,
by
the
multiplycation
of
the
numbers
which
are
already
valued
,
as
by
example
.
If
4.
deniers
Parisis
,
bee
worthe
5.
deniers
Tournois
,
&
10.
deniers
tournois
,
be
worth
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
number
that
you
would
knowe
)
by
4.
deniers
parisis
,
&
by
10
deniers
tournois
which
are
the
number
that
giue
the
value
,
&
they
make
320
:
then
multiplie
5.
deniers
tournois
,
by
12
deniers
of
sauoy
(
which
are
the
numbers
already
valued
)
&
thei
make
60
:
lastly
diuide
320.
by
60
and
you
shal
finde
5.
deniers
⅓
parisis
,
so
muche
are
the
deniers
of
Sauoye
worth
.
Parisis
.
|
tournois
.
|
tournois
.
|
sauoy
.
|
sauoy
.
|
4.
d.
|
.5
d.
|
.10
d.
|
.12
d.
|
.8
d.
|
|
3
2
|
0
|
par.
|
|
|
|
6
|
0
|
(
5.
d.
⅓
.
|
|
|
4.
In
the
fourthe
parte
of
the
rule
of
thre
compounde
:
the
fyrst
number
and
the
last
are
always
semblant
and
of
one
denomination
,
and
the
question
of
this
rule
,
is
alwayes
from
the
last
number
to
the
last
sauing
one
.
Whereof
there
is
a
rule
which
is
thus
.
You
must
multiplye
that
number
which
you
woulde
knowe
,
by
the
numbers
that
are
alreadye
valued
,
and
diuide
the
product
of
the
same
,
by
the
multiplication
which
commeth
of
the
numbers
that
giue
the
value
,
as
by
example
.
If
4.
deniers
Parisis
,
bee
worth
5.
Deniers
Tournois
,
and
10.
Deniers
Tournois
,
be
worthe
12.
Deniers
of
Sauoy
,
I
demaunde
how
many
Deniers
of
Sauoy
,
are
15.
Deniers
Parisis
worth
.
Aunswere
:
Multiplye
15.
Deniers
Parisis
that
you
woulde
knowe
,
by
5.
Deniers
Tournois
,
&
by
12.
Deniers
of
Sauoye
,
which
are
the
numbers
alreadye
valued
,
and
thei
make
900.
Diuide
the
same
by
4.
times
10.
which
are
the
numbers
that
doe
giue
the
value
,
and
you
shal
finde
22.
Deniers
½
of
Sauoye
,
so
much
are
the
15.
Deniers
Parisis
worth
.
Parisis
.
|
tournois
.
|
tournois
.
|
Sauoy
.
|
Parisis
.
|
4.
d.
|
.5
d.
|
.10
d.
|
.12
d.
|
.15
d.
|
|
12
|
|
|
|
|
90
|
0
|
Sauoy
.
|
|
|
44
|
0
|
(
22.
d.
½
|
|
The
thirde
Chapter
treateth
of
questions
of
the
trade
of
Marchaundise
.
IF
31.
Deuonsh
.
dosens
do
cost
me
100.
li.
15.
shil
.
What
shall
4.
dosens
cost
?
Aunswere
:
fyrst
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
quotient
wil
be
260.
s.
The
which
diuide
againe
by
20.
and
therof
commeth
13.
li.
Dosens
.
|
|
|
Dosens
.
|
31.
|
100.
li.
|
15.
sh.
|
4.
d.
|
|
20
|
|
|
|
2015
|
|
|
|
4
|
|
|
|
8060.
|
|
|
|
1
|
|
|
|
28
|
|
|
|
8060
(
260.
|
|
|
3111
|
|
|
|
33
|
|
|
If
foure
Dosens
be
worth
13.
pound
.
What
are
31.
Dosens
worthe
by
the
price
?
Aunswere
:
Multiply
31.
by
13.
and
therof
commeth
403.
The
which
you
shall
diuide
by
4.
and
thereof
commeth
and
thereof
commeth
100.
li.
¾
,
which
¾
are
15
.
s
.
and
so
much
are
31.
Dosens
worth
as
before
.
Dosens
.
|
li.
|
Dosens
.
|
4.
|
.13
.
|
.31
.
|
|
|
13
|
|
|
393
|
|
|
1
|
|
|
403.
|
|
4
03
|
|
|
444
(
100.
li.
¼
|
|
If
49.
elles
be
worth
2.
li.
4.
s.
11.
d.
what
are
18.
elles
worth
by
the
price
?
First
you
must
bring
2.
li.
4.
s.
11.
d.
all
into
pence
,
in
multiplying
2.
li.
by
20.
maketh
40.
adde
thereto
4.
shil
.
they
make
44.
s.
the
which
multiply
by
12.
d
&
thei
make
528.
d.
whereunto
adde
11.
d
all
is
539.
d.
the
which
539.
d.
muste
be
your
second
number
in
the
rule
of
3.
then
multiply
539.
by
18.
&
therof
commeth
9702.
diuide
the
same
by
49.
&
you
shal
haue
in
your
quotient
198.
d.
the
which
diuide
by
12.
&
you
shal
finde
16.
s.
6.
d.
so
much
are
the
18.
elles
worthe
.
Elles
:
|
|
Elles
.
|
49.
|
.2
.
li.
4.
sh.
11.
d.
.18
|
|
20
|
539
|
|
44
|
162
|
|
12
|
54
|
|
99
|
90
|
|
44
|
|
|
539
|
9702
|
13
|
1
|
427
|
76
|
386
|
198
(
16.
sh.
6.
d
|
9702
(
198.
|
122
|
4399
|
1
|
44
|
|
If
18.
elles
be
worth
16
.
s
.
6.d
.
what
are
49.
elles
worth
by
that
price
?
Auns
.
bring
16
.
s
.
6.d
.
into
pence
,
in
multiplying
16.
by
12.
and
thereof
commeth
198.d
.
with
the
6.
d.
added
to
it
,
then
multiplye
198
by
49.
the
product
will
be
9702.
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
much
are
the
49.
elles
worth
.
Elles
.
|
|
Elles
.
|
18.
|
.16
.
sh.
6.
d.
|
.49
|
|
12
|
198
|
|
32
|
392
|
|
166
|
441
|
|
198
|
49
|
|
|
9702
|
17
|
1
|
446
|
151
|
9702
(
539.
|
●39
(
44.
shill.
|
1888
|
122
|
●1
|
1
|
If
a
yarde
of
Veluet
cost
19
.
s
.
what
shall
¾
of
a
yarde
cost
?
Aunswere
:
sette
down
your
numbers
thus
.
If
.
Then
multiplye
1.
times
19.
by
3.
and
therof
commeth
57.
for
your
diuidende
,
or
number
to
be
diuided
.
The
which
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
much
are
the
¾
of
a
yarde
worth
after
19.
shil
.
the
yarde
,
as
by
practise
followeth
.
1/1
|
19/1
|
¾
|
1
1
|
|
|
|
57
(
14.
sh.
¼
|
|
|
|
44
|
Or
otherwise
by
the
rules
of
practise
:
first
for
2/4
of
a
yarde
which
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
numbers
togither
,
&
19.
shil
.
|
9.
sh.
6.d
.
|
4.
.9
.
|
14.
.
3.d
.
|
you
shall
haue
14
.
s
.
3.d
.
as
aboue
is
said
,
and
as
appeareth
here
in
the
margent
.
If
¾
of
a
yarde
of
Veluet
do
cost
14.
shil.
3.d
.
What
shall
1.
yarde
coste
set
your
numbers
downe
thus
:
if
.
Reduce
14.
¼
into
a
fraction
,
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
wil
be
19.
shil
.
so
much
is
the
yarde
of
veluet
worth
.
|
57
|
|
1
|
¾
|
14
¼
|
1/1
|
10
|
|
|
|
228
(
19.
shil
.
|
|
|
|
122
|
|
|
|
1
|
Or
otherwise
by
the
rule
of
practise
:
you
shall
take
the
½
parte
of
14.
sh.
3.d
.
and
adde
it
with
the
same
14.
sh.
3.d
.
and
you
shall
haue
19.
shill.
as
before
.
14.
shil
.
|
3.
d.
|
4.
|
.9
.
d.
|
19.
shil
.
|
.0
.
d.
|
If
one
ell
of
Hollande
clothe
be
worth
5
.
s
.
what
are
⅔
worth
after
the
rate
?
Aunswere
,
say
thus
if
.
Then
multiply
2.
times
5.
one
time
,
and
therof
commeth
10.
for
your
diuidende
:
likewise
multiply
thre
times
1.
one
time
,
thei
make
3.
for
your
diuisor
,
then
diuide
10.
by
3.
&
thereof
commeth
30
.
s
½
which
⅓
is
worth
4.
pence
,
&
so
much
are
the
⅔
of
an
ell
worth
.
1/1
|
5/1
|
⅔
|
1
|
|
|
|
|
10
|
(
3.
shil
.
⅓
|
|
|
|
3
|
|
Or
otherwise
,
by
the
rule
of
practise
:
take
first
the
⅓
of
5
.
s
.
for
the
⅓
of
an
ell
,
which
is
1
.
s
.
8.d
.
Likewise
,
for
the
other
⅓
of
an
ell
take
againe
another
⅓
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
.
5.
shill.
|
|
1.
|
.8
|
1.
|
.8
|
3.
shill.
|
4
d.
|
If
⅔
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
.
First
reduce
3
⅓
all
into
thirdes
,
and
it
will
be
.
10/3.
Then
multiply
1.
times
10.3
.
times
,
and
thereof
commeth
30.
for
your
diuidend
.
Likewise
multiplie
1.
times
3.2
times
,
youre
quotient
wil
be
6.
then
diuide
30.
by
6.
&
you
shall
haue
5.
s.
so
much
is
the
ell
of
Hollande
clothe
worth
.
|
10
|
|
30
|
2
|
3
⅓
|
1/1
|
6
(
5.
sh.
|
Or
otherwise
by
practise
,
take
the
½
of
3
.
s
.
4.d
.
which
is
1
.
s
.
8.d
.
&
adde
it
to
the
same
3
.
s
.
4.d
.
and
therof
wyll
come
5
.
s
.
as
before
.
For
the
⅓
of
5
.
s
.
is
as
much
as
the
½
of
3.
sh.
|
4.
|
1.
|
8.
|
5.
sh.
|
0.d
.
|
3
.
s
.
4.
d.
which
was
the
price
that
the
⅔
of
an
elle
did
cost
,
as
appeareth
.
If
one
ell
cost
me
17
.
s
.
what
shal
.
15.
elles
⅛
part
coste
?
which
⅛
is
halfe
a
quarter
of
an
elle
.
Aunswere
:
saye
,
of
.
First
reduce
15
⅛
into
eight
partes
,
and
they
make
121/8
then
multiplie
121.
by
17.1
time
,
and
thereof
commeth
2057.
for
your
diuidende
.
Likewise
multiply
8.
times
1.
1.
time
,
and
your
quotient
wyll
bee
8.
for
your
diuisor
,
then
diuide
2057.
by
8.
and
you
shall
fynde
257.
sh.
⅛
,
whiche
is
12.
li.
17.
shil.
1.d
.
½
and
so
muche
are
the
15
elles
⅛
worth
,
as
by
practise
doth
appeare
.
Or
otherwise
,
for
10
sh.
take
the
½
of
15
which
is
7
li.
10
sh.
then
for
5
sh.
take
the
½
of
7.
li.
10
.
s
.
which
is
3.
li.
15
.
s
.
thirdly
for
2
.
s
.
take
the
⅕
of
7.
li.
10
.
s
.
bicause
the
⅕
of
10.
sh.
is
2.
sh.
Fourthlye
,
for
the
⅛
of
the
ell
,
you
shal
take
the
⅛
of
17
.
s
.
which
is
2
.
s
.
1.d
.
½
.
Lastlye
,
adde
all
these
summes
togither
,
and
then
shal
you
find
12.
li.
17
.
s
.
1.d
.
½
as
before
,
and
as
appeareth
more
plainly
in
the
margent
.
15.
|
.
⅛
.
|
|
17
.
s
.
|
|
|
7.
|
.10
.
|
.
|
3.
|
.15
.
|
.
|
1.
|
.10
.
|
.
|
|
2.
|
1
½
.
|
12.
li.
17
.
s
.
1.d
.
½
|
If
25.
elles
be
worthe
2.
li.
3
.
s
.
4.d
.
what
are
.
18
elles
¾
worth
by
that
price
?
Aunswere
:
first
put
3
.
s
.
4.d
.
into
that
part
of
a
li.
and
you
shal
haue
⅙
then
say
,
if
25/1
giue
me
2.
li.
⅙
what
shall
18
¾
giue
:
put
the
whole
number
into
his
broken
,
and
then
multiplie
1.
times
13.
by
75.
the
product
will
be
975.
the
whiche
you
shall
diuide
by
25.
times
6.4
.
times
,
which
maketh
600.
Then
diuide
975.
by
600.
and
your
quotient
will
be
1.
li.
and
375.
remaineth
,
the
which
375.
you
shall
multiplie
by
20.
thereof
commeth
7500.
diuide
the
same
by
600.
your
quotient
wil
be
12
.
s
.
and
300.
remaineth
the
which
abbreuiated
bringeth
½
which
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
bicause
that
12.
elles
½
is
the
½
of
25.
elles
,
therfore
take
the
½
of
2.
li.
3
.
s
.
4.d
.
which
is
1.
li.
1
.
s
.
8.d
.
then
for
6.
elles
¼
take
¼
of
2.
li.
3
.
s
.
4.d
.
or
else
the
½
of
the
last
product
(
that
is
to
say
of
1.
li.
1
.
s
.
8.d
.
)
which
is
all
one
,
&
adde
them
togither
,
so
shal
you
haue
1.
li.
12
.
s
.
6.d
.
as
before
.
.2
.
|
.3
.
|
.4
.
|
.1
.
|
.1
.
|
.8
.
|
|
.10
.
|
.10
.
|
1.
li.
|
12
.
s
.
|
6.d
.
|
If
15.
yardes
be
worth
32
.
s
.
what
are
halfe
a
yarde
or
halfe
a
quarter
or
else
⅝
of
a
yarde
worth
.
Aunswere
:
say
,
if
15/1
giue
32/1
what
wyll
⅝
giue
?
Multiplie
1
times
32.
by
5.
and
diuide
the
product
by
15.
times
1.
shil
.
and
4.
remaineth
,
which
is
⅓
of
a
shil
.
that
is
to
say
4.d
.
and
so
much
are
the
5/2
of
a
yarde
worth
.
Or
otherwise
,
se
what
the
yarde
is
worth
after
the
maner
aforesaid
in
the
other
examples
,
&
you
shal
find
that
the
yard
is
worth
2
.
s
.
1.d
⅗
of
the
which
number
take
first
the
½
for
4/8
which
is
1
.
s
.
0
d.
⅘
,
of
the
which
number
,
take
the
¼
for
the
other
⅛
which
is
3
d.
⅕
,
adde
these
two
numbers
togither
,
and
you
shal
finde
the
⅝
to
be
worth
1
.
s
.
4.d
.
as
before
is
said
2.
sh.
|
1.d
.
|
⅗
.
|
1.
|
0.
|
⅘
.
|
1.
sh.
|
4.d
.
|
0.
|
If
13.
els
⅚
,
be
worth
27
.
s
.
what
are
10.
elles
⅔
worth
by
that
price
?
Aunswere
:
say
if
13.
⅚
giue
27/1
,
what
shal
10.
⅔
giue
:
put
the
whole
numbers
into
their
broken
,
&
you
shall
finde
83/6
,
27/1
,
&
32/3.
Then
multiplie
6.
times
27.
by
32.
&
thereof
commeth
5184.
the
which
number
you
shall
diuide
by
83.
times
1.
thre
times
,
and
you
shall
finde
20.
sh.
68/83
which
is
worth
9.d
.
69/83
part
of
a
penny
.
If
two
yardes
½
be
worth
4
.
s
.
8.d
.
what
are
8.
yards
¼
worth
?
Aunswere
:
put
the
8.d
.
into
the
part
of
a
shilling
,
which
wil
be
⅔
then
reduce
the
whole
numbers
into
their
broken
,
and
they
will
stande
thus
.
5/2
,
14/3
,
33/4
,
then
multiplie
two
times
14.
by
33.
and
diuide
the
product
by
5.
times
3.4
.
times
,
&
you
shall
finde
15
.
s
.
4.d
.
⅘
,
so
much
are
the
eyght
yardes
¼
worth
.
If
one
kersey
bee
worth
2.
li.
6
.
s
.
8.d
.
how
many
kerseys
shall
I
bie
for
36.
li.
3
.
s
.
4.d
.
after
that
rate
?
Aunswere
:
put
6s.
8d
.
into
the
part
of
a
li.
&
you
shall
haue
2.
li.
⅓
for
the
first
number
in
the
rule
of
3.
and
1.
ell
for
the
second
number
:
then
put
3
.
s
.
4.d
.
into
the
part
of
a
li.
and
you
shall
finde
36.
li.
⅙
.
for
the
third
number
,
then
will
your
3.
numbers
in
the
rule
of
3.
stande
thus
.
.
Therefore
reduce
the
whole
numbers
into
their
broken
,
&
you
shal
haue
.
Then
multiply
3
times
1.
by
.217
.
&
therof
will
come
651.
for
your
diuidende
.
Likewise
,
multiply
7
times
1.
by
6.
&
the
product
therof
will
be
42.
Then
diuide
651.
by
42.
and
you
shall
finde
15.
½
.
So
many
kerseys
of
2.
li.
6
.
s
.
8.d
.
the
peece
,
shall
you
haue
for
36.
li.
3
.
s
.
4.
d.
¶
The
4.
Chapter
treateth
of
losses
and
gaines
,
in
the
trade
of
Marchaundise
.
If
13.
yards
⅓
be
worth
22.
li.
10
.
s
.
how
shall
I
sel
the
yarde
to
gaine
⅓
,
or
to
make
of
3.4
?
which
is
all
one
?
Auns
.
say
by
the
rule
of
3.
if
3.
be
come
of
4.
or
if
3.
yelde
4.
what
will
22.
½
yelde
:
multiplie
&
diuide
and
you
shal
fynde
30.
li.
Then
say
gaine
by
the
rule
of
3.
if
13.
yardes
⅓
doe
giue
30.
li.
aswell
of
principal
as
of
gaine
:
what
wil
1.
yard
be
worth
by
the
price
?
Multiply
and
diuide
,
and
you
shall
find
2.
li.
5
.
s
.
and
for
that
price
must
the
yarde
be
solde
to
gaine
the
⅓
,
or
to
make
of
3.4
.
|
|
45
|
40.
|
|
5/1
|
4/1
|
22
½
|
13
⅓
30/1
|
1/1
|
Or
otherwise
,
take
the
⅓
part
of
22.
li.
10
.
s
.
which
is
7.
li.
10
.
s
.
that
shall
you
adde
with
22.
li.
10
.
s
.
and
you
shal
haue
30.
li.
as
before
.
22.
|
.10
.
s.
|
7.
|
.10
.
|
30.
|
.00
.
|
Then
diuide
30.
by
13.
⅓
,
&
you
shall
fynde
2.
li.
5
.
s
.
as
aboue
is
said
.
If
one
yarde
bee
worth
.27
.
sh.
6.d
.
for
how
much
shal
16.
yards
⅔
be
solde
to
gaine
2
.
s
.
vpon
the
pound
of
money
,
that
is
to
say
:
vpon
20
.
s
.
Answere
,
adde
2.
vnto
20.
and
you
shal
haue
22
,
then
say
:
if
20
.
s
.
of
principall
,
doe
giue
22
.
s
.
as
wel
of
principal
as
gaine
:
how
much
wil
27
.
s
.
6.d
.
principall
yelde
.
Multiplie
and
diuide
&
you
shall
finde
30
.
s
.
¼
:
then
saye
againe
by
the
rule
of
3.
if
one
yarde
do
giue
me
30
.
s.¼
(
which
is
aswel
the
principal
as
the
gaine
)
what
shall
.16
.
yardes
⅔
giue
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
.
which
is
all
one
.
|
|
55.
|
|
121
|
50
|
20/1
|
22/1
|
27
½
.
|
1/1
|
30
2/4
|
16
⅔
|
If
10.
yardes
⅔
be
worthe
25.
li.
10
.
s
.
For
how
much
shal
2
yardes
¼
be
sold
to
gaine
after
10.
li.
vpon
the
100.
li.
of
money
?
Aunswere
:
say
if
100.
of
principall
yelde
110.
as
well
principall
as
gaine
,
how
muche
will
25.
10
.
s
.
yelde
me
?
Multiply
&
diuide
,
and
you
shall
finde
28.
li.
1
.
s
.
Then
saye
if
10.
yardes
⅔
do
yelde
me
28.
li.
1.
sh.
aswel
of
principal
as
of
gaine
,
how
much
shal
two
yardes
¼
yelde
me
?
multiply
&
diuide
&
you
shall
finde
5.
li.
18
.
s
.
4.d
.
1/12
,
for
so
much
shall
the
2.
yardes
¼
be
solde
to
gaine
after
10.
li.
vpon
the
100.
li.
of
mony
.
|
|
51
|
|
|
|
100/1
|
110/1
|
25
½
|
10
⅔
|
28
1/20
|
2
¼
|
And
although
that
in
these
questions
of
gaine
and
losse
,
sometimes
the
fyrst
number
is
not
like
vnto
the
thirde
number
,
that
is
to
saye
,
of
the
same
denomination
:
as
one
woulde
saye
:
if
20
.
s
.
gaine
2.
shil
.
what
shall
50.
li.
gaine
?
or
25.
li.
&c.
Or
if
20.
li.
do
gain
2.
li.
What
shall
25
.
s
.
gaine
mee
,
or
what
shall
27.
sh.
½
gaine
?
Yet
neuerthelesse
,
the
rule
is
not
therfore
false
.
For
if
20
.
s
.
doe
gaine
2
.
s
:
20.
li.
shall
gaine
2.
li.
&
20.d
.
shall
gaine
2.d
.
likewise
20.
crownes
shall
gaine
2.
crownes
,
and
so
of
all
other
:
therefore
it
is
to
be
vnderstand
,
that
the
first
number
in
these
reasons
is
presupposed
to
bee
semblable
to
the
thirde
.
When
one
Marchant
selleth
wares
to
another
,
and
hee
giueth
to
the
byer
2.
vpon
15
:
howe
much
shall
the
byer
gaine
vpon
the
100.
after
the
rate
?
Aunswere
:
say
if
15.
giue
17.
what
shall
100.
giue
?
Multiplie
and
diuide
,
and
you
shall
finde
113
⅓
,
so
the
byer
getteth
after
the
rate
of
13
⅓
vpon
the
100.
If
one
northen
dosen
cost
me
3.
li.
5.
s.
&
I
sel
the
same
againe
for
3.
li.
12
.
s
.
6.d
.
how
much
doe
I
gaine
vpon
the
pounde
of
money
after
that
rate
?
Auns
.
say
if
3.
li.
¼
doe
giue
3.
li.
⅝
what
shal
20/8
giue
,
put
the
whole
number
into
their
broken
&
you
shal
haue
13/4
,
29/8
,
20/1
,
then
multiply
4.
times
29.
by
20.
&
thereof
commeth
2320.
for
your
number
that
is
to
be
diuided
,
likewise
multiply
13.
times
8.
1
time
,
&
therof
commeth
104.
Then
diuide
2320.
by
104.
&
you
shall
finde
22
.
s
.
4/13.
So
I
shal
get
2
.
s
.
4/13
vpon
20
.
s
.
or
vpon
the
li.
of
money
.
If
a
yarde
of
cloth
cost
me
7
.
s
.
8.d
.
&
afterwarde
I
deliuer
out
13.
yardes
¼
,
for
4.
li.
13
.
s
.
4.d
.
I
woulde
knowe
whether
I
doe
winne
or
lose
,
&
how
much
vpon
the
100.
li.
of
money
?
Aunswere
:
see
first
at
7
.
s
.
8.d
.
the
yard
,
what
the
13.
yardes
¼
shall
coste
,
and
you
shal
finde
5.
li.
1
.
s
.
7.d
.
And
I
sold
them
but
for
4.
li.
13
.
s
.
4.d
.
so
that
I
doe
lose
vpon
the
13.
yardes
¼
the
summe
of
8
.
s
.
3.d
.
Then
for
to
knowe
howe
much
is
lost
vpon
the
100
:
saye
by
the
rule
of
three
,
if
5.
li.
1
.
s
.
7.d
.
doe
lose
8
.
s
.
3.d
.
What
will
100.
lose
?
Fyrst
,
put
1.
shil.
7.d
.
into
the
part
of
a
li.
and
it
will
be
19/240.
Likewise
put
8
.
s
.
3.d
.
into
the
part
of
a
li.
and
it
is
33/80.
Then
will
your
numbers
stand
thus
5
19/240
,
33/80
,
100/1
,
put
the
whole
into
his
broken
,
and
then
multiply
and
diuide
,
so
you
shal
finde
8.
li.
1184/9752
which
is
worth
.2
.
sh.
5.
d.
169/1219
and
so
muche
is
loste
vpon
the
100.
li.
of
money
.
|
1219.
|
|
|
5
|
19/240.
|
38/80
|
100/1.
|
More
,
if
12.
yardes
½
of
scarlet
bee
solde
for
30.
li.
15
.
s
.
vpon
the
which
is
gained
after
the
rate
of
11
1/9
vpon
the
100.
I
demaunde
what
the
yarde
dyd
cost
at
the
first
.
Aunswere
:
from
30.
li.
15
.
s
.
substract
his
1/10
part
which
is
3.
li.
1
.
s.6.d
.
and
there
resteth
27.
li.
13
.
s.6.d
the
which
number
multiplied
by
2.
bringeth
55.
li.
7
.
s
.
of
the
which
is
11.
li
one
shilling
and
foure
pence
.
Then
take
againe
the
⅕
of
the
saide
11
pounde
1.
shil.
4.
pence
,
which
is
2.
pounde
4.
shillinges
thre
pence
.
9/25.
And
so
muche
did
the
Elle
cost
at
the
fyrst
pennye
.
30.
li.
|
15.
sh.
|
|
|
3.
|
1.
sh.
|
6.d
.
|
|
27.
|
13.
|
6.
|
|
2.
|
|
|
|
55.
|
7.
|
0.
|
|
11.
|
1.
|
4.
|
⅘
.
|
2.
|
4.
|
3.
|
9/25.
|
More
,
if
15.
yardes
¾
of
scarlet
doe
cost
me
32.
li.
13
.
s
.
4.d
.
And
I
sell
the
yarde
againe
for
2.
li.
whether
doe
I
winne
or
lose
,
and
howe
much
vpon
the
pounde
of
money
.
Aunswere
:
Loke
what
the
15.
yardes
¾
are
worth
at
2.
li.
the
yarde
,
and
you
shall
finde
that
they
are
worth
31.
li.
10.s
.
But
they
did
cost
32.
li.
13
.
s
.
4.d
.
so
that
there
is
lost
vpon
the
whole
1.
li.
3.s.4.d
.
Then
,
to
know
how
much
is
lost
vpon
the
li.
say
by
the
rule
of
thre
,
if
32.
li.
⅔
doe
lose
1.
li.
⅙
:
what
will
⅛
lose
?
that
is
to
say
,
what
will
1.
li.
lose
?
reduce
the
whole
numbers
into
their
broken
,
&
then
multiplie
&
diuide
,
so
shall
you
finde
21/588.
part
of
a
li.
Then
multiplye
21.
by
240.
bicause
so
many
pence
are
in
a
li.
&
diuide
the
product
by
588.
so
shall
you
finde
8.d
.
336/588
which
being
abbreuiated
doe
make
7/4
,
&
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
.
whereupon
is
lost
after
the
rate
of
10
.
s
.
vpon
the
100.
I
demaunde
what
12.
yardes
½
of
the
same
tissue
did
cost
?
Aunswere
:
adde
vnto
3.
li.
15.
hys
owne
1/10
part
,
which
is
7
.
s.6.d
.
and
al
amounteth
to
4.
li.
2
.
s
.
6.d
.
then
loke
what
the
12.
yardes
½
wil
amount
vnto
,
after
4.
li.
2
.
s
.
6.d
.
&
you
shal
finde
that
they
will
come
to
51.
li.
11
.
s
.
3.d
.
so
much
did
the
12.
yardes
½
cost
.
3.
li.
15
.
s
.
|
12.
|
|
½
|
7
.
s
.
6.d
.
|
4.
li.
2
.
s
.
6.d
.
|
4.
li.
2s.
6d
.
|
48.
|
.00
.
|
.0
.
|
|
1.
|
.10
.
|
.0
.
|
|
2.
|
.01
.
|
.3
.
|
|
51.
li.
|
11
.
s
.
|
3.d
.
|
More
,
if
I
sell
one
wilshire
white
for
6.
li.
12
.
s
.
whereupon
I
doe
gaine
after
the
rate
of
2
.
s
.
vpon
the
li.
of
money
,
that
is
to
saye
,
vpon
20
.
s
.
I
demaunde
what
11.
peeces
of
the
same
whites
did
cost
mee
?
Aunswere
:
abate
from
6.
li.
12
.
s
.
(
which
is
132
.
s
.
)
hys
1/11
part
,
&
thereof
commeth
12
.
s
.
and
there
remaineth
120
.
s
.
or
6.
li.
Then
see
at
6.
li
the
cloth
,
what
the
11.
clothes
are
worth
66.
li.
so
much
did
the
11.
clothes
coste
.
132.
sh.
|
11
|
12.
sh.
|
6
|
120.
sh.
|
66.
li.
|
If
I
sell
10.
elles
½
of
Holland
for
22
s.6.d.
whereupon
I
do
lose
after
the
rate
of
2
.
s
.
vpon
the
li.
of
money
.
I
demaunde
what
the
ell
did
cost
mee
?
Aunswere
:
say
by
the
rule
of
3.
if
18.
giue
20
.
s
.
what
will
22
.
s.6.d
.
giue
?
Multiplie
&
diuide
,
&
you
shall
fynde
25
.
s
.
Then
diuide
25
.
s
.
by
10.
½
,
&
therof
commeth
2s.
4
d.
4/7
:
So
much
did
the
el
cost
.
If
I
sell
one
clothe
for
5
li.
wher
vpon
I
do
lose
after
10
vpon
the
100
,
I
demaunde
how
muche
I
should
lose
or
gayne
vpon
the
100
,
in
case
I
had
solde
the
same
for
5
li.
10
shil
.
Aunswere
:
saye
,
if
90
yelde
100
,
howe
muche
wyl
5.
li.
giue
?
Multiplie
&
diuide
,
&
you
shall
finde
5.
li.
5/9
:
then
say
againe
by
the
rule
of
thre
,
if
5.5/9
come
to
5.
½
,
what
wyll
100.
come
vnto
?
Multiplie
&
diuide
,
&
you
shal
finde
99.
li.
which
being
abated
from
100.
there
wyll
remaine
1.
li.
and
so
much
is
lost
vpon
the
100.
90.
|
100.
|
5.
|
5
5/9
|
5
½
|
100/1
|
¶
The
5.
Chap.
treateth
of
lengthes
&
breadthes
of
tapistry
,
and
other
clothes
.
IF
a
peece
of
tapistry
be
5
elles
¾
longe
,
and
4
elles
⅔
in
breadth
,
how
mani
elles
square
doth
the
same
pece
conteine
?
Aunswere
:
Multiplie
the
length
by
the
breadth
,
that
is
to
saye
5.
¾
by
4.
⅔
,
and
thereof
commeth
26.
elles
⅚
so
many
elles
square
doth
the
same
peece
conteine
.
More
,
if
a
peece
of
Tapistrie
doe
conteine
32.
ells
square
,
and
the
same
being
in
length
6.
elles
¼
.
I
demaunde
how
many
elles
in
breadth
the
same
peece
doth
conteine
.
Ans.
diuide
32.
elles
by
6
¼
and
thereof
commeth
5.
3
/
25
:
So
many
elles
dothe
the
same
peece
conteine
in
breadth
.
More
,
a
peece
of
clothe
beyng
13.
yardes
⅓
in
length
,
and
5
quarters
1/2
in
breadth
,
how
many
yardes
of
⅔
&
½
broade
will
the
same
peece
make
?
Answere
:
see
what
parte
of
a
yarde
,
the
5/4
and
½
be
,
and
you
shall
finde
that
they
make
1
yarde
⅜
.
Then
multiply
13.
yardes
⅓
by
1
yarde
⅜
and
you
shall
haue
18.
yardes
⅓
in
square
the
which
you
must
diuide
by
⅔
&
½
that
is
to
saye
by
⅚
,
(
bicause
that
⅔
,
½
being
brought
into
1
fraction
maketh
⅚
)
&
you
shall
finde
22.
yards
:
So
many
yardes
of
⅔
&
½
large
doth
the
same
peece
conteine
.
More
,
a
marchant
hath
bought
4.
yardes
⅔
of
cloth
being
syxe
quarters
½
broade
to
make
him
a
gowne
the
which
he
will
line
thorowout
,
wyth
black
Say
of
three
quarters
of
a
yarde
broad
,
I
demaund
how
much
Say
he
must
bye
?
Answere
:
Multiply
the
length
of
the
cloth
,
by
the
breadth
,
that
is
to
say
4
⅔
by
1.
⅝
,
(
which
is
the
syx
quarters
½
)
and
therof
commeth
7.
yardes
7/12
,
the
which
diuide
by
¾
&
you
shall
finde
ten
yardes
1/9.
So
many
yardes
of
Say
must
he
haue
to
line
the
same
4.
yards
⅔
of
cloth
of
6.
quart.
½
broad
.
More
,
at
6
.
s
.
8.d
.
the
ell
square
,
what
shall
a
peece
of
tapistre
cost
me
,
which
is
fiue
ells
½
long
and
4.
ells
¼
broade
?
Aunswere
,
multiply
5.
½
by
4.
¼
and
therof
commeth
23.
ells
⅜
square
:
then
say
by
the
rule
of
three
,
if
one
ell
square
cost
me
6
.
s
.
8.d
.
what
shal
23.
⅜
cost
?
Multiplie
and
diuide
,
and
you
shall
finde
7.
li.
15
.
s
.
10.d
.
so
much
the
saide
peece
of
tapistrie
did
cost
.
Or
otherwise
,
by
the
rules
of
practise
,
take
the
⅓
of
23.
3/8
:
and
you
shall
finde
7.
li.
15
.
s
.
10.d
.
as
aboue
is
saide
.
More
,
a
peece
of
Hollande
clothe
conteining
42.
elles
⅔
flemishe
,
how
many
elles
englishe
doe
they
make
?
Here
must
you
fyrst
note
that
100.
els
flemishe
,
doe
make
but
60.
elles
englishe
,
and
so
consequentlye
fiue
elles
flemishe
doe
make
but
3.
els
english
.
Therefore
say
by
the
rule
of
3.
if
5.
els
flemishe
doe
make
three
ells
english
,
how
many
elles
englishe
will
42.
els
⅔
flemishe
make
.
Multiplye
&
diuide
,
so
shall
you
finde
25.
elles
⅗
englishe
,
and
so
many
elles
englishe
doth
42.
2/3
flemishe
conteine
,
the
like
is
to
bee
done
of
all
others
.
More
,
I
haue
bought
a
peece
of
Tapistrie
,
being
5.
elles
¾
longe
,
and
4.
elles
⅔
broade
measure
of
Flaunders
,
I
demaunde
howe
many
elles
square
it
maketh
Englishe
measure
?
Aunswere
.
First
,
forasmuch
as
three
ells
english
are
worth
5
elles
flemishe
,
therefore
put
3
elles
english
into
hys
square
,
in
multiplying
3.
by
him self
which
maketh
9
:
likewise
multiplye
5.
in
hym selfe
squarely
,
and
it
wil
be
25.
Then
multiplye
5
¾
which
is
the
length
of
the
peece
,
by
4
2/3
which
is
the
breadth
,
&
therof
commeth
26
elles
⅚
square
:
them
say
by
the
rule
of
three
,
if
25
elles
square
of
flemishe
measure
,
be
worth
9
elles
square
of
englishe
measure
,
what
are
26
elles
flemish
⅚
worth
?
multiplie
&
diuide
,
and
you
shall
finde
that
they
are
worth
nine
elles
33/80
square
of
english
measure
.
More
at
3
s.
6
d.
the
ell
flemish
what
is
the
englishe
ell
worth
after
that
rate
.
Answere
,
:
saye
if
5.
elles
flemishe
bee
worth
three
ells
english
,
what
is
1
ell
flemishe
worth
?
multiply
and
diuide
,
&
you
shall
fynde
⅗
of
an
englishe
ell
.
Then
saye
by
the
rule
of
3
,
if
⅗
of
an
englishe
ell
,
be
worth
3
s.
6
d.
what
is
1.
englishe
ell
worth
?
multiplie
and
diuide
,
and
you
shall
finde
5
s.
10d
.
so
much
shall
the
englishe
ell
be
worth
.
More
at
6
s.
8d
.
the
flemishe
ell
square
,
what
is
the
englishe
ell
worth
.
Answere
,
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
ell
:
Then
saye
,
if
9/25
of
an
englishe
ell
be
worth
6
s.
8
d.
what
is
one
square
ell
englishe
worth
?
multiplie
and
diuide
,
and
you
shall
fynde
18
s.
6d
.
2/9
,
so
much
shal
one
englishe
ell
square
be
worth
.
¶
The
sixt
Chapter
treateth
of
the
reducing
of
the
paumes
of
Genes
into
english
yardes
,
wherof
foure
Paumes
maketh
one
englishe
yarde
.
I
Haue
bought
97.
paumes
½
of
Genes
veluet
,
&
I
would
know
howe
many
yardes
they
wyll
make
?
Aunswere
,
Diuide
97.
½
by
4.
and
you
shall
haue
24.
yardes
⅜
.
So
many
yards
doe
the
97.
paumes
½
conteine
.
Or
otherwise
,
take
some
other
number
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
3/2
as
before
.
More
,
at
two
shillings
7.d
.
the
paume
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
1/1
yarde
worth
?
Multiplie
&
diuide
,
and
you
shall
finde
ten
shillings
4.d
.
So
much
is
the
englishe
yarde
worthe
.
Or
otherwise
,
multiply
4.
paumes
(
which
is
one
yarde
)
by
two
shillings
7.
pence
,
and
you
shall
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
?
Aunswere
,
saye
:
by
the
rule
of
3.
if
257.
½
paumes
be
worth
20.
⅚
,
what
are
4/1
paumes
worth
.
Multiply
and
diuide
,
and
you
shall
fynde
100/309
part
of
a
pounde
,
which
is
worth
6
.
s
.
5.
pence
,
5●/103
:
so
much
is
one
yard
worthe
.
¶
The.
vij
.
Chapter
treateth
of
marchaundise
solde
by
waight
.
AT
9.d
.
½
the
ounce
,
what
is
the
li.
waight
worth
?
Answere
,
say
if
3/2
giue
9.
½
what
will
16/1
giue
multiply
and
diuide
,
&
you
shal
finde
12
.
s
.
8.d
.
so
much
is
the
yarde
worth
?
Or
otherwise
,
by
the
rules
of
practise
for
syxe
pence
,
take
the
½
of
16.
which
is
8
.
s
.
then
for
3.d
.
take
the
¼
of
16
.
s
.
which
is
4
.
s
.
Finally
,
for
the
halpenye
,
take
16.
ob
.
which
are
8.d
.
adde
all
these
numbers
togither
and
you
shall
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
multiplying
.
112.
li.
by
16.
ounces
&
you
shall
haue
1792.
ounces
,
them
say
by
the
rule
of
3.
if
:
Multiplie
and
diuide
,
and
you
shall
finde
18816
d.
which
do
make
78.
li.
8
s.
and
so
much
are
the
112.
li.
worth
after
10.d
¼
the
ounce
.
At
12
.
s
.
8d
.
the
li.
waight
,
what
is
the
ounce
worth
?
Answere
:
put
12
.
s
.
8d
.
into
pence
,
and
you
shall
haue
152.
pence
:
then
say
by
the
rule
of
3.
if
16.
ounces
cost
152d
.
what
shall
1.
ounce
coste
,
multiplie
and
diuide
,
and
you
shall
finde
9.d
.
½
,
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
.
2d
.
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.
10
.
s
.
all
into
shillings
and
you
shall
haue
650
.
s
.
Then
say
,
by
the
rule
of
three
,
if
multiply
and
diuide
,
and
you
shal
finde
,
6
.
s
.
6.d
.
so
much
is
the
li.
worthe
.
If
one
pound
waight
of
saffron
do
cost
me
18
.
s
.
8.d
.
what
shal
355.
li.
10.
ounces
cost
me
by
yᵉ
same
price
?
Aunswere
saye
by
the
rule
of
3.
if
.
Multiply
and
diuide
,
&
you
shal
finde
331.
li.
18
.
s
.
4.d
.
so
much
are
the
355.
li.
ten
ounces
worth
.
Briefe
rules
of
waight
.
WHo
that
multiplieth
the
pence
that
1.
li.
waight
is
worth
by
5.
and
diuideth
the
product
therof
by
.
12.
hee
shal
finde
how
many
pounds
in
money
the
quintall
is
worth
,
that
is
to
say
,
how
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
deuideth
the
product
by
5.
shall
fynde
how
many
pence
the
pounde
waight
is
worth
.
¶
Example
.
AT
seuentene
pence
the
pounde
waight
,
what
is
the
100.
pounde
waight
worth
?
Aunswere
,
Multiplye
17.
by
5.
and
thereof
commeth
85.
diuide
the
same
by
12.
and
you
shall
finde
7.
pound
1/12
,
which
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
worthe
?
Aunswere
,
Multiplie
13.
by
12.
amounteth
to
156.
the
which
diuide
by
5.
and
you
shal
finde
31.d
.
⅕
which
is
2
.
s
.
7.d
.
⅕
and
so
much
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
worth
,
by
6.
And
diuideth
the
product
by
12.
hee
shall
fynde
how
many
poundes
in
money
the
120.
elles
are
worth
,
which
120.
elles
wee
count
but
for
a
C.
And
contrariwise
,
hee
that
multiplieth
the
poundes
in
money
that
the
120.
elles
are
worth
by
12.
and
diuideth
the
multiplication
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
ell
worthe
?
Aunswere
,
Multiplie
nine
pound
by
twelue
,
and
therof
commeth
108.
the
which
diuide
by
6.
and
you
shall
finde
18.d
.
so
much
is
one
ell
worth
.
The
like
is
to
be
done
of
all
maner
of
wares
,
which
are
sold
after
120.
for
the
hundred
.
¶
Briefe
Rules
for
our
hundreth
waight
here
at
London
,
which
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
how
many
pounds
in
money
the
112.
li.
waight
is
worth
.
ANd
contrariwise
,
hee
that
multiplieth
the
poundes
in
money
that
112.
li.
is
worth
by
60.
and
diuideth
the
product
by
28.
shall
finde
how
many
pence
one
li.
waight
is
worth
.
¶
Example
.
AT
nine
pence
the
pound
waight
,
what
is
the
112.
li.
waight
worth
?
Aunswere
:
multiplie
9.d
.
by
28
,
and
thereof
commeth
252
,
the
which
diuide
by
60.
&
you
shall
finde
4.
li.
12/60
which
being
abbreuiated
is
1/5
of
a
pounde
,
which
is
worthe
4
.
s
.
And
thus
the
112.
li.
is
worth
4.
pound
4.
shil
.
At
8.
li.
the
112.
li.
waight
,
what
is
1.
li.
waight
worth
?
Answere
,
Multiplie
8.
li.
by
60.
and
thereof
commeth
480
,
the
which
diuide
by
28.
&
you
shall
finde
17.d
.
1/7
:
so
much
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
vpon
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
12.
li.
what
are
987.
li.
waight
worth
?
multiply
&
diuide
,
&
you
shal
finde
113.
li.
23/26
which
is
worth
17
.
s
.
8.d
.
4/13.
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
.
Answere
,
see
first
by
the
rule
of
three
,
what
the
100.
pound
is
worth
saying
,
if
Multiplie
and
diuide
,
&
you
shal
finde
33.
li.
⅓
then
adde
4.
li.
vnto
100.
&
they
are
104.
then
say
againe
by
the
rule
of
3.
if
a
104.
li.
be
solde
for
33.
li.
⅓
for
how
much
shall
345.
li.
½
be
solde
?
multiply
&
diuide
,
and
you
shal
finde
110.
li.
14
.
s
.
8.d
.
12/13
So
much
shal
the
345.
li.
½
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
&
Cloffe
?
Answere
,
Multiply
780.
by
4.
and
therof
commeth
3120.
The
which
diuide
by
100.
and
you
shall
haue
31.
li.
⅕
abate
31.
⅓
from
780.
and
there
wyll
remaine
748
⅘
.
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
.
18/25
,
and
so
much
shall
the
780.
li.
cost
,
in
rebating
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
tare
and
cloffe
?
Answere
.
Fyrst
,
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
say
by
the
rule
of
3.
if
105.
be
giuen
for
100.
for
how
much
shall
the
100.
be
giuen
?
Multiply
and
diuide
&
you
shal
finde
95.
5/21
:
and
he
which
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
rebateth
5.
vpon
the
100.
loseth
more
by
5/21
vppon
the
100.
than
the
other
which
gaue
5.
vpon
the
100.
for
tare
and
cloffe
.
If
100.
of
Allam
doo
cost
mee
.
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
of
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
1.
li.
multiplye
&
diuide
,
and
you
shall
haue
3.d
.
26/50
which
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
how
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.
⅙
what
will
100/1
yelde
?
Put
the
whole
numbers
into
theyr
broken
,
then
multiplie
and
diuide
,
&
you
shal
finde
104.
36/41
from
the
which
substract
100.
and
there
resteth
4.
li.
36/41
so
much
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
giue
?
Put
the
whole
number
into
their
broken
.
Then
multiplie
and
diuide
&
you
shall
finde
89.
1/16
the
which
you
must
substract
from
100.
and
there
wyll
remaine
10.
li.
15/16
,
so
much
is
loste
vpon
the
100.
li.
of
money
.
More
if
the
li.
waight
doe
cost
mee
3
.
s
.
2.d
.
&
I
sell
it
againe
for
4
.
s
.
4.d
.
how
muche
shall
I
gaine
vpon
20
.
s
.
Answere
:
say
if
3.
⅙
giue
4.
⅓
what
shal
20/1
giue
,
Multiplie
and
diuide
&
you
shall
fynde
27
.
s
.
7/19
:
out
of
the
which
abate
20
.
s
.
and
there
will
remaine
7.
shillings
.
7/19
,
which
is
4.d
.
4/19
:
and
so
muche
is
gained
vpon
the
pounde
of
money
that
is
to
say
vpon
20
.
s
.
If
the
pounde
waight
doe
coste
me
4
.
s
.
4.d
.
and
I
sell
it
againe
for
3
.
s
.
2.d
.
I
demaunde
howe
muche
I
shall
lose
vpon
the
pounde
of
money
?
that
is
to
saye
vpon
twenty
shillings
.
Answere
:
say
,
if
4.
⅓
giue
but
3.
⅙
what
wil
20/1
giue
,
multiply
&
diuide
&
you
shal
finde
14
.
s
.
8/13
the
which
you
must
abate
from
20
.
s
.
&
there
wil
remaine
5
.
s
.
5/13
which
5/13
,
is
worth
4.d
.
8/13
of
a
pennye
and
so
much
is
loste
vpon
the
pounde
of
money
.
¶
The.
ix
.
Chapter
treateth
of
certeine
questions
,
done
by
the
double
rule
,
and
also
by
the
rule
of
three
compounde
.
WHen
the
quarter
of
wheate
,
doth
cost
6
.
s
.
8.d
.
the
loafe
of
breade
waying
20.
ounces
is
solde
for
a
ob.
I
demaund
that
if
the
quarter
of
wheat
did
cost
ten
shillings
,
for
how
much
shall
the
loafe
of
breade
be
solde
,
that
wayeth
16.
ounces
?
Aunswere
:
by
the
fyrst
part
of
the
rule
,
of
3.
compound
which
is
mentioned
in
the
thirde
part
of
thys
booke
,
and
in
the
seconde
Chapter
of
the
same
.
Therfore
say
by
the
same
first
part
of
the
rule
of
3.
compound
,
if
Then
multiplye
the
fyrst
number
by
the
seconde
,
and
the
product
therof
shal
be
your
diuisor
.
Likewise
multiplie
the
other
three
numbers
the
one
by
the
other
,
and
the
product
thereof
shal
be
your
diuidende
:
as
thus
,
first
multiply
6.
⅔
by
20/1
,
and
thereof
commeth
400/3
for
your
diuisor
,
then
multiply
½
by
10/1
and
the
product
therof
by
16/1
,
so
you
shall
haue
160/2
for
your
number
that
is
to
be
diuided
,
then
diuide
160/2
by
400/3
,
and
thereof
commeth
480/800
the
which
being
abbreuiated
bringeth
⅗
of
a
peny
:
and
for
that
price
must
the
loafe
of
bread
be
solde
,
which
wayeth
16.
ounces
,
and
the
quarter
of
wheate
being
worth
ten
shillings
.
Or
otherwise
by
the
rule
of
3.
at
two
times
.
First
saye
if
20/1
ounces
giue
,
½
what
wyll
16/1
ounces
gyue
?
Multiplie
and
diuide
,
and
you
shall
fynde
⅖
of
a
pennye
.
Then
saye
againe
,
if
6.
2/7
doe
giue
mee
⅖
,
what
will
10/1
giue
?
Multiplye
and
diuide
,
and
you
shall
fynde
⅗
of
a
pennye
,
as
afore
is
sayde
.
When
the
cariage
of
one
hundreth
wayghte
of
marchaundise
50.
miles
doth
cost
5s
.
what
shall
the
cariage
of
500
waight
coste
me
for
16
miles
?
Answere
,
By
the
fyrst
part
of
the
rule
of
3
compound
,
saying
if
Multiply
100
by
50
the
product
wyl
be
5000
,
which
shal
bee
your
diuisor
.
Then
multiply
5
tymes
500
by
16
and
therof
commeth
40000
for
your
diuidend
.
Therfore
diuide
40000
by
5000
and
you
shall
finde
8
s.
so
muche
shall
coste
the
carriage
of
500
wayghte
16
miles
.
Or
otherwise
by
the
double
rule
of
three
,
that
is
to
saye
by
the
rule
of
thre
at
two
times
:
first
say
if
50
miles
do
paye
5
s.
what
shall
16
miles
paye
?
Multiply
and
diuide
,
&
you
shall
find
1
s.
⅗
.
Then
say
,
agayne
,
if
100
waight
do
cost
me
1
s.
⅗
what
shal
500
wayght
cost
?
Multiply
and
diuide
,
and
you
shall
finde
8
s.
as
before
.
When
the
cariage
of
100.
pounde
waight
of
Marchaundise
84.
miles
,
doth
cost
me
six
shillings
,
how
many
miles
may
I
haue
64.
pound
waight
caried
for
three
.
s.
4.d
.
Aunswere
,
by
the
second
part
of
the
rule
of
three
compounde
:
saye
if
100/1
|
14/1
|
6/1
|
64/1
|
3
⅓
.
|
Then
multiplie
the
fourth
number
64/1
by
the
thyrde
number
6/1
,
and
thereof
commeth
304/1
for
your
diuisor
.
Likewise
multiplie
3
⅓
by
100/1
,
and
by
14/1
and
you
shall
haue
in
the
product
14000/3
:
then
diuide
14000/3
by
384/1
and
you
shall
fynde
72.
miles
11/12
of
a
mile
.
So
many
miles
shall
the
64.
li.
waight
be
caried
,
for
three
shillings
4.d
.
Otherwise
by
the
rule
of
three
,
at
two
times
:
Fyrst
say
,
if
100.
waight
doe
cost
me
6
.
s
.
what
shall
64.
pound
waight
cost
?
Multiplye
and
diuide
,
and
you
shall
finde
three
shillings
.21/25.
Then
saye
,
if
3.
21/25
bee
payed
for
84.
miles
cariage
:
for
howe
many
miles
shall
3
.
s
.
⅓
be
payed
?
Multiplye
&
diuide
and
you
shall
finde
72.
miles
.
11/12.
If
100.
horses
in
100.
dayes
do
spende
180.
quarters
of
otes
:
howe
manye
quarters
of
otes
wil
350.
horses
spend
in
150.
dayes
?
Answere
:
by
the
fyrste
part
of
the
rule
of
three
compounde
:
multiply
180.
times
350.
by
150.
and
diuide
the
product
by
100.
times
100
:
and
you
shall
finde
945.
quarters
.
So
many
quarters
of
Otes
will
350.
horses
spende
in
150.
dayes
.
Or
otherwise
by
the
rule
of
3.
at
two
times
:
fyrst
say
,
if
100.
dayes
doe
yelde
mee
180.
quarters
of
otes
:
what
shall
150.
dayes
yelde
:
multiplye
and
diuide
,
and
you
shall
finde
270.
quarters
:
then
say
againe
,
if
100.
horses
do
spende
270.
quarters
of
Otes
,
howe
many
quarters
of
otes
wyl
350.
horses
spend
?
Multiply
and
diuide
,
and
you
shall
finde
945.
quarters
as
before
.
¶
The
tenth
Chapter
treateth
of
the
rule
of
Fellowship
,
wythout
any
time
limited
.
THe
rule
of
felowship
is
thus
:
you
must
set
down
eche
mans
summe
of
money
that
he
laieth
into
company
,
A
Rule
.
euery
one
directly
vnder
the
other
,
the
which
you
shall
adde
altogither
,
&
the
totall
sum
of
all
their
whole
stocke
beyng
thus
assembled
,
shal
be
your
common
diuisor
,
to
the
finding
out
of
euery
mans
part
of
the
gaine
.
Then
shall
you
multiplye
the
gaine
,
or
the
losse
,
by
eche
mans
portion
of
money
that
he
layde
in
,
&
diuide
the
products
by
the
sayde
diuisor
:
so
shal
you
haue
in
your
quotient
euery
mans
part
of
the
gaine
,
or
else
of
the
losse
,
if
any
thing
be
lost
.
¶
Example
1.
Twoo
Marchaunts
haue
made
companye
togither
,
the
first
laide
in
500.
li.
The
seconde
put
in
300.
li.
and
with
occupying
thei
haue
gained
64.
li.
I
demaunde
how
much
eche
man
shal
haue
of
the
same
gaines
according
to
the
money
that
he
laide
in
.
Aunswere
:
Adde
500.
&
300.
both
togither
,
which
are
the
percels
that
they
laide
in
,
and
therof
commeth
800.
for
your
diuisor
:
then
say
by
the
rule
of
three
,
if
800.
li.
(
which
is
their
stock
)
do
gaine
64.
li.
what
shal
500.
li.
gaine
?
(
which
is
the
fyrst
mans
money
that
hee
laied
in
)
multiplie
&
diuide
and
you
shall
finde
40.
li.
for
the
firste
mans
parte
of
the
gaine
:
then
say
if
800.
giue
64.
what
will
300.
giue
?
Multiplye
and
diuide
,
and
you
shal
finde
24.
li.
for
the
seconde
mans
part
of
the
gaine
.
500
|
|
|
|
300
|
800
|
64
|
500.
|
800
|
|
|
|
|
800
|
64
|
300.
|
Or
otherwise
,
put
500.
li.
which
is
the
fyrst
mans
money
that
hee
layed
in
,
ouer
the
800.
li.
which
is
the
whole
,
stocke
,
and
you
shall
haue
500/800
which
being
abbreuiated
,
do
make
⅝
,
&
such
part
of
the
gaine
shal
the
fyrst
man
take
,
that
is
to
say
⅝
of
64.
li.
which
is
40.
li.
And
consequentlye
,
by
the
same
maner
,
the
seconde
shal
take
the
⅜
of
64.
which
is
24.
pound
for
his
part
of
the
gaine
as
before
.
2.
Twoo
Marchaunts
haue
companied
togither
,
the
fyrst
put
in
640.
li.
and
he
taketh
⅝
partes
of
the
gaine
.
I
demaunde
what
the
seconde
Marchaunt
layed
in
?
Aunswere
,
Seing
that
the
fyrst
Marchaunt
taketh
⅝
of
the
gaine
,
it
followeth
that
the
seconde
must
haue
⅜
which
is
the
rest
,
&
therfore
say
by
the
rule
of
three
,
if
⅝
of
the
gaine
,
which
the
fyrste
man
taketh
,
did
lay
into
the
stock
640/1.
How
much
shall
the
⅜
of
the
gaine
laye
in
,
which
is
the
seconde
mans
gaine
?
Multiply
and
diuide
,
&
you
shall
find
384.
li.
so
much
ought
the
second
man
to
lay
into
company
.
3.
Twoo
Marchauntes
haue
companied
togither
,
the
fyrst
man
layed
in
640.
li.
and
the
seconde
hath
layed
in
so
much
,
that
he
must
haue
60.
li.
for
his
part
of
100.
li.
which
thei
haue
gained
.
I
demaunde
howe
much
the
seconde
man
did
laye
into
companye
?
Aunswere
:
seing
that
the
second
man
taketh
60.
li.
of
the
gaine
,
it
followeth
that
the
fyrst
must
haue
but
40.
pounde
.
Therefore
say
by
the
rule
of
three
,
if
40.
li.
do
lay
in
640.
li.
what
shal
60.
li
lay
in
?
Multiply
and
diuide
,
and
you
shall
finde
960.
pounde
,
so
much
did
the
seconde
marchaunt
lay
in
.
4.
Two
marchaunts
haue
companied
togither
,
the
first
laide
in
83.
li.
6
.
s
.
8.d
.
the
seconde
put
in
170.
duckets
:
&
thei
haue
gained
100.
li.
of
the
which
the
fyrst
man
muste
haue
60.
li.
I
demaund
what
the
ducket
was
worth
?
Answere
,
seing
that
the
first
man
must
haue
60.
li.
it
followeth
that
the
seconde
must
haue
40.
li.
therefore
say
by
the
rule
of
thre
if
60.
li.
of
gaine
that
the
first
man
taketh
did
lay
in
83.
li.
6
.
s
.
8.d
.
of
principall
,
howe
much
shall
40.
li.
of
gaine
put
in
,
multiplye
&
diuide
,
and
shall
find
55.
li.
5/9
:
so
much
are
the
170.
duckets
worth
.
Then
put
55.
li.
5/9
into
shillings
,
and
you
shall
haue
1111
.
s
.
1/9
them
to
know
what
the
ducket
is
worth
,
saye
by
the
rule
of
three
,
if
a
170/1
gyue
1111.
1/9
,
what
will
1/1
giue
?
Multiplye
and
diuide
,
&
you
shall
fynde
6
.
s
.
6.d
.
22/51
,
so
much
is
the
ducket
worth
.
5.
Two
Marchauntes
haue
companied
togither
,
the
seconde
man
laide
in
more
by
30.
li.
than
did
the
first
man
:
and
they
gained
120.
li.
of
the
which
the
first
man
ought
to
haue
50.
li.
I
demaund
what
eche
of
them
did
lay
in
.
Answere
,
from
120.
li.
abate
50.
li.
and
there
resteth
70.
li.
for
the
second
mans
part
:
so
that
by
this
meanes
the
seconde
man
(
bicause
he
laide
in
30.
li.
more
than
the
first
man
did
)
taketh
20.
li.
more
of
the
gaine
:
&
therfore
say
by
the
rule
of
3.
if
20.
li.
of
gaine
did
lay
in
30.
li.
of
principall
,
how
much
shall
50.
li.
lay
in
?
Multiplye
and
diuide
,
and
you
shall
finde
75.
li.
so
much
did
the
firste
man
lay
in
,
and
consequentlye
the
seconde
layd
in
105
li.
6.
Two
marchaunts
haue
companied
togither
,
the
second
hath
layd
in
twise
so
muche
as
the
firste
man
dyd
,
and
10
li.
more
:
and
they
gayned
100
li.
of
the
which
,
the
firste
ought
to
haue
32
li.
for
his
part
:
I
demaund
how
much
eche
of
them
dyd
lay
into
company
?
Answere
,
If
it
were
not
for
the
10
li.
that
the
second
man
layd
in
more
:
he
should
haue
had
but
64
li.
of
the
gain
which
is
the
double
of
the
first
mans
parte
.
But
bicause
he
layd
in
10
li.
more
,
hee
hath
foure
pounde
more
of
the
gayne
,
and
therefore
saye
by
the
rule
of
three
,
if
4
li.
of
gayne
did
laye
in
10
li.
of
principall
,
(
which
was
ouer
and
aboue
the
double
of
the
first
mannes
laying
in
)
what
shall
32
li.
of
gaynes
lay
in
?
which
is
the
firste
mannes
parte
of
the
gaynes
that
hee
taketh
.
Multiplye
and
diuide
,
and
you
shall
finde
80
li.
for
the
first
mannes
laying
in
:
and
consequently
170
li.
for
the
second
mans
portion
that
he
layed
in
.
7.
Two
marchaunts
haue
companied
togither
,
and
they
haue
gayned
100
li.
of
the
which
the
first
must
haue
after
the
rate
of
10
vpon
the
100
li.
and
the
second
must
haue
after
the
rate
of
15
li.
vpon
the
100
li.
I
demaunde
how
muche
eche
of
them
oughte
to
haue
?
Aunswere
,
Put
10
li.
for
the
fyrst
mans
laying
in
,
and
15
li.
for
the
second
mannes
laying
in
.
Adde
10
li.
and
15
li.
togither
,
and
they
make
25
li.
Then
put
10
ouer
25.
and
it
is
10/25
which
being
abbreuiated
are
⅖
.
Therfore
he
that
taketh
10
li.
vpon
the
100
li.
must
haue
the
⅖
of
the
gayne
,
which
is
40
li.
Then
put
15
ouer
25.
and
it
is
15/25
which
being
abbreuiated
are
⅕
.
Therefore
the
second
must
haue
⅗
of
the
100
li.
which
is
60
li.
8.
Twoo
Marchauntes
haue
companied
togither
,
the
fyrst
laide
in
46.
li.
18
.
s
.
and
the
seconde
laide
in
33.
li.2.s.
so
they
haue
gained
30.
li.
I
demaund
how
much
euery
man
shall
haue
for
his
part
of
the
gaine
?
Answere
:
Adde
46.
li.
18
.
s
.
and
33.
li.2.s.
both
togither
and
you
shall
finde
80.
li.
for
your
common
diuisor
:
then
say
if
80.
li.
which
is
all
their
stocke
do
gaine
30.
li.
what
will
46.9/10
gaine
,
which
is
the
fyrste
mans
laying
in
:
Multiplie
&
diuide
,
and
you
shall
finde
17.
li.
11
.
s
.
9.d
.
for
the
first
mans
part
of
the
gaine
.
Then
say
again
,
if
80.
li.
do
gaine
30.
li.
what
will
33.
li.
1/10
gaine
,
which
was
the
seconde
mans
,
laying
in
:
multiply
and
diuide
,
and
you
shall
fynde
12.
li.
8
.
s
.
3.d
.
for
the
seconde
mans
part
of
the
gaine
.
And
after
the
same
maner
shall
you
doe
,
in
case
there
were
three
or
foure
Marchaunts
that
would
companye
togither
:
Adding
all
theyre
summes
of
money
(
which
they
laye
into
the
stock
)
into
one
total
summe
:
which
shal
be
your
common
Diuisor
:
and
then
worke
with
the
rest
,
as
is
taught
in
the
former
Questions
of
the
rule
of
companie
.
9.
Three
Marchaunts
haue
companied
togither
,
the
first
laide
in
I
know
not
how
much
:
the
seconde
did
put
in
20.
peeces
of
cloth
,
and
the
thirde
hath
layde
500.
pounde
.
So
at
the
ende
of
their
company
,
their
gaines
amounted
vnto
a
thousand
pounde
,
wherof
the
fyrst
man
ought
to
haue
350.
pound
,
and
the
seconde
must
haue
foure
hundred
pounde
.
Now
I
demaunde
how
much
the
first
man
did
lay
in
,
and
for
how
much
the
20.
peeces
of
clothe
were
put
into
company
?
Aunswere
.
Seing
that
the
fyrste
and
the
seconde
marchaunts
must
haue
750.
li.
for
their
parts
of
the
gaine
.
Then
the
thirde
man
must
haue
the
rest
of
the
thousande
pound
which
is
250.
li.
And
therefore
say
by
the
rule
of
thre
,
if
250.
of
gaine
,
become
of
500.
li.
of
principall
:
of
howe
much
shall
come
350.
li.
of
gaine
?
which
the
fyrst
man
taketh
,
multiplye
and
diuide
and
you
shall
finde
700.
li.
So
muche
did
the
first
man
laye
in
:
then
say
if
250.
li.
of
gaine
be
come
of
500.
li.
principall
,
of
howe
much
will
come
400.
li.
which
is
the
gaine
that
the
seconde
man
taketh
.
Multiply
and
diuide
,
&
you
shal
finde
800.
li.
For
so
much
were
the
20.
peeces
of
cloth
layde
into
company
.
10.
Three
Marchaunts
haue
gained
100.
li.
the
fyrst
muste
haue
the
½
,
the
seconde
must
haue
⅓
:
And
the
third
must
haue
¼
.
I
demaund
how
much
euerye
man
must
haue
of
the
gaine
?
Aunswere
,
Reduce
½
,
⅓
,
¼
,
into
a
common
denomination
,
after
the
order
of
the
second
reduction
in
fractions
,
&
you
shall
finde
12/24
,
for
the
½
:
8/24
,
for
the
⅓
:
and
6/24
,
for
the
¼
:
Then
take
12
for
the
first
mans
laying
in
,
8.
for
the
second
mannes
laying
in
:
and
6
for
the
thirde
mannes
laying
in
.
The
which
three
numbers
being
added
together
shall
be
your
common
diuisor
,
which
do
make
26.
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
finde
46
li.
2/13
for
the
fyrste
mannes
part
of
the
gaine
.
30.
li.
10/13
for
the
second
mannes
parte
:
and
23
li.
1/13
,
for
the
third
mannes
parte
.
11.
Two
marchaunts
haue
gayned
100
li.
the
firste
muste
haue
½
and
5
li.
more
:
the
second
must
haue
⅓
and
4
li
more
:
I
demaunde
how
muche
eche
of
them
shall
haue
?
Aunswere
,
From
100
abate
5
and
4.
so
ther
wil
remayn
91.
then
take
the
½
of
100.
li.
which
is
50
li.
for
the
first
mans
laying
in
:
Likewyse
,
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
which
number
adde
5
,
and
all
is
59.
li.
⅗
for
the
first
mans
part
.
Likewyse
multiplye
.
91.
by
33.
⅓
:
and
diuide
by
83.
⅓
,
&
you
shal
finde
36.
li.
⅖
vnto
the
which
adde
4
:
and
you
shal
haue
fourty
pound
,
⅖
for
the
seconde
mans
part
.
12.
Twoo
Marchauntes
haue
gained
a
hundred
pound
,
the
first
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
?
Aunswere
,
Adde
4.
&
2.
with
100.
&
they
make
106.
Then
take
as
before
is
saide
50.
pounde
,
for
the
first
man
,
&
33.
⅓
for
the
seconde
,
adde
them
bothe
togither
,
and
they
be
83.
which
shal
be
your
diuisor
.
Then
multiplie
106.
by
50.
and
diuide
the
product
by
83.
⅓
,
so
thereof
commeth
63.
li.
⅗
.
From
the
which
abate
the
foure
pounde
lesse
that
the
fyrst
man
taketh
,
and
then
is
there
remaining
59.
pound
,
⅗
for
hys
parte
.
Likewise
multiplie
106.
by
33.
⅓
and
diuide
by
83
⅓
&
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
laieth
in
,
must
be
multiplied
by
the
time
that
it
remaineth
in
company
:
and
of
that
which
commeth
therof
you
shal
make
their
new
layings
in
for
eche
of
them
:
and
then
multiplye
the
gaines
by
euery
one
of
them
seuerally
,
the
which
you
shall
diuide
by
all
their
new
layings
in
added
togither
,
and
you
shall
haue
proporcionally
eche
mans
part
of
the
gaine
according
to
his
laying
in
.
¶
Example
.
1.
Two
Marchaunts
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
dyd
put
450
li.
the
fyrste
of
Ianuary
:
hys
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
750
li.
but
at
the
first
daye
of
Maye
:
so
that
his
money
remayned
in
companye
but
8
monethes
.
Therefore
multiplye
his
750
li.
by
8.
and
therof
commeth
6000
for
hys
new
laying
in
:
Then
adde
5400.
with
6000.
and
they
make
11400
for
youre
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
&
so
much
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
gaynes
?
Answere
,
Multiply
640
li.
by
12.
monethes
that
his
money
abideth
in
the
companye
,
and
therof
commeth
7680
li.
for
his
laying
in
.
And
so
muche
oughte
the
seconde
mannes
laying
in
to
be
,
for
bycause
he
taketh
½
of
the
gaine
:
But
for
that
,
that
he
putteth
in
nothing
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
ought
the
seconde
marchaunt
to
laye
in
the
first
of
Aprill
,
to
the
ende
that
he
may
take
the
one
moyty
of
the
gaynes
.
3.
Three
Marchauntes
haue
companied
togither
,
the
firste
layed
in
the
firste
of
Marche
100
li.
The
seconde
laide
in
the
first
of
Iune
so
much
money
,
that
of
the
gaine
,
hee
must
haue
the
⅓
parte
:
and
the
thirde
laide
in
the
fyrst
of
Nouember
so
much
money
,
that
of
the
gaines
he
must
haue
likewise
⅓
and
thei
continued
in
company
,
vntil
the
next
Marche
folowing
.
I
demaunde
howe
much
the
seconde
and
the
thirde
Marchaunts
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
hys
laying
in
:
and
so
much
ought
the
seconde
and
the
thirde
marchaunt
eche
of
them
to
lay
in
:
Bicause
they
parte
the
gaynes
by
thyrdes
.
But
for
that
,
that
the
seconde
Marchaunt
putteth
in
nothing
tyll
the
fyrst
of
Iune
,
hys
money
can
bee
in
companye
but
nine
monethes
.
Therefore
diuide
1200.
by
nine
monethes
,
and
therof
commeth
133.
⅓
.
And
so
much
ought
the
seconde
Marchaunt
to
laye
in
:
Then
,
forasmuch
as
the
thirde
Marchaunt
,
dyd
laye
in
nothing
vntil
the
fyrst
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
third
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
fyftie
pounde
,
the
fyrste
of
Marche
:
And
the
thyrde
put
in
a
Iewell
the
fyrste
of
Iulye
:
And
at
the
yeares
ende
,
they
had
gained
foure
hundred
crownes
:
of
the
which
,
the
fyrste
marchaunt
must
haue
fifty
crownes
,
and
the
seconde
muste
haue
80.
I
demaunde
what
the
Ducket
was
worth
,
and
at
what
price
the
Iewell
was
valued
,
whych
the
thyrde
Marchaunt
layde
in
?
Aunswere
:
the
firste
mannes
money
is
1200
as
afore
is
sayde
,
and
hee
taketh
50
crownes
of
the
gayne
:
therefore
say
,
if
fifty
crownes
of
gayne
be
come
of
1200
,
which
was
his
stock
,
of
how
muche
shal
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
bee
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
finde
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.
(
which
is
the
third
mannes
laying
in
)
by
6.
monethes
that
his
Iewell
remained
in
companye
,
and
you
shall
finde
1080
Duckets
:
and
for
that
price
was
the
Iewell
put
into
company
.
5.
Three
Marchauntes
haue
companied
togither
:
the
first
layed
in
the
first
of
Ianuary
100
li.
and
the
firste
of
Aprill
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
agayne
50
li.
And
at
the
yeres
end
,
they
found
that
they
had
gained
160
li.
I
demaunde
how
muche
euery
man
shall
haue
?
Answere
,
Multiply
100
li.
which
the
first
man
layed
,
by
12
monethes
,
and
therof
commeth
1200.
li.
from
that
number
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
thirde
man
hath
layed
in
,
by
6.
monethes
,
and
therof
commeth
900.
from
the
which
number
abate
thre
times
50.
and
they
are
150
:
so
there
resteth
750.
for
the
thirde
mans
laying
in
.
Then
procede
with
the
reste
,
as
in
the
first
Question
of
the
rule
of
felowship
with
tyme
,
in
adding
1020
,
1100.
and
750.
altogither
,
which
shall
be
your
Diuisor
:
Then
multiply
160
by
1020.
by
1100.
and
by
750
,
&
diuide
at
euery
time
by
your
Diuisor
,
which
is
by
all
theyr
layings
in
added
togither
,
and
they
make
2870
,
so
you
shall
fynde
56.
248/287
:
for
the
fyrst
man
,
61.
93/287
for
the
seconde
,
and
41.
233/287
for
the
thirde
man
.
6.
Two
Marchaunts
haue
companied
togither
,
the
firste
hath
put
in
960.
pounde
,
for
the
space
of
12.
monthes
,
and
he
ought
to
haue
8.
pounde
vpon
the
hundred
pounde
of
the
gaine
.
The
seconde
hath
layed
in
1120.
li.
for
the
space
of
eight
monethes
,
&
he
ought
to
haue
after
12.
pounde
vpon
the
100.
pound
of
the
gaine
.
And
at
the
yeres
ende
,
they
haue
gained
eyght
hundred
pounde
.
I
demaunde
how
much
eche
of
them
shal
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
92160.
for
the
fyrst
mans
laying
in
:
then
multiplye
the
1120.
that
the
second
hath
layed
in
by
eyght
monethes
,
and
that
which
commeth
therof
you
shall
multiplie
againe
by
12.
and
you
shall
finde
107520.
for
the
seconde
mans
laying
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
399
3/13
for
the
first
man
:
and
430.
li.
10/13
for
the
second
man
.
¶
The
rule
of
company
,
betwene
Marchaunts
and
their
Factours
.
7.
The
estimation
of
the
bodye
or
persone
of
a
Factour
,
is
in
suche
proportion
to
the
stocke
,
which
the
Marchaunt
layeth
in
:
as
the
gayne
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
profite
,
the
persone
of
the
sayd
Factour
shal
be
esteemed
200
li
,
And
if
the
Factour
do
take
but
the
⅓
of
the
gaine
,
he
should
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
Marchant
layeth
in
,
that
is
to
say
100
li.
And
if
the
Factour
did
take
the
⅖
of
the
gayne
,
then
the
Marchaunt
shall
take
the
residue
,
which
are
⅗
of
the
gain
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
?
Multiplie
200
by
2
and
diuide
by
3
so
you
shall
finde
133
⅓
Otherwise
,
consider
that
the
Factoure
taketh
the
⅔
of
that
whiche
the
Marchaunte
taketh
.
And
therefore
take
the
⅔
of
200
,
and
you
shall
fynde
133
⅓
as
before
:
and
so
much
is
the
persone
of
the
Factoure
esteemed
to
bee
worth
.
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
gain
:
I
demaund
for
how
much
shal
his
person
be
estemed
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
estemed
twise
as
much
as
the
stock
that
the
marchant
layeth
in
,
which
shoulde
haue
but
⅓
of
the
gaine
:
for
⅔
vnto
⅓
,
is
in
double
proportion
.
Therefore
double
two
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
which
the
marchaunt
taketh
:
then
the
estimation
of
his
person
,
with
his
laying
in
should
be
estemed
but
the
halfe
of
the
which
the
marchaunt
layeth
in
:
take
therfore
the
½
of
200.
li.
which
is
100.
li.
from
the
which
abate
fourty
pounde
,
and
the
rest
which
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
shuld
haue
but
the
¼
of
the
gaine
,
&
yet
hee
would
haue
the
⅔
,
I
demaunde
how
much
he
shoulde
put
in
of
ready
money
,
besides
hys
person
?
Aunswere
,
seing
that
his
person
gaineth
the
¼
,
al
the
whole
laying
in
,
shall
gaine
the
rest
that
is
to
saye
the
¾
:
nowe
for
bicause
¼
is
the
⅓
of
¾
therfore
his
person
shal
be
estemed
the
⅓
of
all
the
laying
in
.
Take
then
the
⅓
of
240.
and
you
shall
haue
80.
for
the
estimation
of
his
person
,
and
for
that
,
that
he
wil
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
must
lay
in
the
ouerplus
,
which
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
seruice
shal
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
gain
,
hys
persone
ought
to
bee
esteemed
as
muche
as
½
of
the
stock
which
the
Marchaunte
layeth
in
,
that
is
to
say
the
½
of
1200
li.
which
is
600
li.
The
reason
is
,
bycause
the
⅓
of
the
gayne
that
the
Factoure
taketh
,
is
the
½
of
the
⅔
of
the
gaine
that
the
Marchaunt
taketh
.
11.
A
Marchaunt
hath
deliuered
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
he
shal
take
the
⅖
of
the
gayn
:
I
demaund
for
how
much
his
persone
was
esteemed
?
Aunswere
,
Forasmuch
as
the
Factour
taketh
the
⅖
of
the
gaine
,
he
taketh
the
⅔
of
that
which
the
Marchant
taketh
,
for
⅖
are
the
⅔
of
⅗
:
and
therefore
the
Factours
laying
in
ought
to
be
800.
pounde
,
which
is
the
⅔
of
1200.
pound
,
that
the
marchaunt
layed
in
:
Then
abate
500.
pounde
,
which
the
Factour
did
lay
in
,
from
800.
pounde
,
which
should
be
hys
whole
stock
and
there
remaineth
thre
hundred
pounde
for
the
estimation
of
hys
person
.
12.
More
,
a
marchaunt
hath
deliuered
vnto
his
factour
a
thousand
pound
vpon
such
condition
,
that
the
Factour
for
hys
paines
and
seruice
,
shall
haue
the
gaines
of
200.
pounde
,
as
though
he
layde
so
much
in
of
ready
money
:
I
demaunde
what
portion
of
the
gain
,
the
saide
Factour
shal
take
?
Answere
:
See
what
parte
the
200.
li.
(
which
the
Factour
layed
in
)
is
of
1200.
which
is
the
whole
stocke
of
their
company
,
&
you
shall
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
shoulde
haue
the
gaine
of
two
hundred
pound
of
the
stock
,
which
the
marchant
layeth
in
,
that
is
to
say
,
of
the
thousand
pound
.
Then
should
the
Factour
take
the
⅕
part
of
the
gaine
.
For
200.
li.
is
the
⅕
of
a
1000.
pounde
.
¶
The
xj
.
Chapter
treateth
of
the
Rules
of
barter
.
TWoo
Marchants
wil
chaunge
their
marchandise
,
the
one
with
the
other
.
The
one
of
them
hath
cloth
of
7
.
s
.
1.d
.
the
yarde
to
sell
for
ready
money
,
but
in
barter
he
will
sell
it
for
8
.
s
.
4.d
.
The
other
hath
Sinamon
of
4
.
s
.
7.d
.
the
li.
to
sell
for
ready
money
.
I
demaund
how
he
shall
sell
it
in
barter
to
the
ende
he
be
no
loser
?
Answere
,
say
,
if
7.
1/12
(
which
is
the
price
that
the
yard
of
cloth
is
worth
in
redy
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
li.
of
Synamon
is
worth
in
ready
money
,
reduce
the
whole
numbers
into
their
broken
,
and
then
multiply
and
diuide
,
and
you
shall
finde
5
.
s
.
4.d
.
12/17
parts
,
of
a
peny
,
and
for
so
much
shall
he
sell
the
pounde
of
Synamon
in
barter
.
2.
Two
Marchaunts
wil
chaunge
their
marchaundise
the
one
with
the
other
,
the
one
of
them
hath
Chaumlets
of
two
pounde
18
.
s
.
4.d
.
the
peece
to
sell
for
ready
money
,
and
in
barter
he
wyll
sell
the
peece
for
4.
li.
3
.
s
.
4.d
.
the
other
hath
fine
capps
of
35
.
s
.
10.d
.
the
dossen
to
sell
in
barter
.
I
demaund
what
the
dossen
of
caps
did
cost
in
redy
money
?
Answere
,
say
if
4.
li.
3
.
s
.
4.d
.
which
is
the
ouerprice
of
the
peece
of
Chamlet
,
become
of
2.
li.
18
.
s
.
4.d
.
which
was
the
iust
price
of
the
same
,
of
what
shall
come
35
.
s
.
10.d
.
which
is
the
ouerprice
of
the
dossen
of
cappes
?
Multiply
and
diuide
,
&
you
shall
finde
25
.
s
.
1.d
.
and
so
much
are
the
dossen
of
capps
worth
in
redy
money
.
3.
Two
Marchaunts
wil
chaunge
their
marchaundise
the
one
with
the
other
:
the
one
of
them
hath
Fustians
of
18
.
s
.
4.d
.
the
pece
to
sell
for
readye
money
,
and
in
barter
hee
will
sell
the
peece
for
26
.
s
.
8.d
.
The
other
hath
tapistry
of
15.d
.
the
ell
to
sell
for
readie
money
,
and
in
barter
hee
wyll
sell
it
for
20.d
.
the
ell
:
I
demaunde
which
of
them
gaineth
,
and
how
much
vpon
the
hundred
pounde
of
money
?
Aunswere
:
saye
if
18
.
s
.
⅓
(
which
is
the
iust
price
of
the
peece
of
Fustian
)
bee
solde
in
barter
for
26
.
s
.
⅔
:
for
howe
much
shall
1
.
s
.
¼
(
which
is
the
iuste
price
of
the
ell
of
tapistry
)
bee
solde
in
barter
?
Multiplie
and
diuide
,
&
you
shal
finde
21.d
.
9/11.
And
he
doth
ouersel
it
but
for
20.d
.
so
that
of
21.d
.
9/11
:
he
maketh
but
20.d
.
And
therefore
saye
by
the
rule
of
three
,
if
the
seconde
marchaunt
,
of
21
9/11
,
do
make
but
20/1
how
much
shall
he
lose
vpon
the
100/1
?
Multiplie
and
diuide
,
and
you
shall
finde
91.
⅔
,
the
which
being
abated
from
a
hundred
there
wil
remaine
8.
⅓
.
And
after
the
rate
of
8.
⅓
.
doth
the
seconde
marchant
lose
vpon
the
100.
And
consequently
,
the
first
marchaunt
,
of
20.d
.
maketh
21.d
.
9/11
:
and
therefore
saye
againe
by
the
rule
of
three
,
if
the
first
marchaunt
of
20/11
,
do
make
21.9/11
how
much
shall
he
gaine
vpon
100/1
?
Multiplie
and
diuide
,
and
you
shall
finde
109.
li.
1/11.
Thus
the
fyrst
gaineth
after
the
rate
of
9.
li.
1/11
:
vpon
the
hundred
pounde
of
money
.
For
your
better
vnderstanding
of
these
Questions
,
you
must
note
that
when
one
marchaunt
gaineth
of
an
other
after
the
rate
of
ten
pound
vpon
the
hundred
pound
he
gaineth
the
1/10
of
his
owne
principall
,
and
the
other
which
loseth
after
the
rate
of
9.
1/11
vpon
the
hundred
he
loseth
the
1/11
of
his
principal
.
And
it
may
be
proued
thus
:
When
one
marchaunt
will
sell
hys
wares
vnto
another
,
which
wares
stande
him
but
in
100.
li.
&
hee
will
sell
them
for
110.
li.
he
,
of
his
100.
li.
maketh
110.
li.
wherfore
hee
gaineth
after
10.
li.
vpon
the
100.
which
is
the
1/10
of
hys
principall
,
and
the
other
which
byeth
wares
for
110.
li.
that
cost
but
110.
pound
of
the
110.
pound
he
maketh
but
100.
li.
And
therefore
say
by
the
rule
of
thre
,
if
110.
become
of
100.
of
howe
much
shall
come
100
?
Multiplie
and
diuide
,
and
you
shall
finde
90.10/11
,
the
which
abate
from
100
:
and
there
resteth
9.1/11
is
the
1/11
of
hys
principal
that
the
seconde
loseth
vpon
the
100.
as
afore
is
saide
.
And
therefore
,
who
so
that
wil
know
what
one
Marchant
gaineth
of
another
,
either
after
the
rate
of
tenne
vpon
the
hundred
,
which
is
the
1/10
of
of
hys
principall
,
or
else
after
the
rate
of
twenty
vpon
the
hundred
which
is
the
⅕
,
or
of
any
other
parte
,
and
that
he
would
likewise
knowe
what
part
the
other
loseth
of
hys
principall
:
hee
must
take
for
the
numeratour
of
the
broken
number
of
hym
that
loseth
,
as
much
as
for
him
that
gaineth
,
them
adde
the
numerator
and
the
denominator
(
of
the
broken
number
of
hym
that
gaineth
)
both
togither
,
&
make
thereof
the
denominator
of
the
broken
number
of
him
that
loseth
,
&
then
shall
you
haue
the
part
of
him
that
loseth
,
as
by
exaumple
,
of
him
that
gayneth
after
ten
.
li.
vpon
the
100.
li.
which
is
the
1/10
of
hys
principall
:
take
the
numeratour
which
is
1.
and
make
that
the
numerator
of
the
broken
number
of
him
that
loseth
,
then
adde
1.
which
is
the
numerator
of
the
fraction
of
him
that
gaineth
with
ten
,
whych
is
hys
denominator
,
&
you
shall
haue
11.
for
the
denominatour
of
the
fraction
of
him
that
loseth
.
Then
put
one
ouer
the
11.
and
so
you
shal
haue
1/11.
Thus
it
appereth
when
one
marchant
gaineth
of
another
after
ten
vppon
the
hundred
,
he
gaineth
the
1/10
of
his
principall
,
and
the
other
loseth
9.
1/11
which
is
the
1/11
of
his
principall
.
And
yf
hee
would
gaine
after
20.
vpon
the
hundred
which
is
the
⅕
of
hys
principall
,
the
other
shoulde
lose
16.
⅖
which
is
the
⅙
of
hys
principall
,
and
so
is
to
bee
vnderstande
of
all
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
sell
for
readye
money
,
and
in
barter
he
will
sell
the
peece
for
23
.
s
.
4.d
.
&
yet
hee
will
gaine
moreouer
after
ten
pounde
vpon
the
hundred
pound
.
The
other
hath
woll
of
50
.
s
.
the
hundred
to
sell
for
readye
money
.
I
demaund
how
hee
shall
sell
the
C.
of
woll
in
barter
?
Aunswere
,
say
if
20
.
s
.
10.d
.
which
is
the
iust
price
of
the
peece
of
Sey
,
be
solde
in
barter
for
23
.
s
.
4.d
.
for
how
much
shall
50
.
s
.
(
which
is
the
iust
price
of
the
C.
of
woll
)
be
solde
in
barter
?
Multiplie
&
diuide
,
&
you
shal
finde
56
.
s
.
Then
for
bicause
the
first
marchant
gaineth
after
10.
li.
vpon
the
C.
li.
he
maketh
of
his
C.
li.
10.
li.
&
consequently
the
seconde
marchaunt
maketh
of
110.
li.
but
100.
li.
And
therfore
say
,
if
the
seconde
marchaunt
of
110.
doe
make
but
100.
how
much
shall
he
make
of
56
:
Multiplie
&
diuide
,
&
you
shal
find
50
.
s
.
10.d
.
10/11
of
a
peny
,
and
for
so
much
shall
he
sell
the
hundred
of
woll
in
barter
.
5.
More
,
twoo
Marchauntes
wyll
chaunge
their
marchaundise
,
the
one
with
the
other
,
the
one
of
them
hath
Taffeta
,
of
16.
crownes
the
peece
to
sell
for
redie
money
,
and
in
barter
he
will
sell
the
peece
for
twenty
crownes
,
and
yet
he
wyl
gaine
moreouer
after
ten
pounde
,
vpon
the
hundred
pound
.
The
other
hath
ginger
of
3
.
s
.
9.d
.
the
pounde
waight
,
to
sel
in
barter
.
I
demaunde
what
the
pounde
dyd
coste
in
readye
money
?
Aunswere
:
saye
if
twenty
crownes
which
is
the
surprice
of
the
peece
of
Taffata
,
become
of
16.
crownes
the
iuste
price
,
of
how
much
shall
come
.
3
.
s
.
9.d
.
which
is
the
price
of
the
ouerselling
the
pound
of
Ginger
?
Multiply
&
diuide
,
and
you
shall
finde
3
.
s
.
Then
,
for
bicause
that
the
Marchaunt
of
Taffeta
wil
gayne
after
the
rate
of
ten
vpon
the
hundred
:
say
if
100.
doe
giue
110.
what
shall
3
.
s
.
giue
?
Multiplye
and
diuide
,
and
you
shall
finde
3
.
s
.
3.d
.
⅗
and
so
much
dyd
the
pounde
of
Gynger
cost
in
readye
money
.
6.
More
,
two
marchaunts
wyll
chaunge
their
marchaundise
the
one
with
the
other
,
the
one
of
them
hath
Worsteds
of
25
.
s
.
the
peece
to
sell
for
ready
money
,
and
in
barter
,
hee
wyll
sell
the
peece
for
33
.
s
.
4.d
.
and
yet
hee
loseth
after
ten
vpon
the
hundred
:
the
other
hath
waxe
of
3.
li.
6
.
s
.
8.d
.
the
hundred
to
sell
for
readye
money
.
I
would
know
howe
he
should
sell
hys
waxe
in
barter
?
Aunswere
:
say
if
25
.
s
.
which
is
the
iuste
price
of
the
peece
of
Worsted
bee
solde
in
barter
for
33
.
s
.
4.d
.
for
how
much
shall
three
pounde
6
.
s
.
8.d
.
be
solde
,
which
is
the
iuste
price
of
the
hundred
of
waxe
.
Multiplie
and
diuide
,
and
you
shall
fynde
4.
li.
4/9
which
is
8
.
s
.
ten
pence
,
⅔
then
for
bycause
that
the
Marchaunt
of
Worsteds
,
loseth
after
ten
vpon
the
hundred
:
Of
a
hundred
hee
maketh
but
90.
And
therefore
,
saye
:
If
90.
giue
100.
what
giueth
4.
pounde
.
4/9
?
Multiplie
and
diuide
,
and
you
shall
fynde
4.
76/81
which
is
worthe
18
.
s
.
9.d
.
5/27
,
and
for
so
much
shall
he
sell
the
100.
of
Ware
in
barter
.
7.
More
,
two
Marchaunts
wyll
chaunge
their
marchaundise
the
one
wyth
the
other
,
the
one
of
them
hath
Worsteds
of
5.
pounde
6.
shillinges
,
eight
pence
the
peece
to
sell
for
ready
money
,
and
in
barter
he
will
sell
the
peece
for
6.
pounde
,
13.
shillings
.
4.d
.
and
yet
he
loseth
after
ten
vppon
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
ready
money
?
Answere
:
say
if
6.
pound
.
⅔
which
is
the
ouerprice
of
the
peece
of
Worsted
,
become
of
5.
pound
,
⅓
which
is
the
iuste
price
of
the
same
,
of
how
much
shall
come
two
shillings
9.
pence
.
⅓
.
Multiplie
and
diuide
,
&
you
shal
finde
2.
2/9
which
is
2.d
.
⅔
then
for
bicause
that
the
marchaunt
of
Worsteds
loseth
after
ten
vpon
the
hundred
,
of
a
hundred
he
maketh
but
90.
and
therefore
say
if
100.
giue
but
90.
how
much
shal
2
.
s
.
2/9
giue
?
Multiplie
and
diuide
and
you
shall
fynde
2
.
s
.
and
so
much
cost
the
pound
of
Muske
in
ready
money
.
Other
Rules
of
Barter
,
wherein
is
giuen
some
part
in
ready
money
.
WHen
a
Marchaunt
ouer
selleth
hys
marchaundise
&
he
wyll
giue
also
some
part
of
hys
ouerprice
in
ready
money
as
the
½
the
⅓
or
the
¼
&c.
He
must
substract
the
same
parte
of
money
from
the
iuste
price
,
and
also
from
the
ouer
price
of
hys
marchaundise
:
and
the
two
numbers
that
remaine
after
the
substraction
is
made
,
shal
be
that
two
first
numbers
in
the
rule
of
three
and
the
iuste
pryce
of
the
seconde
marchaunt
shal
be
the
third
,
to
know
how
much
he
shall
ouersell
the
part
of
his
marchaundise
.
8.
Two
Marchaunts
wyll
chaunge
theyre
marchaundise
the
one
wyth
the
other
,
the
one
of
them
hath
fyne
woll
at
fiue
pound
the
hundred
,
to
sel
for
ready
money
,
and
in
barter
he
wil
sell
it
for
six
pounde
,
and
yet
hee
wyll
haue
the
⅓
in
ready
money
.
The
other
hath
cloth
of
13
.
s
.
4.d
.
to
sel
for
ready
money
.
I
would
know
how
hee
shall
sell
the
same
in
barter
?
Aunswere
:
take
the
⅓
of
6.
li.
which
is
the
ouerprice
of
the
100.
of
wolle
,
&
you
shall
haue
2.
li.
the
which
abate
from
5.
li.
which
is
the
iust
price
of
the
100.
of
wolle
&
from
6.
li.
which
is
the
ouerprice
,
and
there
shal
rest
3.
li.
and
4.
li.
for
the
two
first
numbers
in
the
rule
of
three
,
then
take
13
.
s
.
4.d
.
which
is
the
iust
price
of
a
yarde
of
cloth
for
the
thirde
number
:
Then
multiply
and
diuide
&
you
shal
finde
17
.
s
.
9.d
.
⅓
:
for
so
muche
shall
the
seconde
sell
his
cloth
in
barter
.
9.
More
,
twoo
Marchauntes
wyll
chaunge
their
marchaundise
the
one
with
the
other
,
the
one
of
them
hath
waxe
of
thre
pound
6
.
s
.
8.d
.
the
C.
to
sel
for
readie
money
,
and
in
barter
he
will
sell
the
same
for
4.
li.
3
.
s
.
4.d
.
&
yet
he
will
haue
the
¼
in
redy
money
:
and
the
other
hath
fine
Crimson
sattine
of
15
.
s
.
the
yarde
to
sel
in
barter
.
I
demaund
what
it
is
worth
in
redy
money
.
Answere
,
Take
the
¼
of
4.
li.
3
.
s
.
4.d
.
and
abate
it
from
4.
li.
3
.
s.4.d
.
and
from
three
pounde
6
.
s
.
8.
pence
,
and
there
resteth
3.
li.
2
.
s.6.d
.
&
2.
li.
5
.
s
.
10.d
.
for
the
two
first
numbers
in
the
rule
of
three
,
and
15
.
s
.
for
the
thirde
number
which
is
the
ouerprice
of
the
yarde
of
sattine
.
Then
multiply
and
diuide
,
and
you
shall
fynde
11
.
s
.
And
so
much
did
the
yarde
of
Sattine
cost
in
ready
money
.
10.
Twoo
Marchaunts
will
chaunge
their
marchaundise
the
one
wyth
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
hee
wyll
gaine
after
ten
vpon
the
hundred
,
and
yet
he
will
haue
the
one
halfe
in
ready
money
:
and
the
other
hath
leade
of
3.
halfepence
the
li.
to
sel
for
redy
money
.
I
demaund
how
hee
shal
sell
the
pounde
in
barter
?
Answere
:
See
first
at
ten
vppon
the
hundred
,
what
the
three
pound
⅓
wil
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
whych
he
demaundeth
in
ready
money
,
is
36.
shillings
and
8.
pence
,
the
whych
being
abated
from
fyftye
shillings
,
and
also
from
three
pounds
13.
shillings
4.
pence
,
there
shall
reste
13.
shillings
4.
pence
,
and
one
pound
16
.
s
.
8.d
.
for
the
twoo
firste
numbers
in
the
rule
of
three
,
which
you
muste
put
al
into
halfepence
,
and
thre
halfepence
for
the
thirde
number
,
and
then
multiply
and
diuide
,
&
you
shall
finde
4.d
.
⅛
,
and
for
so
much
shall
he
sell
the
pounde
of
leade
in
barter
.
11.
More
,
twoo
marchauntes
wyll
chaunge
their
marchaundise
the
one
with
the
other
,
the
one
of
them
hath
steele
of
16
.
s.8.d
.
the
hundred
waight
to
sell
for
ready
money
,
&
in
barter
he
will
sell
it
for
25
.
s
.
and
yet
hee
loseth
after
ten
vpon
the
hundred
,
but
hee
wyll
haue
the
½
in
readye
money
,
the
other
hath
yron
of
6
.
s
.
8.d
.
the
hundred
to
sell
in
barter
,
I
demaunde
what
it
did
coste
in
readye
money
?
Aunswere
:
say
if
a
hundred
come
but
to
90.
how
much
shall
25
.
s
.
come
to
?
Multiplie
and
diuide
,
and
you
shall
fynde
22
.
s.6.d
.
of
the
which
number
,
take
the
½
which
is
11
.
s
.
3.d
.
&
substract
it
from
22
.
s.6.d
.
and
from
16
.
s.8
.
pence
and
there
shall
rest
11
.
s.3.d
.
and
5
.
s.5
.
pence
,
for
the
two
fyrst
numbers
in
the
rule
of
thre
,
and
6
.
s
.
8.d
.
which
is
the
ouerprice
of
a
hundred
of
yron
for
the
thyrde
number
,
then
multiplie
and
diuide
,
and
you
shal
finde
3
.
s
.
2.
pence
,
14/27
:
&
so
much
did
the
hundred
of
yron
cost
in
ready
money
.
12.
More
,
twoo
marchaunts
wyll
chaunge
their
marchaundise
,
the
one
with
the
other
,
the
one
of
them
hath
seyes
of
20
.
s
.
10.d
.
the
peece
to
sel
for
redy
money
,
&
in
barter
he
wyll
sell
the
peece
for
21
.
s
.
&
hee
wyll
haue
the
¼
in
ready
money
:
The
other
hath
capps
of
35.
shillings
the
dossen
to
sell
for
redy
money
:
but
hee
wyll
gaine
after
ten
vpon
the
hundred
.
I
demaunde
how
he
shall
sell
the
same
caps
in
barter
?
Aunswere
:
saye
if
a
hundred
bee
worth
110.
What
shal
35
.
s
.
be
worth
,
which
is
the
iust
price
of
the
dossen
of
cappes
?
Multiplie
and
diuide
,
and
you
shall
fynde
38.
shillings
6.
pence
.
Then
take
the
¼
of
25.
which
is
6
.
s
.
3.d
.
and
substract
it
from
20
.
s
.
10.d
.
&
from
25
.
s
.
and
there
shall
rest
14
.
s
.
7.d
.
and
18
.
s
.
9.d
.
for
the
twoo
fyrste
numbers
in
the
rule
of
three
,
and
38
.
s
6.d
.
which
is
the
iust
pryce
wyth
hys
gaine
of
the
dossen
of
cappes
,
for
the
thirde
number
:
then
multiplie
and
diuide
,
and
you
shall
finde
49
.
s
.
6.d
.
and
for
so
much
hee
shall
sell
the
dossen
of
caps
in
barter
.
¶
The
12.
Chapter
treateth
of
the
exchaunging
of
money
from
one
place
to
another
.
FIrst
,
you
must
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
.
s
.
Flemishe
,
and
20
,
shillings
Flemishe
doe
make
1.
li.
de
gros
.
1.
If
I
deliuer
in
Flaunders
,
500.
li.
Flemishe
,
at
19
.
s.6
.
de
gros
that
is
to
say
at
19
.
s
.
6.d
.
Flemishe
,
to
receaue
20
.
s
.
at
London
,
I
demaunde
how
much
I
shal
receaue
sterling
at
London
for
the
sayde
5.
hundred
pounde
Flemishe
?
Answere
,
:
Say
,
if
19
½
giue
20/1
,
what
will
500/1
giue
?
Multiplie
&
diuide
,
and
you
shal
fynde
512.
li.
16.s
.
4.
pence
12/13
of
a
pennye
.
And
so
much
sterling
shal
I
receaue
in
London
for
my
500.
li.
Flemishe
.
2.
If
I
deliuer
in
London
375.
li.
sterling
,
to
receaue
in
Andwerp
21
.
s.9.d
.
de
gros
,
that
is
to
say
Flemishe
,
for
euery
pounde
sterling
.
I
demaunde
how
many
poundes
Flemish
I
shall
receaue
in
Andwerpe
,
for
the
sayde
375.
li.
sterling
?
Answere
,
say
if
20/1
giue
21.
¾
:
what
will
375/1
giue
?
Multiplie
&
diuide
,
and
you
shall
finde
407.
li.
So
many
pounds
Flemish
shal
I
receaue
for
the
said
375.
li.
ster
.
in
Andwerpe
.
3.
If
I
take
vp
money
at
Andwerp
after
19
.
s
.
6.d
.
flemishe
to
pay
for
the
same
at
London
20
.
s
.
ster
.
and
when
the
day
of
paiment
is
come
,
I
am
forced
to
rechaunge
the
same
,
and
to
take
vp
money
againe
here
in
London
to
repaye
the
same
,
so
that
for
twentye
shillings
,
which
I
take
vp
here
,
I
must
repay
,
19.
shillings
9
d.
at
Andwerpe
.
I
demaunde
whether
I
doe
winne
or
lose
,
and
how
much
vpon
the
100.
li.
of
money
?
Answere
,
Say
.
if
19.
3/4
giue
19.
½
,
what
will
100/1
giue
?
Multiplie
and
diuide
,
and
you
shall
finde
98.
58/79
,
the
which
beyng
abated
from
a
hundred
there
wil
remaine
1.
21
/
79.
And
so
much
doe
I
lose
vpon
the
100.
pounde
of
money
.
4.
If
I
take
vp
at
London
20.
shill.
sterling
to
pay
at
.
Andwerpe
21.
s.
8.
d.
Flemishe
,
and
when
the
day
of
payment
is
come
.
I
am
constrained
to
take
vp
money
againe
at
Andwerpe
wherwith
to
repay
the
foresaide
sum
:
and
there
I
doe
receaue
22.
shillings
.
Flemishe
to
pay
20.
shill.
at
London
.
Nowe
I
demaunde
whether
I
doe
winne
or
lose
and
howe
much
vpon
the
100.
li.
of
money
after
the
rate
?
Answere
,
say
if
21.
⅔
giue
22/1.
What
wil
1000/1
giue
?
Multiply
&
diuide
,
and
you
shall
finde
101.
7/13
,
from
the
which
abate
100.
and
there
wyll
remaine
1.
7/13
,
and
so
much
shal
I
gaine
vpon
the
100.
li.
of
money
.
The
exchaunge
from
London
into
Fraunce
,
is
not
lyke
as
it
is
into
Flaunders
but
is
deliuered
by
the
French
crowne
,
which
is
worth
50.
souse
Tournois
the
peece
.
And
here
muste
you
note
that
in
Fraunce
they
make
theyr
accompte
by
Deniers
Tournois
,
wherof
12.
maketh
one
souse
Tournois
,
and
20.
sou
.
Tournois
maketh
1.
li.
Tournois
,
which
they
cal
a
Liuer
,
and
the
Frenche
Crowne
is
currant
amonge
Marchaunts
for
51.
souse
Tournois
,
but
by
exchaunge
it
is
otherwise
,
for
thei
wyll
deliuer
but
50.
sou
.
Tourneys
,
which
is
.
2.
li.
10.
sou
.
Tournois
for
a
crowne
,
or
at
such
price
as
the
takers
vp
of
money
can
agree
wyth
the
deliuerer
.
As
by
Example
.
5.
If
I
deliuer
340.
li.
ster
.
here
in
London
after
6.
s.
4.
d.
sterling
the
crowne
,
to
receaue
at
Roan
,
or
at
Parris
50.
sou
.
Tournois
for
euery
crowne
,
I
would
knowe
how
many
Liures
Tournois
I
shall
receaue
there
for
my
340.
li.
ster
.
Answere
:
say
if
6.
s.
⅓
doe
giue
mee
2.
li.
½
.
Tour.
what
will
6800/1
shil
.
giue
(
which
is
the
340.
li.
reduced
into
shillings
)
multiplye
and
diuide
,
and
you
shall
fynde
2684.
Liures
4/19
which
is
worth
4.
s.
4/19
Tournois
,
and
so
much
shall
I
receaue
in
Roan
or
Parris
for
my
340.
li.
sterling
.
6.
If
I
deliuer
in
Parris
or
Roan
,
or
elsewhere
in
Fraunce
1250.
Liures
Tournois
,
at
50.
sou
.
Tournois
the
crowne
to
receaue
for
euerye
suche
crowne
,
6.
s.
3.
d.
sterling
at
London
.
I
demaunde
how
much
sterling
money
I
shal
receaue
at
London
for
my
1250.
pound
Tournois
.
Answere
:
say
,
if
two
pounde
,
½
doe
giue
mee
6.
shil
.
¼
,
what
will
1250/1
giue
?
Multiplye
&
diuide
,
and
you
shall
fynde
3125.
shil
.
sterling
,
which
maketh
156.
pounde
,
5.
shillings
sterling
.
And
so
manye
poundes
shall
I
receaue
at
London
,
for
the
sayde
1250.
pounde
Tournois
,
after
6.
shillings
three
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
bynde
togither
diuers
percelles
of
sundrie
prices
,
and
to
knowe
how
much
you
must
take
of
euery
percell
,
according
to
the
numbers
of
the
Question
.
¶
Example
.
1.
A
Goldsmith
hath
three
sortes
of
Golde
.
The
fyrste
is
worth
thirtye
Crownes
the
pounde
waight
:
The
seconde
is
worth
36.
Crownes
.
And
the
thyrde
is
worth
45.
crownes
,
and
of
these
three
sortes
hee
wyll
make
a
Scepter
of
syxe
pound
waight
,
which
shall
be
worth
fourtye
Crownes
the
pounde
.
I
demaunde
how
much
hee
must
take
of
euery
sorte
?
Aunswere
:
fyrst
you
must
set
downe
the
numbers
wherof
you
shall
make
the
Alligation
(
which
are
30.36.38
.
&
45.
orderly
the
one
vnder
the
other
,
as
yf
you
should
make
of
them
an
addition
:
and
the
common
number
wherunto
you
will
reduce
them
,
shall
you
set
on
the
left
hand
,
which
common
number
in
this
example
is
40.
Then
marke
what
summes
bee
lesser
than
that
common
number
,
and
which
be
greater
,
and
with
a
draught
of
your
pen
,
euermore
lynke
two
numbers
togither
,
so
that
the
one
be
lesser
than
that
common
number
,
and
the
other
greater
than
he
.
For
two
greater
nor
two
smaller
numbers
may
not
be
lynked
togither
,
for
they
wil
either
be
lesser
,
or
else
greater
than
the
common
number
:
but
one
greater
number
,
and
one
smaller
may
bee
so
mixed
,
that
they
wyl
make
the
common
number
.
And
two
greater
or
twoo
smaller
numbers
,
can
neuer
make
the
common
number
in
due
order
,
as
hereafter
shall
appeare
.
After
that
you
haue
thus
lynked
them
,
then
marke
howe
much
eche
of
the
lesser
numbers
is
smaller
than
the
common
number
,
and
that
difference
shal
you
set
against
the
greater
numbers
,
which
be
linked
with
those
smaller
,
eche
of
them
with
his
match
still
on
the
right
hande
.
And
likewise
you
must
set
the
excesse
of
the
greater
numbers
against
the
lesser
which
Bee
combined
with
them
.
Then
shal
you
adde
all
those
differences
into
one
summe
,
which
shal
be
the
fyrst
number
in
the
rule
of
three
,
and
the
second
number
shal
be
the
whole
massye
peece
that
you
wyll
haue
of
all
the
perticulers
,
the
thirde
summe
shall
be
eche
difference
by
it selfe
,
and
by
them
shall
you
finde
out
the
fourth
number
,
declaring
the
iust
portion
of
euery
particuler
in
that
mixture
,
as
nowe
by
the
former
exaumple
,
I
wyll
make
it
playne
.
The
prices
seuerall
.
|
The
differences
.
|
|
The
common
price
or
number
.
|
40
|
30
|
5.
|
A
|
36
|
2.
|
B
|
42
|
4.
|
C
|
45
|
10
|
D
|
|
|
|
21.
|
|
21.
|
.6
.
|
.5
.
|
21.
|
.6
.
|
.2
.
|
21.
|
.6
.
|
.4
.
|
21.
|
.6
.
|
.10
.
|
Here
in
this
former
example
,
you
see
that
I
haue
set
downe
the
seueral
prices
,
whych
bee
30.36.42.45
.
and
haue
linked
togither
30.
with
45.
and
36.
with
42.
The
common
price
40.
I
haue
set
on
the
left
syde
,
&
the
difference
of
it
from
euery
seuerall
price
,
I
haue
sette
on
the
righte
hande
,
against
that
summe
with
the
which
it
is
lynked
.
So
the
difference
of
30.
from
fourtie
is
ten
,
which
I
set
against
45.
that
hee
is
lynked
with
all
,
and
the
difference
of
45.
about
40
is
5.
which
I
haue
set
againste
.
30.
So
likewyse
,
the
difference
of
42.
aboue
40.
is
2.
that
I
haue
set
against
36.
And
the
difference
betweene
36.
and
40.
(
which
is
4.
)
I
haue
set
againste
42.
Then
I
adde
all
those
differences
togither
&
they
make
21.
which
I
make
the
fyrst
number
in
the
Rule
of
thre
,
and
syxe
the
seconde
number
,
which
is
the
waight
of
the
Scepter
of
Golde
,
and
the
thyrde
number
shall
bee
euerye
particulers
difference
.
Then
I
worke
by
the
Rule
of
three
:
saying
if
twenty
and
one
(
which
is
the
differences
added
togither
)
do
giue
mee
syx
pound
,
which
t
s
the
waight
of
the
Scepter
,
what
shall
fyue
giue
,
which
is
the
fyrste
difference
?
Multiplye
and
diuide
,
and
you
shall
fynde
one
pounde
3/7
:
so
much
muste
I
haue
of
the
fyrste
price
.
Then
doe
likewyse
with
the
rest
and
you
shall
finde
4/7
of
the
seconde
price
,
one
pound
,
1/7
of
the
thirde
price
,
and
2.
li.
6/7
of
the
fourth
,
the
which
foure
summes
being
added
togither
,
doe
make
6.
li.
which
is
the
totall
that
I
would
haue
.
And
now
to
proue
if
the
pryces
doe
agree
,
you
shall
doe
thus
:
Fyrst
multiply
this
totall
summe
6.
by
the
common
price
40.
and
it
wyll
make
240.
crownes
,
which
you
shall
keepe
by
it selfe
.
And
afterward
multiply
euery
seueral
summe
of
waight
by
the
price
belonging
to
the
same
waight
,
and
if
that
summe
doe
agre
with
the
fyrst
that
you
kept
by
it self
,
then
is
your
work
well
done
,
as
here
one
pound
,
3/7
is
the
waight
of
the
sorte
of
golde
which
is
of
30.
crownes
price
.
Then
multiplye
30.
by
1.
li.
3/7
,
&
it
maketh
42.
crownes
6/7
,
which
you
shall
set
downe
.
Then
multiplye
4/7
(
which
is
the
wayght
of
the
seconde
sorte
of
gold
)
by
36.
which
is
the
price
of
the
same
&
thereof
commeth
20.
crownes
4/7
:
so
againe
1.
li.
1/7
multiplied
by
42.
doth
make
48.
crownes
.
And
last
of
al
2.
li.
6/7
multiplied
by
45.
maketh
128.
crownes
4/7.
All
these
added
togither
doth
make
240.
crownes
,
agreable
to
the
former
summe
of
40.
multiplied
by
6
And
thus
I
maye
affirme
that
thys
worke
is
well
done
.
2.
A
Tauerner
hath
foure
sortes
of
wyne
,
of
foure
seuerall
prices
,
the
fyrst
of
eight
pence
the
Gallonde
,
the
second
of
ten
pence
the
gallonde
,
the
thirde
of
15.
pence
and
the
fourthe
of
18.
pence
.
And
he
wil
mingle
one
punchen
with
all
these
sortes
,
so
that
the
Gallonde
shall
bee
worth
but
twelue
pence
.
I
demaunde
how
many
Gallondes
he
muste
take
of
euery
sorte
?
Aunswere
:
First
suppose
the
punchen
to
holde
some
certaine
measure
,
as
to
conteine
84.
gallonds
and
then
the
forme
wyll
be
after
thys
sort
,
as
you
see
hereafter
folowing
.
If
15.
doe
giue
18.
What
will
.
3.
|
They
make
|
16
⅘
of
the
first
.
|
What
will
.
6.
|
33
⅗
of
the
seconde
|
What
will
.
4.
|
22
⅖
of
the
third
.
|
What
will
.
2.
|
11
⅕
of
the
fourth
|
|
|
84
|
¶
The
14.
chapter
treateth
of
the
Rule
of
falsehode
,
or
false
positions
.
THe
rule
of
falsehode
is
so
named
,
not
for
that
it
teacheth
any
deceit
or
falsehode
,
but
that
by
fained
numbers
taken
at
al
aduentures
,
it
teacheth
to
finde
out
the
true
number
that
is
demaunded
.
And
this
(
of
all
the
vulgare
Rules
which
are
in
practise
)
is
the
most
excellent
:
this
rule
hath
two
partes
the
one
is
of
one
false
position
alone
:
the
other
is
of
two
positions
as
hereafter
shal
appere
Those
questions
which
are
done
by
false
positions
,
haue
theyr
operations
,
in
a
maner
like
vnto
that
of
the
rule
of
three
,
but
only
that
in
the
rule
of
three
,
we
haue
thre
numbers
knowen
,
and
here
in
thys
rule
wee
haue
but
one
(
I
meane
that
commeth
in
operation
)
vnto
the
likenes
whereof
wee
muste
diuise
two
other
,
the
one
multiplying
,
and
the
other
diuiding
,
as
by
example
.
1.
I
haue
deliuered
to
a
banker
a
certeine
summe
of
poundes
in
money
,
to
haue
of
hym
by
the
yeare
6.
li.
vpon
the
100.
li.
And
at
the
ende
of
10.
yeares
,
he
paide
mee
500.
li.
for
al
both
principal
and
gaine
.
I
demaund
how
much
was
the
principall
summe
that
I
deliuered
at
the
fyrst
.
Here
you
see
that
there
are
diuers
termes
:
but
the
chiefe
to
worke
with
all
is
500.
pound
which
commeth
of
the
other
numbers
,
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
number
at
pleasure
,
and
wyth
the
same
let
vs
make
oure
discourse
,
euen
as
though
it
were
the
principall
summe
that
wee
seeke
for
.
As
by
Example
.
Suppose
that
I
deliuered
him
at
the
first
200.
li.
the
which
were
worth
to
me
in
ten
yeres
.
120.
li.
after
the
rate
of
6.
li.
vpon
the
hundred
pounde
.
Then
120.
li.
added
with
200.
li.
Do
make
but
320.
li.
and
I
must
haue
500.
pound
.
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
formed
artificially
,
which
are
200.
and
320
:
in
such
sort
,
that
320.
ought
to
haue
such
proportion
to
200
,
as
500.
hath
vnto
the
number
that
I
seeke
:
that
is
to
saye
,
vnto
the
true
principall
summe
,
then
must
I
haue
recourse
vnto
the
rule
of
thre
,
after
this
sorte
,
saying
.
If
320.
li.
become
of
200.
li.
of
howe
much
shall
come
500.
li.
Multiply
500.
by
200.
and
thei
are
100000.
the
which
you
shall
diuide
by
320.
li.
and
thereof
commeth
312.
li.
½
which
is
the
summe
that
I
deliuer
at
the
first
.
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
bee
opened
,
the
water
wyll
auoide
cleane
in
one
houre
,
at
the
seconde
it
wyll
auoide
in
twoo
houres
,
and
at
the
thyrde
it
will
require
three
houres
.
Nowe
I
demaunde
in
what
space
wil
it
auoide
,
all
the
cocks
being
set
open
.
Suppose
that
it
will
auoide
in
halfe
an
houre
,
that
is
to
say
,
in
30.
minutes
.
Then
must
there
auoide
at
the
fyrste
cocke
the
½
,
which
is
thirtye
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
whych
summes
being
added
togither
dooe
make
55.
pipes
,
but
it
shoulde
bee
60.
pipes
.
Therefore
I
say
by
the
rule
of
three
,
if
55.
pipes
doe
voide
in
30.
minutes
:
in
how
many
minuts
wil
60.
pipes
voide
?
Multiplie
and
diuide
,
&
you
shall
finde
32.
minutes
40/55.
And
in
that
space
will
the
water
auoide
if
all
the
cockes
be
set
open
.
¶
Of
the
Rule
of
twoo
false
positions
.
A
Rule
.
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
work
in
stede
of
the
true
number
,
as
the
question
doth
import
,
and
if
you
see
that
you
haue
missed
.
Then
is
the
last
number
of
the
work
either
to
great
or
to
little
,
that
shal
you
note
for
to
be
the
fyrst
errour
,
in
the
which
you
haue
missed
with
the
signe
of
more
,
or
lesse
,
which
signes
shal
be
noted
with
these
figures
,
4
:
—
.
This
figure
4
:
betokeneth
more
,
&
this
plaine
line
—
signifieth
lesse
,
that
is
to
say
the
one
signifieth
to
much
,
&
the
other
to
little
:
than
begin
againe
,
&
take
another
number
,
which
shall
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
(
&
this
must
alwayes
be
obserued
)
&
keepe
the
twoo
products
:
then
if
the
signes
be
both
like
,
that
is
to
say
,
either
both
to
much
,
or
both
to
little
,
abate
the
lesser
product
from
the
greater
,
and
likewise
,
you
shall
substract
the
lesser
errour
from
the
greater
,
&
by
the
remaine
of
those
errours
,
you
shall
diuide
the
residue
of
the
products
:
the
quotient
shal
be
the
true
number
that
you
seeke
for
.
But
if
the
twoo
signes
be
vnlyke
,
that
is
to
saye
the
one
to
much
and
the
other
to
lyttle
,
than
shall
you
adde
those
productes
togither
so
shall
you
also
adde
both
the
errours
togither
,
and
by
the
summe
of
those
errours
,
diuide
the
totall
summe
of
both
the
products
:
the
quotient
shall
bee
lykewise
the
true
number
that
the
question
seketh
,
and
thys
is
the
whole
rule
,
as
by
exaumple
.
3.
A
man
lying
at
the
point
of
death
,
sayde
that
hee
had
in
a
certaine
Coffer
a
hundred
Duckets
,
the
whych
he
bequeathed
to
three
of
his
frendes
by
hym
named
,
after
this
sort
.
The
first
must
haue
a
certeine
portion
,
the
seconde
must
haue
twise
so
many
as
the
fyrst
abating
eyght
Duckets
:
and
the
thyrde
must
haue
thre
times
so
many
as
the
fyrst
,
lesse
by
15.
Duckets
.
Now
I
demaunde
howe
many
euerye
of
them
must
haue
.
Aunswere
:
Fyrst
I
doe
imagine
that
the
fyrst
man
had
30.
Duckets
,
then
by
the
order
of
the
question
the
seconde
should
haue
52
:
&
the
third
75.
These
thre
summes
being
added
togither
do
make
157
:
&
I
should
haue
but
100
:
so
that
this
fyrst
errour
is
to
much
by
57.
then
I
note
a
parte
the
fyrst
position
30.
wyth
his
errour
57.
to
much
after
this
sort
30.457
.
Therfore
I
prosecute
my
worke
and
I
suppose
that
the
fyrste
had
24.
then
by
the
order
of
the
question
,
the
second
should
haue
40.
and
the
thyrd
57
:
these
three
summes
being
added
togither
,
do
make
121.
&
I
must
haue
but
100.
so
the
seconde
errour
is
to
much
by
21.
Therfore
I
note
24.421
.
vnder
the
30.457
.
as
you
may
se
in
the
margent
of
the
next
syde
following
.
Then
I
multiplye
crossewayes
,
30.
(
whych
is
the
fyrst
position
)
by
twelue
whych
is
the
seconde
errour
,
and
thereof
commeth
630.
lykewyse
I
multiplye
twenty
&
4.
(
whych
is
the
seconde
position
)
by
57.
which
is
the
fyrst
errour
,
and
I
fynde
1368
:
then
bicause
the
signes
of
the
errours
are
both
lyke
:
that
is
to
saye
to
much
,
I
must
therfore
substract
630.
from
1368.
&
there
will
remain
738
which
is
the
diuidende
:
again
I
must
substract
the
lesser
errour
from
the
greater
,
that
is
to
wit
,
21.
out
of
57.
and
there
wil
remaine
36.
which
shall
bee
my
diuisor
.
Thys
done
I
diuide
738.
by
36.
and
the
quotient
wyll
be
20.
½
.
The
which
20.
½
is
the
iust
number
of
the
Duckets
that
the
first
man
had
for
his
parte
,
so
consequently
the
seconde
man
had
33.
Duckets
,
and
the
thyrde
46.
½
,
as
by
proofe
in
the
margent
,
may
appere
.
|
1
1.
|
|
73
8.
|
20
½
|
36
6.
(
20.
½
|
33
|
3
|
46.
½
|
|
100.
|
|
The
like
number
will
also
appere
,
in
case
the
errours
were
both
to
litle
,
as
in
making
the
twoo
positions
by
18.
and
20.
where
you
shal
fynde
the
two
errours
both
to
lyttle
,
the
fyrst
wyll
bee
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
fynde
that
the
first
errour
will
be
21.
to
muche
,
and
the
seconde
wyll
be
three
to
little
:
Therfore
I
multiply
24.
by
3.
crossewaies
,
therof
commeth
72.
Lykewise
I
multiplye
twentye
by
21.
the
product
will
be
420
:
These
twoo
summes
72.
and
420
,
I
adde
both
togither
bycause
the
signes
of
the
errours
bee
vnlyke
,
&
they
make
492.
the
which
shall
bee
my
diuidende
,
and
againe
,
I
adde
the
the
lesser
errour
3.
wyth
the
greater
errour
21.
and
they
make
24.
for
my
diuisor
,
them
diuiding
492
by
24.
the
quotient
wil
be
20
½
:
as
in
the
margent
doth
plainly
appeare
.
And
now
bicause
you
shall
not
forget
thys
part
of
the
rule
,
learne
thys
briefe
remembraunce
folowing
.
The
signes
both
like
substraction
do
require
.
And
vnlike
signes
addition
will
desire
.
The
meaning
whereof
is
thus
yf
both
the
errours
haue
lyke
sygnes
,
them
must
the
diuidende
and
the
diuisor
be
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
,
hauinge
to
them
bothe
,
but
one
couer
,
the
weyght
whereof
is
5.
ounces
,
if
the
couer
bee
put
to
the
lesser
cuppe
,
it
wyll
bee
in
double
proportion
vnto
the
waight
of
the
greater
,
and
the
couer
being
put
to
the
greater
cuppe
,
wyll
bee
in
triple
proportion
,
vnto
the
waight
of
the
lesser
.
I
demaunde
what
was
the
waight
of
euerye
cuppe
?
Suppose
that
the
lesser
cuppe
dyd
wayghe
seauen
ounces
,
then
wyth
the
couer
it
must
waigh
twelue
,
and
thys
waight
should
be
in
double
proportion
vnto
the
greater
,
therefore
the
greater
muste
waygh
sixe
ounces
adde
vnto
it
5.
ounces
for
the
couer
,
all
will
be
11.
ounces
,
but
it
should
be
21.
for
to
haue
it
in
triple
proportion
,
vnto
7.
which
representeth
the
waight
of
the
lesser
cuppe
:
So
that
this
fyrste
errour
is
to
little
by
10.
which
you
shal
note
after
7.
in
this
sorte
7.
—
10.
After
you
shall
suppose
some
other
number
,
as
9.
&
make
the
lyke
worke
as
before
,
so
shal
you
find
15.
to
little
,
for
the
seconde
errour
,
which
you
shall
put
behinde
9.
and
then
worke
with
the
reste
as
aboue
is
sayde
,
and
you
shall
fynde
that
the
lesser
cup
wayed
three
ounces
,
and
consequentlye
the
greater
foure
ounces
.
5.
One
man
demaunded
of
another
in
a
morning
what
a
clocke
it
was
,
the
other
made
hym
this
aunswere
,
if
you
doe
adde
(
saith
hee
)
the
¼
of
the
houres
which
be
paste
synce
midnight
,
with
the
⅔
of
the
houres
which
are
to
come
vntyll
noone
,
you
shall
haue
the
iuste
houre
,
that
is
to
say
,
you
shall
knowe
what
a
clocke
it
was
:
Suppose
that
it
was
4.
a
clocke
in
the
morning
,
so
shoulde
there
remaine
8.
vntill
none
:
then
I
take
the
¼
of
4.
which
is
1.
and
the
⅔
of
eight
which
is
5.
⅓
,
and
I
adde
them
togither
,
so
I
finde
6.
⅓
and
supposed
but
4.
therefore
this
fyrst
error
is
to
much
by
2.
⅓
,
which
I
note
after
my
position
thus
,
4.42
.
⅓
:
then
againe
I
suppose
anothee
number
,
that
is
to
saye
9.
so
shoulde
remaine
but
3.
houres
vntill
none
,
I
take
the
¼
of
9.
and
the
⅔
of
3.
which
is
2.
¼
&
2.
these
I
adde
togither
and
they
make
4.
¼
:
but
I
supposed
that
it
was
9.
therefore
the
seconde
errour
is
4.
¾
to
little
which
I
note
behinde
my
position
thus
9.44
.
¾
.
And
then
I
multiplie
crossewyse
,
as
before
is
taught
,
and
bicause
the
sygnes
of
the
errours
are
vnlike
,
that
is
to
saye
,
the
one
to
much
,
and
the
other
to
lyttle
,
therefore
in
thys
worke
I
muste
adde
the
productes
,
and
they
wyll
bee
40.
Lykewyse
I
adde
the
errours
,
and
they
be
7.1/12.
Then
I
diuide
40.
by
7.1/12
,
and
thereof
commeth
5.
houres
11/17
,
and
that
houre
it
was
in
the
mornyng
.
¶
The
15.
Chapter
treateth
of
sportes
,
and
pastime
,
done
by
number
.
IF
you
would
know
the
number
that
anye
man
doth
thinke
or
imagine
in
hys
minde
,
as
though
you
coulde
deuine
.
Bydde
him
triple
the
same
number
,
then
of
the
product
let
him
take
the
,
½
if
the
number
bee
euen
,
or
else
the
greater
halfe
,
if
the
same
be
odde
,
then
bid
hym
triple
againe
the
saide
½
:
after
say
to
him
that
put
away
,
if
he
can
36.27.18
.
or
9.
from
the
last
number
being
tripled
:
that
is
to
saye
,
cause
hym
subtellye
to
put
awaye
9.
as
many
times
as
is
possible
&
keepe
the
number
secretly
:
&
when
he
can
no
more
take
away
9.
than
to
know
if
that
yet
there
remaine
any
number
,
bid
him
abate
3.2
.
or
one
,
if
he
can
:
this
done
,
see
howe
many
times
9.
you
haue
caused
hym
to
abate
,
for
the
which
keepe
you
in
minde
so
many
times
2.
&
if
that
you
know
that
he
had
any
thing
remaining
besyde
the
nines
,
the
same
shall
also
note
vnto
you
1.
Suppose
that
he
thought
6.
which
being
tripled
is
18.
wherof
the
½
is
9.
the
triple
of
that
is
27
:
nowe
cause
him
to
abate
18
,
or
9.
or
27.
and
againe
9
:
but
then
he
will
say
vnto
you
that
he
cannot
,
bid
him
then
abate
3.
or
2.
or
1.
he
will
say
also
that
he
can
not
:
wherfore
considering
that
you
haue
made
him
to
abate
three
times
9.
iustly
,
you
shal
tell
him
that
he
thought
6.
for
3.
times
2.
maketh
6.
If
he
had
thought
5.
the
triple
thereof
is
15.
whereof
the
greater
½
is
8.
the
triple
of
that
maketh
24.
which
conteineth
two
times
9.
thei
are
worth
4.
&
the
remaine
signifieth
1.
the
which
added
togither
make
5.
which
is
the
number
that
he
thought
.
2.
If
in
any
company
,
one
of
them
hath
a
ringe
vpon
his
finger
,
and
you
would
know
by
maner
of
deuining
,
who
hath
the
same
&
vpon
what
finger
&
what
ioint
:
cause
the
persons
to
sit
down
in
order
,
&
kepe
likewise
an
order
of
their
fingers
:
then
seperate
your self
from
them
in
some
certeine
place
,
and
say
vnto
one
of
the
lookers
on
,
that
he
double
the
number
(
marking
the
order
)
of
him
that
hath
the
ring
:
and
vnto
the
double
bid
hym
adde
5.
and
then
cause
him
to
multiplye
this
addition
by
5.
&
vnto
the
product
byd
him
adde
the
number
of
the
finger
of
the
person
which
hath
the
ring
:
be
it
that
the
same
laste
sum
did
amount
to
89.
then
afterwarde
saye
to
him
that
he
put
after
the
same
last
number
toward
hys
right
hande
a
figure
signifiyng
vpon
which
of
the
ioints
he
hath
the
ring
.
As
if
it
bee
vpon
the
thirde
ioint
,
let
him
put
3.
after
89.
and
it
wil
be
893
:
this
done
,
you
shall
aske
him
what
number
hee
kepeth
,
from
the
which
you
shall
abate
250.
&
you
shall
haue
three
fygures
remaining
at
the
least
.
The
fyrste
towarde
your
lefte
hande
shall
signifye
the
number
of
the
person
which
hath
the
ring
.
The
second
or
middle
fygure
shall
represent
the
number
of
the
finger
.
And
the
last
fygure
towarde
your
right
hande
,
shal
betoken
the
number
of
the
ioint
.
As
if
the
number
which
hee
did
keepe
were
893.
from
that
you
shall
abate
250.
and
there
wyll
remaine
643
:
Which
do
note
vnto
you
,
that
the
syxt
person
hath
the
ring
vpon
the
fourth
finger
,
and
vpon
his
thirde
iointe
.
But
note
that
when
you
haue
made
your
substraction
,
if
there
do
remain
a
cipher
in
the
place
of
tennes
,
that
is
to
say
in
the
seconde
place
,
you
must
then
abate
from
that
figure
which
is
in
the
place
of
hundreds
,
that
is
to
say
from
the
figure
which
is
next
your
left
hand
,
&
that
shal
be
worth
ten
tenths
,
signifiyng
the
tenth
finger
:
as
if
there
should
remaine
703.
you
must
say
that
the
syxt
person
(
vpon
his
tenth
finger
&
vpon
hys
thirde
ioint
)
hath
the
ring
.
3.
And
after
the
same
maner
,
if
a
man
do
cast
three
dice
,
you
may
know
the
poinctes
of
euery
one
of
them
,
for
if
you
do
cause
him
to
double
the
poincts
of
one
die
,
&
vnto
the
double
to
adde
5.
&
the
same
sum
to
multiply
by
5.
&
vnto
the
product
adde
the
poincts
of
one
of
the
other
dice
,
and
behinde
the
number
toward
the
right
hand
,
to
put
the
figure
which
signifieth
the
poincts
of
the
last
die
,
&
then
shal
you
aske
him
what
number
he
kepeth
from
the
which
abate
250.
&
there
wil
remaine
3.
figures
,
which
do
note
vnto
you
the
poincts
of
euery
dye
.
4.
Likewise
if
3.
of
your
companions
,
to
say
,
Peter
,
Iames
,
and
Iohn
that
woulde
(
in
your
absence
)
giue
themself
euery
one
a
contrary
name
:
as
for
example
:
Peter
would
be
called
a
king
,
Iames
a
duke
,
and
Iohn
a
countie
:
And
I
would
deuine
which
of
them
is
called
a
king
,
which
the
duke
,
and
which
the
countie
.
Take
24.
stones
,
or
other
peeces
whatsoeuer
,
&
giue
vnto
Peter
1.
vnto
Iames
2.
and
vnto
Iohn
3.
or
otherwise
.
But
marke
well
vnto
which
of
them
you
haue
giuen
1
,
vnto
which
2
,
and
vnto
whom
3.
Then
leauing
the
18.
stones
(
before
them
)
that
are
remaining
,
you
shall
absent
your selfe
from
theire
syght
,
or
else
tourne
your
face
from
them
,
saying
thus
vnto
them
,
whosoeuer
nameth
hymselfe
a
king
:
for
euery
stone
that
I
gaue
him
let
hym
take
one
of
the
residue
,
and
hee
that
nameth
himselfe
a
duke
for
euerye
stone
that
I
gaue
him
let
him
take
2
of
them
that
remain
,
and
he
that
calleth
himself
a
countie
,
for
euery
stone
that
I
gaue
him
let
him
take
4
:
this
being
done
approche
nere
them
,
and
marke
how
many
stones
are
remaining
:
and
know
this
,
that
there
can
not
remaine
any
other
number
,
but
one
of
these
sixe
,
1
,
2
,
3
,
5
,
6
,
7
,
for
the
which
sixe
numbers
we
haue
chosen
to
euery
of
them
a
seuerall
name
,
which
are
these
:
Angeli
,
Beati
,
Qualiter
,
Messias
,
Israel
,
Pietas
:
eche
of
them
containing
three
Vowelles
,
a
,
e
,
i
,
which
doe
shew
the
names
by
order
:
That
is
to
say
,
A
,
1
|
2
|
1
|
2
|
3
|
3
|
2
|
1
|
3
|
3
|
1
|
2
|
3
|
3
|
2
|
1
|
2
|
1
|
a
|
e
|
a
|
e
|
i
|
i
|
e
|
a
|
i
|
i
|
a
|
e
|
i
|
i
|
e
|
a
|
e
|
a
|
1
|
2
|
3
|
5
|
6
|
7
|
A
|
B
|
Q
|
M
|
I
|
P
|
sheweth
which
is
the
kinge
,
E
,
telleth
which
is
the
duke
and
,
I
,
sheweth
which
is
the
county
:
in
followinge
the
order
how
,
&
to
whom
you
haue
giuen
one
stone
,
to
which
2
,
and
to
which
3.
Then
if
there
doe
remaine
but
one
stone
,
the
first
name
Angeli
,
(
by
these
thre
vowels
a
,
e
,
i
,
)
sheweth
that
Peter
is
the
king
,
Iames
the
duke
,
and
Iohn
the
countie
.
And
if
there
doe
remaine
2
stones
,
the
seconde
name
Beati
,
shal
shew
you
by
these
three
vowels
,
e
,
a
,
i
,
that
Peter
is
the
Duke
,
Iames
the
King
,
and
Iohn
the
countie
.
And
so
of
the
other
as
by
this
Table
doth
plainely
appere
.