Numeration
.
NUmeration
is
accounted
the
first
part
of
Arithmetick
,
and
it
is
to
know
how
to
read
a
Summe
of
figures
expressed
in
writing
;
or
to
write
down
any
Summe
to
be
expressed
.
To
the
doing
of
which
there
are
four
things
necessary
.
First
,
to
know
their
number
,
which
is
Nine
.
Secondly
,
their
shapes
,
which
are
1
2
3
4
5
6
7
8
9.
Of
which
the
first
toward
the
left
hand
ever
signifieth
One
,
the
second
Two
,
&c.
Thirdly
,
to
know
the
value
of
their
places
.
Lastly
,
How
their
proper
signification
i●
altered
thereby
.
The
value
of
their
places
is
thus
,
When
two
,
three
,
or
more
figures
stand
in
one
Summe
,
tha●
is
,
without
any
Point
,
Line
or
Comma
betwixt
them
,
as
321
,
that
place
next
the
right
hand
where
the
figure
1
standeth
,
is
called
the
place
of
Unity
,
or
Unities
,
and
the
figure
1
standeth
in
that
place
onely
for
one
,
and
the
figure
2
when
it
is
found
in
that
first
place
,
stands
only
for
two
;
and
the
like
of
the
rest
.
But
in
the
Summe
321
,
above
expressed
,
the
figure
2
is
in
the
second
place
,
and
every
place
conteines
the
value
of
that
place
before
towards
the
right
hand
ten
times
;
and
therefore
the
figure
2
doth
not
signifie
two
,
but
(
in
this
second
place
)
ten
times
two
,
that
is
Twenty
.
And
so
the
figure
3
if
it
had
been
in
that
place
had
signified
ten
times
three
,
that
is
Thirty
;
but
being
here
in
the
third
place
it
signifies
ten
times
thirty
,
that
is
Three
hundred
.
And
so
the
whole
Summe
321
,
is
to
be
read
,
Three
hundred
twenty
and
one
.
Fourthly
,
It
is
hereby
seen
,
how
their
proper
significations
,
which
were
Three
,
Two
and
One
,
are
altered
by
being
thus
placed
,
and
the
Sum
,
which
otherwise
had
been
but
Six
,
is
Three
hundred
twenty
one
,
as
before
.
In
like
sort
,
if
their
had
been
more
places
,
as
seven
,
the
value
is
quite
through
increased
ten
times
by
being
a
place
behinde
towards
the
left
hand
;
as
in
the
Summe
1̇111̇111̇
,
The
figure
1
in
the
second
place
stands
for
ten
times
one
(
that
is
ten
,
)
in
the
third
for
ten
times
ten
(
which
is
one
hundred
,
)
in
the
fourth
for
ten
hundred
,
(
which
is
called
one
thousand
,
)
in
the
fifth
for
ten
thousand
,
in
the
sixth
for
ten
times
ten
thousand
(
which
is
one
hundred
thousand
)
in
the
last
(
here
the
seventh
)
place
for
ten
hundred
thousand
,
which
is
called
a
Milion
:
and
so
on
,
if
there
were
more
places
,
observing
the
same
order
.
Now
to
read
this
readily
,
make
a
prick
over
the
place
of
Unity
,
another
over
the
third
from
it
,
and
over
every
third
still
towards
the
left
hand
,
for
so
those
points
will
be
over
the
places
of
Unites
,
Thousands
,
and
Milions
;
and
so
beginning
at
the
last
,
that
is
,
at
the
left
hand
,
read
one
Milion
,
and
because
the
three
following
towards
the
right
signifie
properly
one
hundred
and
eleven
,
but
the
prick
belonging
to
them
is
in
the
place
of
thousands
,
call
them
one
hundred
and
eleven
thousand
,
and
three
remaining
being
under
the
point
over
Unity
,
signifie
onely
one
hundred
and
eleven
;
and
all
three
points
read
together
in
one
summe
is
,
One
Milion
,
one
hundred
and
eleven
Thousand
,
one
hundred
and
eleven
.
In
like
manner
,
if
this
number
73598624
were
given
to
be
read
(
according
to
former
directions
)
make
a
prick
over
every
third
figure
,
beginning
with
the
first
figure
towards
the
right
hand
(
which
is
the
place
of
Unity
)
and
then
will
your
number
stand
thus
.
73̇598̇624̇
Then
for
the
ready
reading
thereof
(
because
the
third
prick
signifieth
Millions
)
call
all
the
figures
towards
the
left
hand
,
standing
from
that
prick
,
Milions
,
which
in
this
example
are
7
and
3
,
so
then
this
number
conteins
73
Millions
,
598
Thousand
,
624
which
in
words
at
length
we
read
Seventy
three
Millions
,
five
hundred
ninety
eight
thousand
,
six
hundred
twenty
four
.
Let
thus
much
suffice
concerning
the
planing
of
large
numbers
,
for
the
ready
reading
of
them
,
only
take
these
foure
Tables
following
,
for
illustration
of
what
hath
been
hitherto
delivered
in
words
,
the
very
sight
whereof
is
better
then
a
whole
Chapter
of
information
.
The
first
Table
is
thus
to
be
read
]
One
in
the
first
place
signifies
One.
One
in
the
second
place
signifies
Ten.
One
in
the
third
place
signifies
a
hundred
,
&c.
as
in
the
Table
.
The
third
Table
is
only
certain
rowes
of
figures
set
together
and
orderly
disposed
,
having
the
signification
or
reading
of
the
same
numbers
in
words
at
length
to
them
annexed
,
and
is
only
inserted
for
the
better
satisfaction
of
such
as
shall
doubt
whether
they
perfectly
understand
what
hath
been
before
taught
.
The
fourth
Table
is
much
like
the
second
,
only
it
consisteth
but
of
one
number
and
extends
three
places
farther
then
the
greatest
number
in
the
second
Table
doth
:
viz.
to
twelve
places
which
figures
are
thus
to
be
read
.
736
Millions
of
Millions
,
842
Millions
,
708
Thousand
,
645.
(
1
Table
)
One
in
the
|
first
|
place
signifies
|
1
one
|
One
in
the
|
second
|
place
signifies
|
10
ten
|
One
in
the
|
third
|
place
signifies
|
100
a
hundred
|
One
in
the
|
fourth
|
place
signifies
|
1000
a
thousand
|
One
in
the
|
fifth
|
place
signifies
|
10000
ten
thousand
|
One
in
the
|
sixth
|
place
signifies
|
100000
a
hundred
thousand
|
One
in
the
|
seventh
|
place
signifies
|
1000000
a
Million
|
One
in
the
|
eigth
|
place
signifies
|
10000000
ten
Millions
|
One
in
the
|
ninth
|
place
signifies
|
100000000
a
hundred
Milion
|
(
2
Table
)
8
|
54
|
762
|
3.483
|
97.621
|
243.794
|
8.749.807
|
57.316.248
|
357.846.903
|
Millions
thousands
hundreds
|
(
3
Table
)
number
of
places
|
1
|
8
(
eight
,
|
number
of
places
|
2
|
54
(
fifty
four
,
|
number
of
places
|
3
|
762
(
seven
hundred
sixty
two
,
|
number
of
places
|
4
|
3483
(
three
thousand
,
four
hundred
eighty
three
|
number
of
places
|
5
|
97621
(
ninety
seven
thousand
,
six
hundred
twenty
one
|
number
of
places
|
6
|
243794
(
two
hundred
43
thousand
7
hundred
94
|
number
of
places
|
7
|
8749807
(
eight
milions
,
749
thous
.
8
hundred
and
7
|
number
of
places
|
8
|
57316248
(
fifty
seven
mil.
3
hund.
and
16
thous
.
248
|
number
of
places
|
9
|
35784698
(
three
hundred
fifty
seven
milions
,
eight
(
hundred
46
thous
.
9
hund.
and
3.
|
(
4
Table
)
7
|
hundred
|
milions
of
milions
|
3
|
ten
|
milions
of
milions
|
6
|
one
|
milions
of
milions
|
8
|
hundred
|
milions
|
4
|
ten
|
milions
|
2
|
one
|
milions
|
7
|
hundred
|
thousands
|
0
|
ten
|
thousands
|
8
|
one
|
thousands
|
6
|
hundred
|
hundreds
|
4
|
ten
|
hundreds
|
5
|
one
|
hundreds
|
Addition
.
ADdition
is
the
collecting
or
gathering
together
of
two
or
more
sums
,
either
of
one
or
of
diverse
denominations
,
into
one
sum
,
which
is
called
the
(
Aggregate
)
(
Total
)
or
Grosse
sum
.
)
In
addition
of
numbers
of
one
denomination
,
the
order
is
to
set
the
numbers
to
be
added
one
directly
under
the
other
,
that
is
to
say
Unites
under
Unites
,
Tens
under
Tens
,
Hundreds
under
Hundreds
,
&c.
The
Rule
.
Having
placed
your
numbers
to
be
added
in
due
order
,
one
under
another
,
draw
a
line
under
them
,
and
begin
at
the
lowermost
figure
towards
your
right
hand
,
and
adde
that
to
the
next
figure
above
,
and
the
summe
of
them
to
the
next
figure
above
that
,
proceeding
in
this
order
till
you
have
added
the
whole
line
together
,
which
when
you
have
done
,
consider
how
many
tens
are
contained
in
that
line
,
and
for
every
ten
keep
one
Vnite
in
your
mind
,
to
be
added
to
the
next
row
,
but
if
there
be
any
odde
digits
,
you
must
set
them
beneath
the
stroake
,
just
under
the
line
you
added
together
,
having
thus
finished
the
addition
of
one
line
proceed
to
the
next
,
and
from
thence
to
the
third
,
and
so
forward
,
be
there
never
so
many
.
The
Examples
following
will
make
this
plain
,
Example
1.
Thousands
Hundreds
Tens
Unites
|
7832
|
5609
|
376
|
8547
|
22364
|
Let
the
numbers
given
to
be
added
together
be
7832
,
5609
,
376
,
8547
,
having
thus
placed
them
in
order
one
under
another
as
in
the
Margine
is
done
,
draw
a
line
under
them
,
then
begin
your
addition
,
at
the
lowermost
figure
towards
your
right
hand
,
saying
,
7
and
6
is
13
,
and
9
is
22
,
and
2
is
24
now
because
in
24
,
there
is
two
tens
,
and
4
remaining
,
I
place
the
4
under
the
line
,
and
carry
the
two
zens
to
the
next
row
,
saying
,
2
which
I
carryed
and
4
makes
6
,
and
7
makes
13
,
and
3
makes
16
,
in
which
row
there
is
but
one
ten
conteined
and
6
remaining
,
which
6
I
set
under
the
line
,
and
carry
the
ten
to
the
next
row
of
hundreds
,
saying
,
1
that
I
carryed
and
5
makes
6
,
and
3
makes
9
,
and
6
makes
15
,
and
8
makes
23
,
in
which
23
,
ten
is
conteined
two
times
,
and
3
remaining
,
the
three
I
place
under
the
line
,
and
carry
the
two
tens
to
the
next
row
,
saying
2
which
I
carryed
and
8
makes
10
,
and
5
makes
15
,
and
7
makes
22
,
in
which
ten
is
conteined
two
times
,
and
2
remaining
,
which
2
I
set
under
the
line
,
and
because
there
is
never
another
row
to
be
added
(
to
which
I
should
carry
the
two
tens
)
I
therefore
set
it
down
also
under
the
line
towards
the
left
hand
,
as
you
see
done
in
the
margine
,
so
the
total
or
grosse
sum
of
these
numbers
being
added
together
is
22364.
Example
,
2.
Apple
trees
|
137
|
Pear
trees
|
76
|
Cherry
trees
|
107
|
Plum
trees
|
36
|
Trees
in
all
|
355
|
A
man
hath
in
his
Orchard
136
Apple
trees
,
76
Pear
trees
,
107
Cherry
trees
,
and
36
Plum
trees
,
and
he
desires
readily
to
know
how
many
trees
he
hath
in
all
,
place
your
uumbers
one
under
another
as
in
the
margine
,
and
then
begin
to
adde
them
together
,
at
your
right
hand
,
saying
,
6
and
7
,
make
13
,
and
6
make
19
,
and
6
make
25
,
place
5
under
the
line
,
and
carry
2
to
the
next
row
,
saying
2
and
3
is
5
,
and
7
is
12
,
and
3
is
15
,
place
5
under
the
line
,
and
carry
1
to
the
next
row
,
saying
1
and
1
is
2
,
and
1
is
3
which
3
I
set
under
the
line
,
and
because
there
was
no
tens
conteined
in
that
line
therefore
the
totall
is
355
,
and
so
many
Trees
are
in
the
Orchard
.
Other
Examples
for
Practice
.
Addition
of
numbers
of
diverse
Denominations
.
1
Addition
of
English
Money
The
most
usual
Coyns
used
in
England
are
Pounds
,
Shillings
,
Pence
,
and
Farthings
,
of
which
Coyns
-
4
Farthings
make
1
Peny
thus
charactred
d.
-
12
Pence
make
1
Shilling
thus
charactred
s.
-
20
Shillings
make
1
Pound
.
thus
charactred
li.
for
a
farthing
we
use
q.
THE
RVLE
.
In
the
addition
of
numbers
of
divers
denominations
this
order
is
to
be
observed
,
viz.
Place
all
numbers
of
the
same
denomination
one
directly
under
another
,
as
Pounds
under
pounds
,
shillings
under
shillings
,
pence
under
pence
,
and
farthings
under
farthings
.
Then
draw
a
line
under
them
,
and
begin
your
Addition
with
the
least
denomination
first
,
observing
how
many
times
the
next
greater
denomination
is
contained
in
the
least
,
and
for
every
time
carry
one
unite
to
the
next
denomination
,
as
before
you
did
the
tens
,
setting
down
the
remainder
if
any
be
,
then
adding
the
next
denomination
together
,
take
notice
how
many
times
the
next
greater
denomination
is
contained
in
that
lesser
,
carrying
for
every
time
one
to
the
next
,
thus
proceeding
till
you
have
gone
over
all
the
denominations
,
be
they
never
so
many
.
Example
1.
li.
|
s.
|
d.
|
q.
|
37
|
16
|
9.
|
3.
|
21
|
09.
|
8.
|
1
|
13
|
12
|
9
|
2
|
72
|
19
|
3
|
2
|
Let
the
numbers
to
be
added
together
be
37
li.
16
s.
9
d.
3
q.
21
li.
9
s.
8
d.
1
q.
13
li.
12
s.
9
d.
2
q.
Place
the
numbers
as
in
the
margine
,
draw
a
line
under
them
,
and
begin
with
the
least
denomination
(
which
in
this
example
is
farthings
,
)
first
,
saying
,
2
q.
and
1
q.
is
3
q.
and
3
q.
is
6
q.
which
is
one
peny
and
2
q.
remaining
,
which
2
q.
I
place
under
the
line
,
and
carry
the
one
peny
to
the
next
row
which
is
the
place
of
pence
,
saying
,
one
peny
and
9
d.
is
10
d.
and
8
d.
is
18
d.
which
is
1
s.
and
6
d.
(
Now
against
the
8
I
make
a
prick
with
my
pen
for
my
better
remembrance
to
signifie
that
there
is
one
shilling
to
be
carried
to
the
place
of
shillings
,
)
then
go
on
and
say
6
d.
and
9
d.
is
15
d.
which
is
1
s.
and
3
d.
therefore
against
9.
I
make
a
prick
with
my
pen
,
and
(
because
that
is
the
last
number
)
I
set
down
the
odde
3
d.
under
the
place
of
pence
,
and
(
because
I
finde
two
pricks
in
the
line
of
pence
,
therefore
)
I
carry
2
s.
to
the
place
of
shillings
saying
2
s.
which
I
carryed
,
and
12
s.
is
14
s.
and
9
s.
is
23
s.
which
is
one
pound
,
and
3
s.
remaining
,
I
make
a
prick
against
9
,
and
going
on
,
say
3
s.
and
16
s.
is
19
s.
which
(
being
there
is
no
more
numbers
to
be
added
,
and
being
also
lesse
then
20
s.
)
I
set
under
the
line
,
and
finding
one
prick
in
the
line
of
shillings
,
I
therefore
carry
one
to
the
place
of
pounds
,
saying
,
one
which
I
carryed
and
3
is
4
,
and
1
is
5
,
and
7
is
12.
set
down
the
2
under
the
line
(
as
in
addition
of
numbers
of
one
denomination
)
and
carry
one
to
the
next
row
,
saying
one
that
I
carried
and
1
is
2
,
and
2
is
4
,
and
3
is
7
,
which
being
the
last
I
set
down
,
and
so
the
total
or
grosse
sum
is
72
li.
19
s.
3
d.
2
q.
Example
2
li.
|
s.
|
d.
|
29
|
16.
|
8.
|
32
|
17.
|
9.
|
81
|
13
|
11
|
144
|
08
|
4
|
Let
the
numbers
to
be
added
be
29
li.
16
s.
8
d.
32
li.
17
s.
9
d.
81
li.
13
s.
11
d.
and
let
it
be
required
to
find
the
totall
or
grosse
sum
.
Here
in
this
Example
the
least
denomination
is
pence
,
therefore
I
begin
with
them
,
and
say
,
11
d.
and
9
d.
is
20
d.
which
is
1
s.
and
8
d.
make
a
prick
against
the
9
and
say
8
d.
and
8
d.
is
16
d.
that
is
1
s.
and
4
d.
make
a
prick
against
the
8
and
set
down
the
odde
4
d.
then
(
because
there
are
two
pricks
in
the
line
of
pence
)
you
must
carry
2
s.
to
the
place
of
shillings
,
saying
2
s.
which
I
carry
and
13
s.
is
15
s.
and
17
s.
is
32
s.
which
is
1
l.
12
s.
make
a
prick
agianst
17
,
and
say
12
s.
and
16
s.
is
28
s.
make
a
prick
against
16
,
and
(
because
there
is
no
more
numbers
to
be
added
)
set
down
the
odde
8
s.
under
shillings
,
and
(
being
there
is
two
pricks
in
the
line
of
shillings
)
carry
2
to
the
place
of
pounds
,
saying
2
and
1
is
3
,
and
2
is
5
,
and
9
is
14
,
set
down
4
and
carry
1
to
the
next
line
,
and
say
1
and
8
is
9
,
and
3
is
12
,
and
2
is
14
,
which
(
because
it
is
the
last
)
you
must
set
down
,
so
is
the
totall
or
grosse
sum
144
li.
8
s.
4
d.
Other
Examples
for
Practice
.
li.
|
s.
|
d.
|
q.
|
29
|
18.
|
7
|
3
|
63
|
11.
|
2.
|
1.
|
129
|
4
|
0
|
2
|
3
|
7
|
10
|
1
|
226
|
1
|
8
|
3
|
li.
|
s.
|
d.
|
36
|
2
|
8.
|
29
|
0
|
2
|
31
|
16.
|
9.
|
6
|
2
|
5
|
103
|
2
|
0
|
2
Addition
of
Troy
Weight
.
Troy
weight
is
a
Weight
used
in
England
,
by
the
which
is
weighed
,
Bread
,
Gold
,
Silver
,
Pearl
,
&c.
the
most
usual
denominations
of
which
weight
are
,
Pounds
,
Ounces
Peny
weights
,
and
grains
,
of
which
-
24
Grains
make
1
Peny
weig
.
thus
charactred
pw
.
-
20
Peny
weight
make
1
Ounce
thus
charactred
ou
.
-
12
Ounces
make
1
Pound
thus
charactred
lib
for
a
grain
we
write
gr.
The
Addition
of
Troy
weight
(
and
consequently
of
any
other
weight
or
measure
whatsoever
either
Domestique
or
Forreign
)
differeth
nothing
at
all
from
the
addition
of
Coine
last
taught
,
if
the
affinity
of
one
denomination
to
an
other
be
first
known
,
for
whereas
in
money
because
1●
d.
make
1
s.
you
therefore
observe
how
many
twelves
there
are
in
the
addition
of
your
pence
,
and
for
every
12
you
adde
one
shilling
to
the
place
of
shillings
,
so
in
the
addition
of
Troy
weight
,
knowing
that
24
gr.
make
one
peny
weight
,
you
must
therefore
in
the
addition
of
Grains
of
Troy
weight
observe
how
many
times
24
you
finde
in
your
line
of
Grains
,
and
for
every
24
,
carry
one
to
the
place
of
peny
weights
,
likewise
in
the
addition
of
peny
weights
Troy
,
you
must
consider
how
many
times
20
is
contained
in
your
line
,
and
for
every
20
carry
one
to
the
place
of
ounces
,
(
because
20
peny
weights
make
an
ounce
.
)
Also
in
the
addition
of
Ounces
Troy
,
you
must
observe
how
many
times
12
you
finde
in
your
line
of
ounces
,
and
for
every
12
carry
one
to
the
place
of
pounds
then
lastly
adde
your
pounds
together
,
as
numbers
of
one
denomination
.
Example
.
li.
|
ou
.
|
pw
.
|
gr.
|
7
|
11.
|
13
|
19
|
6
|
07
|
16.
|
19.
|
3
|
07
|
09
|
06
|
18
|
02
|
19
|
20
|
Let
the
numbers
to
be
added
together
be
7
li.
11
ou.
13
pw.
19
gr.
6
li
7
ou.
16
pw.
19
gr.
3
li.
7
ou.
9
pw.
6
gr.
Place
your
numbers
as
in
addition
of
money
,
each
under
other
according
to
their
respective
denominations
,
as
in
the
margine
then
draw
a
line
under
them
and
begin
your
Addition
with
the
least
denomination
first
,
viz.
grains
,
saying
6
gr.
and
19
gr.
is
25
gr.
which
is
one
peny
weight
and
one
grain
,
make
a
prick
against
19
,
and
carry
the
odde
grain
to
the
number
above
,
saying
1
gr.
and
19
gr.
is
20
gr.
which
(
because
it
is
lesse
then
one
peny
weight
)
I
set
under
the
line
,
then
finding
one
prick
in
the
line
of
grains
,
I
(
therefore
)
carry
one
to
the
place
of
peny
weights
,
saying
1
and
9
is
10
,
and
16
is
26
,
which
is
one
ounce
,
and
6
pw
.
make
a
prick
against
16
,
and
say
6
and
13
is
19
,
which
(
being
lesse
then
an
ounce
)
set
under
the
line
,
then
for
the
one
prick
,
carry
1
to
the
place
of
ounces
,
saying
1
and
7
is
8
,
and
7
is
15
,
which
is
one
pound
and
3
ou
.
make
a
prick
at
7
,
and
say
3
and
11
is
14
,
which
is
one
pound
and
2
ounces
,
set
down
the
2
ounces
,
and
for
the
two
pricks
carry
2
pounds
to
the
place
of
pounds
,
saying
2
and
3
is
5
,
and
6
is
11
,
and
7
is
18
,
which
set
under
the
place
of
pounds
,
so
is
your
addition
ended
,
and
the
sum
is
18
li.
2
ou.
19
pw.
20
gr.
Other
Examples
for
Practice
.
i.
|
ou
.
|
pw
.
|
gr.
|
32
|
9.
|
12
|
16
|
17
|
11.
|
6
|
9.
|
34
|
8.
|
15.
|
10
|
8
|
10
|
4
|
7
|
94
|
3
|
18
|
18
|
li.
|
ou
.
|
pw
.
|
gr.
|
0
|
10
|
17.
|
11
|
0
|
6.
|
0
|
5
|
0
|
0
|
16
|
8.
|
0
|
5
|
2
|
19
|
1
|
10
|
16
|
19
|
3
Addition
of
Avoirdupois
little
Weight
.
There
is
another
kind
of
weight
most
commonly
used
in
England
called
Avoirdupois
little
weight
,
by
which
is
weighed
all
sorts
of
wares
or
merchandise
Garbable
,
as
Suger
,
Pepper
,
Cloves
,
Mace
,
&c.
This
weight
is
commonly
divided
into
these
denominations
,
Pounds
,
Ounces
,
and
Drams
,
of
which
-
16
Drams
make
1
Ounce
thus
charact.
cu.
-
16
Ounces
make
1
Pound
thus
charact.
li.
for
a
dram
we
write
dr.
In
the
addition
of
Avoirdupois
weight
,
you
must
observe
the
very
same
method
and
order
as
in
Money
and
Troy
weight
,
having
due
respect
to
the
quantity
of
the
denomination
,
as
in
the
addition
of
drams
to
make
a
prick
at
every
16
,
setting
down
the
remainder
and
for
every
prick
carrying
a
unite
to
the
next
place
.
The
preceding
rules
being
so
copious
in
this
particular
I
shall
forbear
to
make
any
verball
illustration
,
but
only
give
you
some
examples
ready
wrought
,
together
with
the
most
usual
parts
into
which
the
Weights
and
Measures
now
used
in
England
are
divided
into
:
which
to
the
ingenious
will
be
of
most
validity
.
Examples
of
Addition
of
Avoirdupois
little
weight
.
li.
|
ou
.
|
dr.
|
12
|
11.
|
09
|
76
|
05
|
12.
|
32
|
10.
|
00
|
91
|
07.
|
13.
|
32
|
13
|
07
|
246
|
00
|
09
|
li.
|
ou
.
|
dr.
|
06
|
13.
|
07.
|
05
|
09.
|
12
|
06
|
03
|
09.
|
10
|
00
|
00
|
05
|
07
|
09
|
34
|
02
|
05
|
4
Addition
of
Avoirdupois
great
weight
.
There
is
a
weight
commonly
used
in
England
,
by
which
is
weighed
all
commodities
that
are
sold
by
the
hundred
,
as
Corents
,
Wool
,
Flesh
,
Butter
,
Cheese
,
and
the
like
,
the
which
hundred
weight
containeth
112
pounds
,
and
the
hundred
weight
is
divided
into
quarters
,
pounds
,
and
ounces
,
so
that
-
16
Ounces
makes
1
Pound
thus
chaeactred
li.
-
28
Pounds
makes
1
quarter
of
a
C.
thus
chaeactred
qr.
-
4
quarters
makes
1
Hundred
weig
.
thus
chaeactred
C.
for
an
Ounce
we
write
oz.
Examples
of
Addition
of
Avoirdupois
great
Weight
.
C.
|
qr.
|
li.
|
ou
.
|
37
|
03.
|
21
|
12.
|
09
|
01
|
06
|
03
|
33
|
02
|
20.
|
00
|
10
|
00
|
00
|
00
|
12
|
03.
|
07
|
03
|
103
|
02
|
27
|
02
|
C.
|
qr.
|
li.
|
ou
.
|
05
|
01.
|
00
|
07
|
03
|
02
|
18
|
06.
|
00
|
01
|
06
|
08
|
11
|
03.
|
04
|
00
|
06
|
01
|
10
|
05
|
17
|
01
|
11
|
10
|
I
might
further
proceed
to
shew
you
Examples
of
addition
of
common
English
measures
,
viz.
of
long
measures
,
Liquid
measures
,
and
dry
measures
,
as
also
of
Time
,
Motion
,
&c.
but
the
preceding
Examples
being
of
sufficient
extent
,
I
shall
forbear
to
trouble
either
my selfe
or
the
Reader
with
that
which
I
conceive
superfluous
:
Only
,
before
I
leave
Addition
,
I
will
give
you
a
brief
view
of
the
most
usual
measures
in
England
,
which
take
as
followeth
.
And
1
Of
Liquid
Measures
.
Liquid
measures
are
those
in
which
all
sorts
of
Liquid
substances
are
measred
,
of
which
(
according
to
the
Statute
of
12
Hen.
7.
chap.
5.
)
a
Pinte
is
the
least
,
from
which
the
greater
Liquid
measures
are
deduced
,
according
as
is
expressed
in
the
Table
following
.
-
2
Pintes
make
1
Quart
-
2
Quarts
make
1
Pottle
-
2
Pottles
make
1
Gallon
-
8
Gallons
make
1
Firkin
of
Ale
,
Sope
,
or
-
9
Gallons
make
1
Firkin
of
Beer
(
Herings
-
10
½
Gallons
make
1
Firkin
of
Salmon
or
Eels
-
2
Firkins
make
1
Kilderkin
-
2
Kilderkins
make
1
Barrel
-
42
Gallons
make
1
Tierce
of
Wine
-
63
Gallons
make
1
Hogshed
-
2
Hogsheads
make
1
Pipe
or
But
-
2
Pipes
or
Buts
make
1
Tun
of
Wine
.
2
Of
Dry
Measures
.
Dry
Measures
are
these
in
which
all
kind
of
dry
substances
are
measured
,
as
Corn
,
Salt
,
Cole
,
Sand
,
&c.
of
which
a
pinte
is
the
least
.
-
2
Pintes
make
1
quart
-
2
quarts
make
1
Pottle
-
2
Pottles
make
1
Gallon
-
2
Gallons
.
make
1
Peck
-
4
Pecks
make
1
Bushel
Land
measure
-
5
Pecks
make
1
Bushel
Water
measure
-
8
Bushels
make
1
quarter
-
4
quarters
make
1
Chaldron
-
5
quarters
make
1
Wey
.
3
Of
Long
Measures
.
Long
Measure
is
that
by
which
is
measured
Cloth
,
Land
,
Board
,
Glasse
,
Pavement
,
Tapestry
,
&
c●
of
which
measures
(
according
to
the
Statute
of
33
Ed.
1.
and
25
El.
)
a
Barley
corn
is
the
least
.
So
that
-
3
Barley
Corns
make
1
Inch
-
12
Inches
make
1
Foot
-
3
Foot
make
1
Yard
-
3
Foot
9
inches
make
1
Ell
-
6
Foot
make
1
Fadome
-
5
½
Yards
,
or
16
½
Foot
make
1
Pole
or
Perch
-
40
Perches
make
1
Furlong
-
8
Furlongs
make
1
English
mile
.
4
Of
Time
.
Time
consisteth
of
Years
,
Moneths
,
Weeks
,
Dayes
,
Houres
,
and
Minutes
.
So
that
-
60
Minutes
make
1
Hour
-
24
Hours
make
1
Day
natural
-
7
Dayes
make
1
Week
-
4
Weeks
make
1
Moneth
of
28
dayes
-
13
Moneths
,
one
day
6
houres
make
1
Year
.
5
Of
Apothecaries
Weights
.
The
weights
used
by
Apothecaries
are
Grains
,
Scruples
,
Drams
,
and
Ounces
,
of
which
-
20
Grains
make
1
Scruple
thus
charact.
℥
-
3
Scruples
make
1
Dram
thus
charact.
ʒ
-
8
Drams
make
1
Ounce
thus
charact.
℈
-
12
Ounces
make
1
Pound
thus
charact.
li.
By
help
of
these
Tables
,
and
the
rules
and
cautions
before
expressed
,
any
man
may
make
addition
of
any
of
the
abovesaid
measures
one
with
another
,
and
therefore
I
shall
forbear
to
illustrate
them
by
Examples
,
but
leave
them
to
every
mans
own
practice
,
and
thus
I
conclude
Addition
.
The
Proof
of
Addition
.
Having
placed
your
numbers
in
order
,
and
added
them
together
,
and
set
the
Total
under
the
line
,
cut
off
the
upper
number
by
drawing
a
line
with
your
pen
betwixt
that
and
the
others
,
then
adde
all
the
numbers
together
except
the
uppermost
,
and
set
the
Total
of
them
under
the
Total
before
found
,
then
adde
this
last
Total
,
and
the
first
number
which
you
cut
off
with
your
pen
together
,
and
if
the
sum
of
those
two
numbers
be
equal
with
your
Total
sum
first
found
,
then
is
your
work
right
,
otherwise
not
.
Example
.
In
the
first
example
of
whole
numbers
,
the
sums
to
be
added
were
7833
,
5609
,
376
,
and
8547
,
these
numbers
placed
in
due
order
and
added
together
,
the
total
or
grosse
sum
of
them
is
22364
,
now
to
prove
whether
this
Total
be
true
or
not
,
I
cut
off
the
uppermost
number
,
(
to
wit
7832
,
)
with
a
dash
of
the
pen
,
and
I
adde
the
other
three
numbers
together
,
namely
,
5609
,
376
,
and
8547
,
and
the
Total
of
them
is
14732
,
which
number
being
added
to
7832
(
the
number
cut
off
)
the
sum
of
them
is
22364
exactly
agreeing
with
the
Total
first
found
,
cleerly
evidenceth
that
the
addition
was
truly
performed
;
but
if
they
had
disagreed
then
the
work
had
been
erroneous
.
The
like
course
must
be
taken
for
the
proof
of
those
sums
which
have
different
denominations
as
in
Money
and
Weight
,
as
by
the
examples
following
will
appear
.
Other
Examples
proved
.
1
Example
of
Money
.
li.
|
s.
|
d.
|
q.
|
37
|
16
|
9
|
3
|
21
|
9
|
8.
|
1
|
13
|
12
|
9
|
2
|
1
Total
72
|
19
|
3
|
2
|
2
Total
35
|
2
|
5
|
3
|
Proof
72
|
19
|
3
|
2
|
li.
|
ou
.
|
pw
.
|
gr.
|
32
|
9
|
12
|
16
|
17
|
11
|
6
|
9
|
34
|
8
|
15
|
10
|
8
|
10
|
4
|
7
|
94
|
3
|
18
|
18
|
61
|
6
|
6
|
2
|
94
|
3
|
18
|
18
|
There
are
are
other
wayes
to
prove
Addition
,
by
casting
away
of
all
the
nines
in
numbers
of
one
denomination
,
and
of
all
the
twelves
,
twenties
,
and
nines
,
in
pounds
shillings
and
pence
,
&c.
but
this
as
the
most
certain
and
easie
I
embrace
,
and
thus
much
for
Addition
and
the
proofe
thereof
.
Subtraction
.
SUbtrrction
is
the
taking
of
one
or
more
small
sums
out
of
a
greater
,
as
7
s.
out
of
12
s.
or
37
li.
out
of
100
li.
or
137
foot
out
of
983
foot
,
and
the
like
.
As
in
Addition
the
sums
to
be
Added
may
be
either
of
one
or
of
diverse
denminations
,
so
likewise
they
may
be
in
Subtraction
,
and
the
manner
of
placing
them
is
the
same
,
for
you
must
set
Unites
under
Unites
.
Tens
under
Tens
,
Hundreds
under
Hundreds
,
&c.
Example
1
number
given
|
986
|
number
to
be
subtracted
|
234
|
remainder
|
752
|
Of
Subtraction
of
numbers
of
one
denomination
,
let
it
be
required
to
subtract
234
,
out
of
986
,
place
the
numbers
one
unde
the
other
as
you
see
done
in
the
Margine
,
draw
a
line
under
them
and
begin
with
the
first
figure
towards
your
right
hand
,
which
is
4
,
saying
take
4
from
6
,
and
there
remains
2
,
place
2
under
the
line
,
and
go
to
the
next
figure
which
is
3
,
saying
,
take
3
from
8
,
and
there
remains
5
,
place
5
under
the
line
,
and
go
to
the
next
figure
which
is
2
,
saying
take
2
out
of
9
and
there
remains
7
,
place
7
under
the
line
,
and
your
subtraction
is
ended
,
and
it
is
evident
by
the
work
,
that
if
you
take
234
out
of
986
there
will
remain
75●
,
which
you
may
thus
prove
,
for
if
you
adde
the
234
to
752
,
you
shall
find
the
sum
of
that
addition
to
be
986
,
which
is
equall
to
the
whole
sum
from
which
234
was
subtracted
,
Example
.
2.
Let
it
be
required
to
subtact
2976
out
of
96527
,
place
the
numbers
one
under
another
as
in
the
margine
you
fee
done
,
then
draw
a
line
under
them
,
and
beginning
with
the
first
figure
towards
your
left
hand
,
say
take
6
out
of
7
,
and
there
remains
1
,
place
1
under
the
line
and
prodeed
to
the
next
figure
,
saying
7
out
of
2
I
can
not
(
wherefore
you
must
alwayes
adde
10
to
the
number
above
which
in
this
example
is
2
,
and
it
makes
it
12
,
)
therefore
take
7
out
of
12
,
and
there
remains
5
,
place
5
under
the
line
,
and
(
because
you
added
10
to
the
2
to
make
it
12
,
you
must
)
carry
a
unite
to
the
next
figure
,
saying
,
one
which
I
carryed
and
9
is
10
,
take
ten
out
of
5
,
which
I
cannot
,
therefore
I
must
adde
10
to
5
and
it
makes
15
,
and
say
10
out
of
15
,
and
there
remains
5
,
place
5
under
the
line
,
and
because
you
added
10
to
5
to
make
it
15
,
you
must
therefore
carry
a
unite
to
the
next
figure
saying
,
one
which
I
carryed
,
and
2
is
3
,
take
3
out
of
6
and
there
remains
3
,
place
3
under
the
line
,
and
because
there
is
no
more
figures
to
be
subtracted
from
the
number
above
,
you
must
,
say
nothing
fom
9
and
there
remains
9
,
set
the
9
under
the
line
,
and
your
Subtraction
is
ended
.
Other
Examples
for
Practice
.
|
li.
|
reams
of
|
Paper
|
sheep
|
|
Lent
|
5762
|
bought
|
9765
|
from
|
1000
|
Paid
|
378
|
sold
|
6529
|
take
|
394
|
Rests
to
pay
|
5384
|
unsold
|
3236
|
remains
|
606
|
Subtraction
of
Numbers
of
diverse
Denominations
,
1
Of
English
Money
.
In
Subtraction
of
numbers
of
diverse
denominations
,
you
must
observe
the
same
order
as
in
Addition
,
namely
,
to
place
every
number
in
due
order
,
with
respect
to
its
denomination
,
as
pounds
under
pounds
,
shillings
under
shillings
,
&c.
the
greater
number
alwayes
uppermost
,
and
drawing
a
line
under
them
,
begin
with
the
least
denomination
first
,
subtracting
it
from
the
line
above
,
and
setting
the
remainder
under
the
line
as
in
whole
numbers
,
but
if
the
pence
or
shillings
in
the
upper
row
be
smaller
then
those
in
the
neather
row
,
you
must
adde
12
d.
or
20
s.
to
the
smaller
number
,
that
so
subtraction
may
be
made
,
as
by
the
examples
following
will
appear
.
Example
1.
|
li.
|
s.
|
d.
|
Lent
|
269
|
18
|
10
|
Paid
|
38
|
12
|
8
|
rests
|
231
|
6
|
2
|
Let
it
be
required
to
subtract
38
li.
12
s.
8
d.
out
of
269
l.
18
s.
10
d.
Place
your
numbers
as
in
the
margine
,
then
beginning
with
the
least
denomination
first
,
(
which
in
this
Example
is
pence
)
say
8
d.
from
10
d.
,
and
there
remains
2
d.
set
the
2
d.
under
the
line
,
and
proceed
to
the
next
denomination
which
is
shillings
,
saying
take
12
s.
out
of
18
s.
and
there
remains
6
s.
place
6
s
,
under
the
line
,
and
go
to
the
pounds
,
saying
8
out
of
9
and
there
remains
1
,
place
1
under
the
line
,
and
say
3
out
of
6
,
and
there
remains
3
,
then
(
because
there
is
no
more
figures
to
be
subtracted
)
say
nothing
out
of
2
and
there
remains
2
,
which
set
under
the
line
,
so
is
your
subtraction
ended
,
and
the
remainder
is
231
li.
6
s.
2
d.
Example
2.
|
li.
|
s.
|
p.
|
Lent
|
9320
|
10
|
07
|
Paid
|
2628
|
16
|
10
|
Rests
|
6691
|
11
|
8
|
Let
it
be
required
to
subtract
2628
li.
16
s.
10
d.
out
of
9320
li.
10
s.
7
d.
place
the
numbers
in
order
,
and
beginning
with
the
pence
,
say
10
d.
out
of
7
d.
I
cannot
,
(
therefore
I
must
adde
12
d.
(
which
is
one
shillings
)
to
7
d.
and
it
makes
19
d.
)
but
10
d.
out
of
19
d.
and
there
remains
9
d.
set
the
9
d.
under
the
line
,
and
(
because
I
added
12
d.
to
7
d.
)
I
must
therefore
carry
one
to
the
place
of
shillings
,
saying
1
s.
which
I
carryed
,
and
16
s.
is
17.
then
17
s.
from
10
s.
I
cannot
take
,
therefore
,
I
must
adde
20
s.
(
which
is
one
pound
)
to
10
s.
and
it
makes
30
s.
and
17
s.
out
of
30
s.
and
there
remains
13
,
set
13
under
the
line
,
and
carry
one
to
the
place
of
pounds
,
saying
one
which
I
carried
and
8
is
9
,
take
9
of
o
I
cannot
,
but
9
out
of
10
,
and
there
remains
1
,
set
1
under
the
line
,
and
carry
a
unite
to
the
next
place
,
saying
1
which
I
carried
and
2
is
3
,
take
3
out
of
2
I
cannot
,
but
3
out
of
12
,
and
there
remains
9
,
place
9
under
the
line
,
and
carry
1
to
the
next
place
,
saying
1
which
I
carry
and
6
is
7
,
take
7
out
of
3
I
cannot
,
but
7
out
of
13
,
and
there
remains
6
,
place
6
under
the
line
,
and
carry
one
to
the
next
row
,
saying
1
and
2
is
3
,
take
3
from
9
and
there
remains
6
,
place
6
under
the
line
,
so
is
your
Subtraction
ended
,
and
the
remainder
is
6691
li.
13
s.
9
d.
Example
3.
Suppose
a
man
had
lent
to
another
man
1000
pound
,
and
that
the
borrower
had
paid
thereof
at
one
time
127
li.
at
another
time
430
li.
10
s.
and
at
a
third
payment
50
li.
and
the
creditor
would
know
how
much
he
hath
received
,
and
how
much
is
owing
of
his
debtor
.
Place
the
numbers
as
here
you
see
,
|
li.
|
s.
|
d.
|
Money
lent
|
1000
|
00
|
00
|
|
127
|
00
|
00
|
paid
at
several
times
|
430
|
10
|
00
|
|
50
|
00
|
00
|
paid
in
all
|
607
|
10
|
00
|
rests
to
pay
|
392
|
10
|
00
|
first
the
sum
of
money
lent
,
and
draw
a
line
under
it
,
then
set
the
sums
paid
at
several
times
one
under
another
,
and
draw
a
line
under
them
:
Then
adde
all
the
sums
which
have
been
paid
at
several
times
together
,
which
make
607
li.
10
s.
which
is
the
sum
which
the
debtor
hath
paid
in
all
,
then
subtract
this
607
li.
10
s.
from
1000
li.
and
there
will
remain
392
li.
10
s.
and
so
much
is
still
owing
to
the
Creditor
.
Other
Examples
for
Practice
.
|
li.
|
s.
|
d.
|
Lent
|
2601
|
13
|
6
|
Paid
|
98
|
7
|
9
|
Rests
|
2503
|
5
|
9
|
|
li.
|
s.
|
d.
|
Owing
in
all
|
100
|
00
|
00
|
Paid
in
all
|
●6
|
10
|
06
|
Rests
to
pay
|
63
|
10
|
06
|
|
li.
|
s.
|
d.
|
q.
|
Lent
|
3625
|
16
|
08
|
03
|
Paid
at
several
times
|
100
|
00
|
00
|
00
|
Paid
at
several
times
|
336
|
10
|
06
|
02
|
Paid
at
several
times
|
039
|
12
|
09
|
02
|
Paid
at
several
times
|
100
|
00
|
00
|
00
|
Paid
in
all
|
576
|
03
|
04
|
00
|
Rests
to
pay
|
3049
|
13
|
04
|
03
|
The
Proof
of
Subtraction
.
The
Proof
of
Subtraction
is
performed
by
Addition
,
for
adding
the
number
to
be
subtracted
,
to
the
remainder
,
the
Sum
of
them
must
be
equal
to
the
number
given
,
if
you
have
truly
wrought
.
As
in
the
first
example
of
numbers
of
one
denomination
,
the
number
given
is
|
986
|
the
number
to
be
subtracted
is
|
234
|
the
remainer
is
|
752
|
Proof
|
986
|
Adde
the
number
to
be
subtracted
234
,
to
the
remainer
752
,
the
sum
of
them
is
986
,
equal
to
the
number
given
.
Examples
for
Practice
proved
.
|
li.
|
s.
|
d.
|
Lent
|
62
|
18
|
09
|
Paid
|
37
|
19
|
06
|
Rests
|
24
|
19
|
03
|
Proof
|
62
|
18
|
09
|
|
li.
|
s.
|
d.
|
Borrowed
|
100
|
00
|
00
|
Received
|
36
|
13
|
04
|
Due
|
63
|
06
|
08
|
Proof
|
100
|
00
|
00
|
Other
Examples
in
Weight
and
Measure
.
1
Exa
.
in
Troy
weight
|
li.
|
ou
.
|
pw
.
|
gr.
|
Bought
of
Silver
|
07
|
11
|
13
|
19
|
Sold
|
05
|
07
|
03
|
05
|
unsold
|
42
|
04
|
10
|
14
|
Proof
|
07
|
11
|
13
|
19
|
2
Example
in
Avoirdupois
great
weight
.
|
C.
|
q.
|
li.
|
ou
.
|
Bought
|
37
|
03
|
22
|
11
|
Sold
|
13
|
01
|
23
|
06
|
Rests
|
24
|
01
|
27
|
05
|
Proof
|
37
|
03
|
22
|
11
|
3
Example
in
Avoirdupois
little
weight
.
|
li.
|
ou
.
|
dr.
|
Bought
|
84
|
12
|
13
|
Sold
|
26
|
08
|
11
|
Rests
|
58
|
04
|
02
|
Proof
|
84
|
12
|
13
|
4
Example
in
Time
.
|
days
|
ho.
|
m.
|
From
|
364
|
23
|
50
|
Take
|
76
|
09
|
22
|
Rests
|
288
|
14
|
28
|
Proof
|
364
|
23
|
50
|
Questions
performed
by
Addition
and
Subtraction
.
Question
1.
What
number
is
that
which
added
to
376
shall
make
1000
?
Subtract
376
from
1000
,
the
remainder
is
624
,
the
number
sought
.
Question
.
2.
What
number
of
pounds
,
shillings
and
pence
must
be
added
to
36
l.
17
s.
3
d.
to
make
that
sum
up
100
li.
subtract
36
li.
17
s.
3
d.
from
100
li.
the
remainer
is
63
li.
2
s.
9
d.
which
added
to
36
li.
17
s.
3
d.
makes
100
li.
Quest.
3.
In
the
year
of
our
Lord
1440
,
the
famous
art
or
mystery
of
Printing
was
invented
,
I
would
know
how
long
it
is
since
that
time
to
this
yeer
of
our
Lord
1655.
From
1655
,
subtract
1440
,
the
remainer
is
215.
and
so
many
years
are
expired
since
printing
was
invented
.
Question
4.
An
Armie
consisting
of
13721
horse
,
and
26850
foot
,
in
an
engagement
there
were
slain
3760
hors
,
&
7523
foot
,
the
question
is
how
many
were
slain
in
all
,
&
how
many
horse
and
how
many
foot
escaped
.
From
the
1372
horse
which
went
out
,
subtract
the
3760
that
were
slain
,
there
remains
9961
,
and
so
many
horse
escaped
,
Also
from
the
26850
foot
which
went
out
,
subtract
the
7523
which
were
slain
,
and
there
remains
19327
,
the
number
of
foot
which
escaped
,
and
by
adding
the
3760
horse
which
were
slain
,
to
the
7523
foot
that
were
slain
,
their
Total
is
11283
,
and
so
many
were
slain
in
all
.
Multiplication
.
MUltiplication
is
that
part
of
Arithmetick
which
teacheth
how
to
encrease
one
number
by
another
,
so
that
the
number
produced
by
their
Multiplication
shall
contain
one
of
the
numbers
multiplied
so
many
times
as
there
are
unites
contained
in
the
other
.
Multiplication
may
fitly
be
termed
a
Compendium
of
Addition
,
for
that
it
performeth
at
one
operation
the
same
which
to
effect
by
Addition
would
require
many
.
For
instance
,
if
it
were
required
to
know
how
much
7
times
5
is
,
to
perform
this
by
addition
,
I
must
set
seven
fives
,
or
five
sevens
,
one
under
another
,
and
adding
them
together
,
I
shall
finde
that
either
of
their
Totalls
shall
contain
35
,
but
this
by
multiplication
is
performed
with
far
more
brevity
,
as
by
examples
hereafter
shall
appear
.
Before
you
enter
upon
the
practice
of
Multiplication
,
it
is
necessary
to
remember
the
product
produced
by
the
multiplication
of
any
one
of
the
nine
digits
,
by
any
other
of
the
same
,
as
readily
to
know
;
that
4
times
5
is
20
,
6
times
7
is
42
,
2
times
9
is
18
,
7
times
9
is
63.
8
times
9
is
72
,
&c.
Which
this
Table
following
,
will
plainly
declare
,
and
must
be
perfectly
learned
by
heart
before
you
attempt
to
multiply
great
numbers
.
Multiplication
Table
.
-
2
times
2
makes
4
-
2
times
3
makes
6
-
2
times
4
makes
8
-
2
times
5
makes
10
-
2
times
6
makes
12
-
2
times
7
makes
14
-
2
times
8
makes
16
-
2
times
9
makes
18
-
3
times
3
makes
9
-
3
times
4
makes
12
-
3
times
5
makes
15
-
3
times
6
makes
18
-
3
times
7
makes
21
-
3
times
8
makes
24
-
3
times
9
makes
27
-
4
times
4
makes
16
-
4
times
5
makes
20
-
4
times
6
makes
24
-
4
times
7
makes
28
-
4
times
8
makes
32
-
4
times
9
makes
36
-
5
times
5
makes
25
-
5
times
6
makes
30
-
5
times
7
makes
35
-
5
times
8
makes
40
-
5
times
9
makes
45
-
6
times
6
makes
36
-
6
times
7
makes
42
-
6
times
8
makes
48
-
6
times
9
makes
54
-
7
times
7
makes
49
-
7
times
8
makes
56
-
7
times
9
makes
63
-
8
times
8
makes
64
-
8
times
9
makes
72
-
9
times
|
9
|
makes
81.
The
use
of
the
Table
of
Multiplication
,
and
the
manner
how
it
is
to
be
read
.
This
Table
sheweth
what
the
Summe
of
any
two
digits
multiplyed
one
by
another
doth
amount
unto
,
and
is
thus
to
be
read
,
2
times
2
makes
4
,
2
times
3
makes
6
,
2
times
4
makes
8
:
Also
6
times
4
makes
24
,
7
times
8
makes
56
,
8
times
8
makes
(
or
is
)
64
,
9
times
9
is
81
,
&c.
In
Multiplication
there
are
three
terms
commonly
used
,
that
is
to
say
,
-
The
Multiplicand
,
-
The
Multiplier
,
and
-
The
Product
.
The
Multiplicand
is
the
number
to
be
multiplied
.
The
Multiplier
is
the
number
by
which
the
Multiplicand
is
multiplied
,
and
The
Product
is
the
number
which
is
produced
by
the
multiplication
of
the
Multiplicand
and
the
Multiplier
together
.
Thus
,
if
it
were
required
to
multiply
8
by
7
,
here
8
is
the
Multiplicand
,
7
the
Multiplier
,
and
56
is
the
Product
,
for
8
times
7
,
or
7
times
8
,
is
56.
In
Multiplication
it
mattereth
not
which
of
the
two
numbers
given
is
made
the
Multiplicand
,
or
which
the
Multiplier
,
for
the
Product
produced
by
either
will
be
the
same
,
but
the
usual
way
is
to
make
the
greater
number
the
Multiplicand
;
and
the
lesser
number
the
Multiplier
.
THE
RVLE
.
The
numbers
to
be
multiplied
must
be
set
one
under
another
.
viz.
the
Multiplicand
(
or
greater
number
)
.
above
,
and
the
Multiplier
,
(
or
lesser
number
)
below
,
the
last
figure
of
the
Multiplier
under
the
last
figure
of
the
Multiplicand
,
then
draw
a
line
under
them
,
and
(
having
learned
the
preceding
Table
perfectly
by
heart
)
multiplie
every
digit
of
the
Multiplier
,
into
every
digit
of
the
Multiplicand
,
setting
the
several
products
under
the
line
,
then
having
finished
your
Multiplication
,
draw
a
line
,
and
adde
all
the
products
together
,
and
the
Sum
of
those
products
is
the
general
product
of
the
whole
multiplication
,
as
by
the
following
examples
will
oppear
.
Example
1.
Let
it
be
required
to
multiply
736
by
7.
First
,
I
write
down
736
the
Multiplicand
,
and
under
it
7
,
the
Multiplyer
,
and
under
them
I
draw
a
line
,
then
I
multiply
7
into
every
digit
of
the
Multiplicand
,
saying
7
times
6
is
42
,
place
2
under
the
line
under
7
,
and
for
the
four
tens
keep
4
in
mind
,
then
say
again
,
7
times
3
is
21
,
and
4
which
I
kept
in
mind
is
25
,
place
5
under
the
line
,
and
keep
the
two
tens
in
mind
,
then
say
again
7
times
7
is
49
,
and
●
which
I
kept
in
mind
is
51
,
place
1
under
the
line
,
the
5
tens
,
(
because
there
is
no
more
figures
to
be
multiplied
)
I
set
down
under
the
line
also
,
so
is
the
work
ended
,
and
the
product
of
this
multiplication
is
5152.
Example
2.
Let
it
be
required
to
multiply
2417
by
5
,
place
the
numbers
one
under
another
,
and
draw
a
line
under
them
as
in
the
margine
,
then
begin
your
multiplication
,
saying
5
times
7
is
35
,
place
5
under
the
line
,
and
keep
the
three
tens
in
mind
,
then
say
again
,
5
times
1
is
5
,
and
3
which
I
kept
in
minde
is
8
,
place
8
under
the
line
,
and
(
because
it
is
lesse
then
10
,
I
keep
nothing
in
mind
)
then
say
again
,
5
times
is
20
,
place
a
cypher
under
the
line
,
and
keep
the
two
tens
in
mind
;
lastly
,
say
5
times
3
is
15
,
and
2
which
I
kept
in
mind
is
17
,
which
17
being
the
last
number
I
place
under
the
line
,
and
so
is
my
Multiplication
ended
,
and
the
product
is
17085.
¶
You
may
be
satisfied
of
the
truth
of
this
work
if
you
will
take
the
pains
to
set
down
the
Multiplicand
3417
,
five
times
one
under
another
,
and
adde
them
together
,
as
so
many
several
sums
,
so
shall
you
find
the
Total
of
that
addition
,
to
be
17085
,
exactly
the
same
with
the
product
of
this
Multiplication
.
Example
3.
In
the
two
fore-going
examples
,
the
Multiplyer
consisted
but
of
one
digit
,
we
are
now
to
shew
how
multiplication
is
performed
when
the
Multiplier
consists
of
more
then
one
figure
,
therefore
in
this
example
,
let
it
be
required
to
multiplie
5704
by
37
,
place
your
numbers
and
draw
a
line
under
them
as
you
see
in
the
margine
,
then
begin
your
Multiplication
in
this
manner
,
saying
,
7
times
4
is
28
,
set
8
under
the
line
,
and
keepe
the
two
tens
in
mind
,
then
say
7
times
nothing
is
nothing
,
but
the
two
tens
in
mind
is
2
,
set
2
under
the
line
,
then
say
7
times
7
is
49
,
set
9
under
the
line
,
and
keep
4
in
mind
,
then
lastly
,
say
7
times
5
is
35
,
and
4
in
mind
is
39
,
which
being
the
last
number
to
be
multiplied
I
set
down
under
the
line
,
so
is
the
mulplication
of
one
of
the
digits
(
namely
7
)
finished
.
Then
begin
to
multiply
the
second
digit
,
saying
3
times
4
is
12
,
place
2
in
the
second
line
one
place
towards
the
left
hand
,
and
keep
1
in
mind
,
then
say
3
times
nothing
is
nothing
,
but
1
in
mind
is
1
,
set
down
1
by
the
2
in
the
second
line
,
thirdly
,
say
3
times
7
is
21
,
place
1
in
the
second
line
,
and
keep
the
two
tens
in
mind
;
Lastly
,
say
3
times
5
is
15
,
and
2
in
mind
is
17
,
which
17
,
(
because
there
is
no
more
figures
to
be
multiplied
)
I
place
in
the
second
line
also
.
Having
thus
done
I
draw
a
line
under
them
,
and
adde
these
two
lines
together
,
as
in
common
Addition
of
numbers
of
one
denomination
,
saying
8
is
8
,
place
8
under
the
line
,
then
say
2
and
2
is
4
,
place
4
under
the
line
,
then
say
1
and
9
is
10
,
place
a
cypher
under
the
line
,
and
carry
1
to
the
next
place
saying
1
and
1
is
2
and
9
is
11
,
place
1
under
the
line
and
carry
1
to
the
next
row
,
saying
1
and
7
is
8
,
and
3
is
11
,
place
1
under
the
line
,
and
carry
1
to
the
next
place
,
saying
,
1
which
I
carry
and
1
is
2
,
place
2
under
the
line
and
so
is
your
multiplication
ended
,
and
the
product
is
211048.
Example
4.
Let
it
be
required
to
multiply
57325
by
4032
,
place
the
multiplicand
and
multiplierone
under
another
,
&
draw
a
line
asbefore
then
proceed
to
the
multiplication
as
formerly
,
saying
,
first
,
2
times
5
is
10
,
set
down
a
cypher
and
keep
1
in
mind
:
then
2
times
2
is
4
,
and
1
in
mind
is
5
,
place
5
under
the
line
then
2
times
3
is
6
,
set
6
under
the
line
:
then
2
times
7
is
14
,
set
down
4
,
and
keep
1
in
mind
,
then
2
times
5
is
10
,
and
1
mind
is
11
,
which
11
(
being
the
last
)
I
set
down
.
The
multiplication
of
one
of
the
digits
being
finished
,
proceed
to
the
multiplication
of
the
next
,
saying
3
times
5
is
15
,
set
down
5
in
the
second
line
a
place
more
towards
the
left
hand
,
and
keep
1
,
then
3
times
2
is
6
,
and
1
kept
is
7
,
set
down
7
,
then
3
times
3
is
9
,
set
down
9
,
then
3
times
7
is
21
,
set
down
1
and
keep
2
in
mind
,
then
3
times
5
is
15
,
and
2
in
mind
is
17
,
which
being
the
last
set
down
also
.
Two
of
the
figures
of
the
multiplier
being
finished
proceed
to
the
third
,
which
(
in
this
example
)
being
a
cypher
,
you
may
wholly
neglect
,
and
proceed
to
the
multiplication
of
the
fourth
figure
,
onely
remember
to
remove
the
product
of
the
fourth
figure
one
place
more
to
the
left
hand
,
as
in
the
example
you
may
see
,
for
the
cypher
,
though
it
be
not
written
down
,
yet
it
must
keep
its
place
,
and
the
figures
following
must
be
removed
a
place
farther
.
Then
for
the
Multiplication
of
the
fourth
and
last
digit
,
say
4
times
5
is
20
,
set
down
a
cypher
(
under
9
)
and
keep
2
in
mind
,
then
4
times
2
is
8
and
2
in
mind
is
10
,
set
down
a
cypher
and
keep
1
in
mind
,
then
4
times
3
is
12
,
and
1
is
13
,
set
down
3
and
keep
1
,
then
4
times
7
is
28
,
and
1
kept
is
29
,
set
down
9
and
keep
2
,
then
4
times
5
is
20
,
and
2
kept
is
22
,
which
22
(
because
the
multiplicatiou
is
ended
)
set
down
also
.
Having
thus
multiplied
all
the
digits
severally
,
draw
a
line
under
their
products
,
and
adde
them
altogether
as
in
the
former
example
,
so
shall
you
find
their
general
product
to
be
231134400.
Other
Examples
for
Practice
.
Compendiums
in
Multiplication
.
1
If
the
Multiplier
consist
of
cyphers
in
the
last
place
or
places
,
you
may
omit
the
multiplcation
of
them
,
and
place
the
former
figures
of
the
Multiplier
under
the
Multiplicand
,
thus
if
it
were
required
to
multiply
3257
by
2600.
place
the
numbers
as
you
see
in
the
margent
,
then
multiplying
3257
by
26
,
the
Product
will
be
84682
,
to
which
if
you
add
two
cyphers
,
(
because
there
were
two
cyphers
in
the
Multiplier
)
it
will
be
8468200
,
which
is
the
true
product
of
the
multiplication
.
2
If
it
be
required
to
multiply
any
number
by
10
,
100
,
1000
,
10000
,
&c
,
You
have
no
more
to
do
but
to
add
so
many
cyphers
to
the
multiplicand
as
there
are
cyphers
in
the
multiplier
,
thus
if
you
were
to
multiply
365
by
10
,
the
product
will
be
3650
,
or
by
100
,
it
would
be
36500
,
or
by
1000
,
it
would
be
365000
,
or
by
10000
,
it
would
be
3650000
,
&c.
3
If
any
number
given
were
to
be
multiplied
by
5
,
you
may
abreviate
your
worke
thus
,
adde
a
cypher
to
the
Multiplicand
,
take
halfe
that
number
,
and
it
shall
be
the
product
required
,
thus
if
it
were
required
to
multiply
8627
by
5
,
adde
a
cypher
to
the
multiplicand
,
then
it
is
86270
the
halfe
whereof
is
43135
,
which
is
the
product
required
.
The
Proof
of
Multiplication
.
The
most
certain
proof
of
Multiplication
is
by
Division
,
but
because
Division
is
not
yet
known
,
I
will
here
shew
a
neer
way
by
which
Multiplication
may
be
proved
.
Which
is
thus
,
THE
RVLE
.
First
,
take
a
Crosse
as
in
the
Margine
,
then
,
any
sums
being
multiplied
,
you
may
prove
the
truth
of
your
work
in
this
manner
,
(1)
Cast
away
all
the
nines
which
you
can
find
in
the
multiplicand
,
what
remaineth
set
on
the
right
side
of
the
Crosse
.
(2)
Cast
away
also
the
nines
in
the
Multiplyer
,
and
what
remains
set
on
the
left
side
of
the
Crosse
.
(3)
Multiply
the
figure
on
the
right
side
of
the
Crosse
by
that
on
the
left
side
of
the
Crosse
,
and
out
of
that
product
cast
away
the
nines
,
setting
the
figure
remaining
over
the
Crosse
,
then
(4)
Cast
away
all
the
nines
in
the
product
,
and
if
the
figure
remaining
be
the
same
with
that
which
standeth
over
the
Crosse
,
then
is
your
multiplication
truly
performed
,
otherwise
not
.
1
Cast
away
all
the
nines
in
the
Multiplicand
,
saying
4
and
3
is
7
,
and
2
is
9
,
which
being
rejected
there
remains
4
,
which
I
set
on
the
right
side
of
the
crosse
,
then
2
Cast
away
all
the
nines
in
the
Multiplier
,
saying
2
and
3
is
5
,
which
(
being
lesse
then
9
)
I
set
on
the
left
side
of
the
crosse
,
then
3
Multiply
4
by
5
,
saying
4
times
5
is
20
from
which
cast
all
the
nines
,
and
there
remain
2
,
place
2
over
the
crosse
,
and
4
Cast
away
all
the
nines
in
the
Product
,
saying
,
2
and
5
is
7
,
and
4
is
11
,
cast
away
9
,
and
there
remains
2
,
which
exactly
agrees
with
the
figure
over
the
crosse
,
and
demonstrates
that
the
multiplication
is
truly
performed
.
To
multiply
by
any
of
the
nine
Digits
without
charging
the
memory
.
To
multiply
any
number
by
2
,
Either
double
the
number
in
your
mind
,
or
adde
it
,
by
setting
it
down
twice
,
so
57325
produceth
115750.
To
multiply
any
number
by
3
,
To
the
number
given
,
adde
the
double
thereof
,
the
sum
is
the
product
,
so
57325
produceth
171975.
To
multiply
any
number
by
4
,
Double
the
duplication
in
your
mind
,
so
57325
produceth
229290.
To
multiply
any
number
by
5
,
Conceive
a
cypher
added
to
the
given
number
,
and
in
your
mind
half
thereof
is
the
product
,
thus
a
cypher
added
to
57325
,
maketh
it
573250
,
the
halfe
whereof
is
286625.
To
multiply
any
number
by
6
,
Take
half
adding
a
cypher
,
and
adde
to
the
half
,
the
figure
standing
next
before
,
thus
573250
produceth
343950.
To
multiply
any
number
by
7
,
Take
half
and
adde
it
to
the
double
of
the
former
figure
,
supposing
a
cypher
added
as
before
,
so
57325
thus
ordered
,
produceth
401275.
To
multiply
any
number
by
8
,
Double
each
former
figure
,
and
subtract
it
from
the
following
,
so
57325
produceth
458600.
To
multiply
any
number
by
9
,
Suppose
the
number
multiplyed
by
1
,
then
subtract
each
former
figure
from
the
following
,
beginning
with
that
next
before
the
cypher
,
the
remainer
is
the
product
,
so
57325
produceth
515925.
Questions
performed
by
Multiplication
only
.
Question
1.
If
a
piece
of
land
be
236
perches
long
,
and
182
perches
broad
,
how
many
square
perches
are
contained
therein
?
multiply
236
the
length
,
by
182
the
breadth
,
the
product
is
42952
,
and
so
many
square
perches
are
contained
in
such
a
square
piece
of
land
.
Question
2.
In
a
year
there
are
365
dayes
natural
,
and
in
every
day
24
houres
,
how
many
houres
be
there
in
a
year
?
Multiply
365
the
number
of
dayes
,
by
24
the
number
of
houres
,
the
product
is
8760
,
and
so
many
houres
be
there
in
a
year
.
Question
3.
From
London
to
Coventry
it
is
accounted
76
miles
,
how
many
yards
therefore
is
it
from
London
to
Coventry
?
Multiply
1760
(
which
are
the
number
of
yards
contained
in
one
mile
)
by
79
,
the
product
is
133760
,
and
so
many
yards
are
between
London
and
Coventry
.
Division
.
DIvision
is
the
just
contrary
to
Multiplication
,
for
that
turnes
small
denominations
to
greater
,
as
Multiplication
turns
greater
to
smaller
.
Or
(
in
whole
Numbers
,
of
which
only
we
yet
speak
)
Division
is
the
asking
how
many
times
one
Sum
is
contained
in
another
?
and
the
number
which
answereth
to
that
question
is
called
the
Quotient
.
And
the
number
containing
,
which
is
to
be
divided
,
is
called
the
Dividend
.
And
the
number
contained
,
or
by
which
the
Dividend
is
to
be
divided
,
is
called
the
Divisor
.
And
as
often
as
the
Dividend
contains
the
Divisor
,
so
often
doth
the
quotient
containe
Unity
.
The
wayes
of
performing
Division
are
divers
I
will
begin
with
that
which
is
most
used
and
taught
,
which
is
as
followeth
.
THE
RVLE
.
Place
the
Divisor
under
the
Dividend
,
so
that
the
figures
next
to
the
left
hand
stand
directly
one
under
the
other
,
if
the
rest
of
the
Divisor
be
not
greater
:
or
if
all
the
Divisor
be
greater
then
that
above
it
,
then
the
said
Divisor
must
be
devolved
one
place
further
toward
the
right
hand
;
having
so
placed
them
,
try
how
many
times
the
lower
figures
are
contained
in
the
upper
figures
,
and
write
that
figure
which
answereth
that
question
within
a
crooked
line
in
the
margine
of
the
work
,
which
is
called
the
Quotient
,
and
by
it
multiply
the
first
figure
of
the
Divisor
,
and
take
the
product
out
of
the
figures
directly
over
it
,
beginning
the
Subtraction
toward
the
left
hand
;
then
cancel
that
figure
of
the
Divisor
,
and
also
,
that
of
the
Dividend
which
hath
been
already
used
,
with
a
light
dash
of
a
pen
,
and
write
the
remaine
(
when
the
product
of
the
first
figure
multiplied
by
the
quotient
is
subtracted
as
before
)
just
over
the
figure
used
and
cancelled
;
the●
proceed
to
doe
the
like
with
the
second
,
third
,
and
fourth
figure
of
the
Divisor
if
there
be
so
many
;
till
having
cancelled
it
all
,
and
set
the
remaine
orderly
above
the
Dividend
,
you
have
finished
one
work
.
Now
if
the
Dividend
have
still
some
figures
untouched
towards
the
right
hand
,
then
remove
the
Divisor
still
toward
the
right
hand
,
but
one
place
at
a
time
,
and
then
againe
aske
or
try
how
many
times
the
lower
may
be
had
in
the
upper
,
and
write
the
answer
in
the
quotient
whether
it
be
1
or
more
,
(
onely
it
cannot
be
above
9
)
or
nothing
,
then
put
0
in
the
quotient
,
and
multiply
the
Divisor
by
this
new
figure
,
and
subtract
the
product
,
setting
the
remainer
orderly
above
,
as
before
,
this
work
must
be
repeated
by
removing
the
Divisor
still
one
place
towards
the
right
hand
,
until
Vnity
in
the
Divisor
stand
under
Unity
in
the
Dividend
,
and
then
the
work
is
done
.
Example
.
1.
Let
it
be
required
to
divide
4096
by
3
Place
them
thus
Aske
how
many
times
3
in
4
the
quotient
is
1
,
which
is
put
within
a
crooked
line
by
it self
.
Then
in
your
mind
multiply
the
Divisor
3
by
the
quotient
1.
And
having
said
these
words
Once
3
is
3
,
presently
cancel
the
3.
And
having
added
these
words
,
Out
of
4
,
cancel
the
4.
And
after
these
words
,
and
there
remains
1
,
write
1
just
over
the
4
,
as
you
see
here
done
.
Then
remove
the
divisor
one
place
towards
the
right
hand
,
saying
,
how
many
times
3
in
10
,
the
answer
is
3
,
which
write
in
the
quotient
;
and
in
your
mind
multiplie
the
Divisor
3
by
the
quotient
3
,
the
product
is
o
,
wherefore
say
three
times
three
is
nine
out
of
ten
,
and
there
remains
one
,
then
(
having
cancelled
the
10
and
the
3
,
write
over
them
1.
Again
,
remove
the
Divisor
3
one
place
more
,
asking
how
many
times
3
in
19
?
the
answer
being
6
,
write
6
in
the
quotient
,
and
say
6
times
3
is
18
out
of
19
,
and
there
remains
1
,
wherefore
having
cancelled
the
3
,
and
the
19
,
write
1
over
3
,
and
remove
the
Divisor
once
more
,
and
aske
how
many
times
3
in
16
?
answer
is
5
,
which
write
in
the
quotient
,
then
in
mind
say
5
times
3
is
15
out
of
16
,
and
there
remains
1
,
and
cancelling
the
16
and
the
3
,
write
over
3
,
1
:
Now
because
the
Divisor
3
is
advanced
so
far
till
it
is
come
to
stand
under
6
in
the
Dividend
,
which
6
is
the
place
of
Unity
there
,
the
said
Divisor
cannot
be
removed
any
more
,
and
therefore
the
division
is
ended
,
and
the
quotient
being
1365
,
shewes
that
the
Divisor
3
is
contained
in
the
Dividend
4096
,
1365
times
;
and
1
remaining
,
which
1
being
lesse
then
the
Divisor
3
,
doth
not
contain
it
once
,
but
one
third
part
of
once
,
which
after
the
Reader
hath
skill
in
broken
numbers
must
be
joined
to
the
quotient
thus
1365⅓
.
The
best
proof
of
this
Division
is
by
multiplying
the
quotient
into
the
Divisor
,
and
to
the
product
adde
the
remaine
;
then
if
the
work
be
well
done
,
the
sum
shall
be
equal
to
the
Dividend
.
So
1365
multiplied
by
3
produceth
4095
,
to
which
adding
the
remain
1
,
the
sum
4096
is
equal
to
the
Dividend
.
Here
we
divided
only
by
one
figure
3
,
because
the
first
Example
being
easie
and
clear
,
should
be
a
faire
Introduction
to
the
second
,
Note
that
if
the
Divisor
had
been
greater
then
4
,
as
5
,
the
work
must
have
begun
thus
So
the
quotient
would
have
been
819½
,
which
is
one
place
lesse
.
Example
2.
Let
it
be
required
to
divide
1310720
by
4096
,
Place
them
thus
Now
the
question
to
be
asked
is
,
how
many
times
4096
is
there
in
13107
?
to
find
an
answer
to
this
question
,
the
Reader
which
hath
but
an
indifferent
faculty
of
judging
;
may
do
it
,
(
for
the
most
part
)
by
considering
the
first
figure
in
the
divisor
,
as
here
4
,
for
presently
he
knowes
that
4
times
4
is
16
,
which
cannot
be
had
in
13
,
therefore
the
first
figure
in
the
quotient
must
be
lesse
then
4.
Again
,
it
cannot
be
much
lesse
,
because
the
second
place
in
the
Divisor
is
0.
He
may
therefore
venture
on
3
,
And
having
put
3
in
the
quotient
,
say
3
times
4
is
12
,
which
subtract
out
of
13
over
it
thus
,
say
2
out
of
3
,
and
there
remains
1
,
and
1
out
of
1
and
there
remaines
0
,
cancelling
the
0
and
the
13
,
and
set
the
remaine
1
over
the
4
,
as
you
see
.
Then
go
on
saying
3
times
0
is
0
,
out
of
11
,
and
remains
still
11
,
againe
3
times
9
is
27
,
take
7
out
of
10
there
remains
3
,
to
be
set
over
the
0
,
which
is
over
9
,
and
2
with
1
borrowed
(
to
make
the
0
10
,
from
the
which
the
7
was
taken
)
is
3
,
and
3
out
of
11
,
remains
8
,
which
write
over
the
place
of
0
in
the
Divisor
,
cancelling
the
9
in
it
,
and
also
those
figures
of
the
Dividend
110
,
out
of
which
you
have
taken
any
thing
;
then
lastly
,
say
3
times
6
is
18
,
take
8
out
of
7
I
cannot
,
therefore
(
borrowing
10
,
)
say
8
out
of
17
remains
9
,
which
write
over
the
7
,
then
in
the
next
place
,
the
10
borrowed
is
1
,
and
the
10
in
the
18
is
1.
Say
therefore
1
and
1
is
2
,
out
of
3
,
and
the
remain
is
1
,
which
write
over
the
3
,
having
still
cancelled
the
figures
which
you
have
used
orderly
as
you
go
.
Now
the
whole
Divisor
being
cancelled
,
it
must
be
removed
on
one
place
further
,
and
placed
as
here
.
Now
aske
again
how
many
times
4096
in
8192
?
it
will
be
found
(
because
of
the
0
after
the
4
)
as
often
as
4
in
8
,
that
is
two
times
,
put
therefore
in
the
quotient
2
,
and
work
as
before
,
saying
2
times
4
is
8
,
out
of
8
remains
0
,
then
2
times
0
is
0
out
of
1
remains
1
,
then
2
times
9
is
18
,
take
8
out
of
9
remains
1
,
(
which
put
over
the
9
,
)
and
1
out
of
1
in
the
next
place
remains
0
,
then
lastly
,
2
times
6
is
12
,
that
is
,
2
out
of
2
there
remaines
0
,
and
1
out
of
1
,
there
also
remaines
0
,
cancel
and
put
the
remainer
over
as
formerly
.
Now
againe
,
the
Divisor
being
all
cancelled
,
should
be
removed
;
and
aske
how
many
times
4096
in
0000
,
the
answer
is
0
,
which
being
put
in
the
quotient
,
the
work
is
all
done
.
And
the
quotient
320
,
shewes
that
the
Divisor
4096
is
contained
in
the
Dividend
1310720
,
three
hundred
and
twenty
times
.
And
whether
the
Divisor
have
2
,
3
,
5
,
7
,
or
more
places
,
the
working
is
still
like
this
,
not
differing
from
it
at
all
.
Except
for
brevity
in
some
cases
,
where
the
Divisor
toward
the
right
hand
hath
one
Cypher
or
more
;
for
those
Cyphers
may
be
placed
orderly
under
the
Dividend
at
first
,
and
remain
there
till
the
work
be
done
,
with
the
rest
of
the
Divisor
,
which
must
needs
shorten
the
Division
.
As
if
2587645
were
to
be
divided
by
15000
,
place
them
thus
And
say
how
many
times
15
in
25
?
answer
is
1
,
which
write
in
the
quotient
,
and
multiply
in
minde
the
divisor
by
the
quotient
,
saying
once
one
is
one
,
out
of
2
there
remains
1
,
then
cancel
the
2
,
and
the
1
under
it
,
and
write
the
remaining
1
over
it
,
as
here
is
done
,
then
say
once
5
is
5
out
of
5
,
and
there
remains
0
,
therefore
cancel
the
two
5
,
and
over
the
uppermost
write
0.
Now
remove
the
Divisor
one
place
,
and
aske
how
many
times
15
in
108
?
answer
is
7
,
which
write
in
the
quotient
,
and
multiply
,
saying
7
times
1
is
7
,
out
of
10
remains
3
,
which
write
over
the
0
,
then
say
7
times
5
is
35
,
take
5
out
of
8
remains
3
,
which
write
over
8
,
and
3
out
of
3
remains
0.
Now
againe
,
remove
the
Divisor
,
and
aske
how
many
times
15
in
37
?
answer
is
2
,
which
write
in
the
quotient
,
and
say
two
times
15
is
30
,
out
of
37
remaines
7
,
and
the
Division
is
ended
,
the
remaine
being
7645.
Proof
of
this
.
Multiply
the
quotient
|
172
|
by
the
Divisor
|
15
|
|
860
|
|
172
|
The
Product
is
|
2580
|
Before
which
put
the
three
Cyphers
,
And
then
it
is
|
2580000
|
To
which
adde
the
remaine
|
7645
|
The
Total
is
|
2587645
|
Which
is
equal
to
the
Dividend
,
and
therefore
the
work
is
right
.
So
if
one
would
divide
any
sum
by
10
,
100
,
1000
,
10000
,
&c.
he
need
but
cut
off
the
first
,
two
first
,
three
first
,
or
four
first
figures
towards
the
right
hand
,
the
other
figures
shall
be
the
quotient
,
and
those
cut
off
,
the
remaine
.
As
if
1587645
be
divided
by
1000
,
the
quotient
is
1587
,
and
the
remaine
645.
If
one
would
know
how
many
pounds
Sterling
are
in
95670
shillings
;
having
placed
them
thus
,
that
is
,
2
under
9
,
and
0
under
0
,
then
divide
by
2
,
which
is
very
easie
.
So
the
quotient
4783
,
shewes
that
in
the
number
of
shillings
given
,
is
contained
4783
pounds
,
and
10
shillings
remaining
.
The
reason
why
the
Divisor
was
2
,
is
because
there
are
20
shillings
in
one
pound
,
and
therefore
any
number
of
shillings
is
turned
into
pounds
,
by
dividing
by
20
,
that
is
by
2
,
putting
the
cypher
under
unity
,
onely
to
fill
a
place
at
last
.
And
this
way
of
turning
any
number
of
shillings
into
pounds
may
be
easily
effected
by
memory
,
if
you
suppose
the
last
figure
of
your
given
number
to
be
cut
off
with
a
line
,
or
comma
,
and
taking
the
half
of
the
other
figures
.
Thus
,
let
the
given
number
of
shillings
be
5739
;
Imagine
the
last
figure
9
to
be
seperated
from
the
rest
by
a
line
thus
573
9
,
now
by
memory
take
the
half
of
573
,
by
saying
in
your
mind
the
half
of
5
is
2
,
(
and
1
remaining
,
which
makes
the
7
following
17
,
)
the
half
of
17
is
8
,
(
and
1
remaing
)
the
half
of
13
is
6
,
(
and
1
remaing
,
which
1
is
10
s.
to
be
added
to
the
9
,
and
the
whole
is
286
li.
19
s.
In
this
way
of
Division
,
as
in
all
others
,
if
the
remaine
at
last
be
greater
then
the
Divisor
,
the
quotient
is
not
just
,
but
too
little
,
which
may
be
remedied
(
without
beginning
the
work
againe
)
by
dividing
the
remainer
onely
by
the
same
divisor
;
for
thereof
will
arise
a
new
quotient
,
which
added
to
the
former
quotient
,
the
summe
will
be
the
just
quotient
.
So
if
7290
be
divided
by
27
,
the
numbers
being
placed
thus
,
Because
27
can
be
had
in
72
but
twice
,
put
2
in
the
quotient
,
saying
2
times
27
is
54
,
out
of
72
,
and
there
remains
18
which
write
over
72
,
cancelling
as
before
is
shewed
;
then
removing
the
Divisor
,
say
how
many
times
27
in
189
?
answer
is
7
,
but
if
one
should
mistake
,
and
write
5
in
the
quotient
,
and
say
5
times
27
is
135
,
out
of
189
,
remains
54
and
write
it
over
as
before
;
and
remove
the
Divisor
,
and
say
,
how
many
times
27
in
540
?
answer
is
20
,
but
should
not
be
above
9
,
say
therefore
,
9
times
27
is
243
,
out
of
540
,
and
the
remain
is
297
,
which
being
the
last
remain
,
and
greater
then
the
Divisor
,
shewes
the
quotient
259
is
too
little
.
Wherefore
divide
the
last
remain
297
,
By
the
Divisor
27
saying
,
twenty
seven
in
29
once
,
and
write
1
in
the
new
quotient
;
and
say
once
27
is
27
,
out
of
29
,
remains
2
which
write
over
9
,
and
remove
the
Divisor
,
and
say
,
27
in
27
,
justly
once
,
so
write
1
in
the
quotient
,
so
the
quotient
is
11
,
which
added
to
the
former
quotient
259
,
gives
270
,
which
is
the
true
and
whole
quotient
.
A
Second
way
of
Division
.
Although
(
as
I
have
said
)
the
former
way
is
more
used
,
yet
this
may
seem
plainer
and
more
natural
to
some
,
I
will
therefore
give
one
example
of
it
.
Example
.
Let
it
be
required
to
divide
6477734
by
334
,
where
the
first
figure
in
the
quotient
is
easily
seen
1
,
subtract
once
334
out
of
647
,
and
write
the
remain
under
a
line
,
then
see
how
many
times
334
in
3137
,
answer
9
,
by
which
multiply
the
divisor
334
in
another
paper
,
the
product
is
3006
,
which
subtract
out
of
the
remaine
,
then
the
new
remaine
is
131734.
Again
,
how
many
times
334
in
1317
,
answer
is
4
,
for
the
third
figure
in
the
quotient
;
by
which
multiply
the
Divisor
,
the
product
is
1002
,
which
take
out
of
the
later
remaine
,
as
in
the
margine
,
then
will
remain
31534.
Now
try
how
many
times
334
in
3153
,
answer
is
9
,
for
the
fourth
figure
in
the
quotient
,
by
which
multiply
the
divisor
(
in
a
by
paper
)
the
product
is
3006
,
(
as
might
have
been
seen
above
in
the
second
subtraction
)
which
subtracted
out
of
the
later
remaine
,
there
remains
now
1474.
Lastly
,
aske
how
many
times
334
in
1474
,
answer
is
4
,
for
the
last
figure
in
the
quotint
,
by
which
multiplying
the
divisor
334
,
the
product
is
1336
,
which
subtracted
from
1474
,
there
finally
remaines
138
,
which
being
lesse
then
the
divisor
,
shewes
the
division
is
done
.
Proof
of
this
.
The
several
subtractions
and
the
final
remaine
added
together
.
The
Total
equal
to
the
dividend
6477734
If
the
former
(
as
I
have
said
more
usual
)
way
seem
difficult
to
beginners
,
because
the
products
of
the
divisor
into
the
several
figures
in
the
the
quotient
are
not
set
down
but
mentally
made
;
and
also
because
the
subtraction
of
them
begins
towards
the
left
hand
:
and
lastly
,
because
the
remaine
is
still
set
above
:
yet
this
later
way
which
agrees
altogether
with
plain
Subtraction
before
taught
,
I
hope
is
so
plain
,
that
any
diligent
reader
may
acquire
it
,
without
a
Tutor
.
And
yet
for
the
better
satisfaction
and
help
of
the
young
Learner
,
I
will
adde
another
way
or
two
more
of
Division
.
A
Third
way
of
Division
.
There
is
another
kind
of
division
which
is
very
much
used
,
and
is
in
most
request
with
those
who
have
most
occasion
to
divide
great
numbers
,
the
manner
of
working
is
not
much
unlike
the
way
before
taught
,
one
or
two
examples
will
make
it
plain
.
Example
1.
Let
it
be
required
to
divide
162483
by
1321
,
set
down
your
numbers
as
you
see
them
placed
in
the
margine
,
viz.
First
,
set
down
162483
the
dividend
,
then
on
the
left
hand
thereof
set
the
divisor
1321
with
a
crooked
line
between
them
,
then
on
the
right
hand
thereof
make
another
crooked
line
which
must
serve
to
set
the
figures
of
the
quotient
in
,
so
are
your
numbers
placed
in
due
order
,
then
draw
a
line
under
the
dividend
,
and
make
a
prick
under
the
figure
4
,
because
so
far
the
figures
of
the
divisor
would
extend
if
they
had
been
placed
under-neath
the
dividend
,
according
as
in
the
other
examples
,
this
prick
serves
onely
to
shew
how
far
you
have
proceeded
in
your
work
,
and
must
at
every
division
be
removed
a
place
further
,
till
at
length
you
come
to
the
last
figure
of
the
dividend
:
your
numbers
being
thus
placed
with
a
line
under
them
,
you
are
ready
for
the
work
,
which
must
be
performed
according
to
the
directions
of
the
following
rule
.
THE
RULE
.
Demand
how
often
the
divisor
may
be
had
in
the
dividend
,
and
place
that
number
in
the
quotient
,
then
multiply
the
divisor
by
the
quotient
,
and
place
the
product
under
the
line
:
then
subtract
this
product
from
the
dividend
,
and
set
the
remainer
under
the
product
,
then
make
a
prick
under
the
next
figure
of
the
dividend
,
and
bring
that
figure
down
to
the
remainer
,
and
then
proceed
as
before
.
Example
,
Your
numbers
being
placed
as
is
before
directed
,
you
may
begin
your
work
in
this
manner
,
first
,
say
how
many
times
1321
,
can
I
have
in
1624
?
say
once
,
place
1
in
the
quotient
,
by
which
1
multiply
the
divisor
1321
,
beginning
at
the
left
hand
,
saying
,
once
one
is
1
,
place
1
under
the
line
,
then
once
2
is
2
,
set
2
under
the
line
,
then
once
3
is
3
,
place
3
under
the
line
,
lastly
,
once
1
is
1
,
place
1
under
the
line
,
then
subtract
this
1321
,
from
1624
,
and
there
will
remain
303
,
to
this
303
bring
down
the
next
figure
in
the
dividend
,
namely
8
,
so
will
that
number
be
3038
,
under
which
draw
a
line
,
and
repeat
the
same
work
again
,
saying
,
how
many
times
1321
can
I
have
in
3038
,
which
may
be
had
two
times
,
place
2
in
the
quotient
,
by
which
2
multiply
the
divisor
1321
,
saying
2
times
1
is
2
,
place
2
under
the
line
,
then
2
times
2
is
4
,
place
4
under
the
line
,
then
2
times
3
is
6
,
place
6
under
the
line
,
lastly
,
2
times
1
is
2
,
place
2
under
the
line
,
and
subtract
this
2642
from
3038
,
and
there
will
remain
396
,
to
this
396
bring
down
the
next
figure
of
the
dividend
,
which
is
3
,
so
is
this
number
made
3963
,
under
which
draw
a
line
,
and
repeat
the
work
once
again
,
saying
,
how
many
times
1321
can
I
have
in
3963
,
which
may
be
had
3
times
,
by
which
3
multiply
the
divisor
1321
,
saying
3
times
1
is
3
,
then
3
times
2
is
6
,
then
3
times
3
is
9
,
and
lastly
,
3
times
1
is
3
,
which
place
under
the
line
,
and
subtract
it
from
the
line
above
,
which
in
this
example
is
the
same
number
,
therefore
there
remains
nothing
,
and
the
work
is
ended
,
but
if
any
remainer
had
been
,
that
should
have
been
set
under
the
line
,
as
by
the
examples
following
will
appear
.
Other
Examples
for
Practice
.
In
this
Example
where
793058
,
is
divided
by
5624
,
you
may
perceive
that
the
quotient
is
141
,
and
74
remaining
,
so
that
the
real
quotient
is
141
74/5624.
The
Proof
of
this
Division
.
This
kind
of
division
is
proved
by
Addition
,
for
If
you
adde
the
several
products
arising
from
the
multiplication
of
the
several
quotients
into
the
divisor
,
and
also
adde
thereunto
the
remainder
(
if
any
be
)
the
total
of
this
addition
shall
be
equal
to
the
dividend
,
if
there
be
no
errour
in
the
work
.
So
in
the
example
following
,
if
you
adde
4325
the
first
product
,
and
30275
the
second
product
,
and
2796
the
remainder
,
together
,
in
the
same
order
as
they
now
stand
in
the
example
,
you
shall
find
the
Total
of
this
Addition
to
be
76321
,
equal
to
the
dividend
,
which
demonstrates
the
work
to
be
true
.
A
Fourth
way
of
Division
.
There
is
a
fourth
way
of
division
used
by
some
,
not
inferiour
to
any
of
the
preceding
,
for
that
it
is
no
burthen
to
the
memory
,
and
it
is
also
proved
by
Addition
.
The
manner
of
placing
the
figures
is
the
same
with
the
third
kind
of
division
last
taught
;
And
for
the
performance
of
the
work
,
this
is
THE
RVLE
.
First
write
down
the
dividend
,
and
on
the
left
hand
thereof
the
divisor
,
with
a
crooked
line
betwixt
them
,
and
on
the
right
hand
of
the
dividend
make
another
crooked
line
wherein
to
place
the
figures
of
the
quotient
,
then
draw
a
line
under
the
dividend
,
and
also
make
a
prick
under
that
figure
of
the
dividend
under
which
the
last
figure
of
the
divisor
would
fall
,
were
it
to
be
placed
as
in
the
first
kind
of
division
.
This
done
,
demand
how
often
the
divisor
may
be
found
in
those
figures
of
the
dividend
,
and
place
that
digit
in
the
quotient
,
then
by
this
digits
multiply
the
divisor
,
and
set
the
product
of
this
Multiplication
directly
under
the
dividend
,
beginning
at
the
place
where
you
made
the
prick
,
then
subtract
this
product
from
the
figures
of
the
dividend
,
and
place
the
remainder
over
the
dividend
,
cancelling
the
figures
of
the
dividend
as
you
proceed
,
so
is
the
first
figure
of
the
quotient
finished
,
then
make
a
prick
under
the
next
figure
of
the
dividend
,
and
demand
how
often
the
divisor
may
be
found
in
the
last
remainder
,
and
the
other
figure
being
added
thereto
,
which
place
in
the
quotient
,
and
proceed
in
all
respects
as
before
,
till
you
have
pointed
all
the
figures
of
the
dividend
.
Example
,
Let
it
berequiren
to
divide
763258
by
2345
,
place
your
numbers
as
you
see
in
the
margent
,
and
because
there
are
4
figures
in
the
divisor
,
therefore
make
a
prick
under
the
fourth
figure
of
the
dividend
,
which
is
2
,
and
draw
a
line
,
then
begin
your
division
in
this
manner
:
saying
,
First
,
how
many
times
2345
can
I
have
in
7632
,
(
or
how
many
times
2
can
I
have
in
7
)
say
2
times
,
place
3
in
the
quotient
,
by
which
3
multiply
the
divisor
,
saying
3
times
5
is
15
,
place
5
under
the
prick
,
then
3
times
4
is
●●
and
1
is
13
,
place
3
under
the
line
,
then
3
times
3
is
9
,
and
1
is
10
,
place
a
cypher
under
the
line
,
then
3
times
2
is
6
,
and
1
is
7
,
place
7
under
the
line
,
then
subtract
7035
from
7632
,
saying
5
from
12
,
and
there
remains
7
,
place
7
over
2.
(
cancelling
the
5
and
the
2
,
)
and
bear
one
in
mind
,
then
1
and
3
is
4
,
out
of
13
,
there
remains
9
,
place
9
over
3
,
and
cancel
3
and
●
,
then
1
which
I
carried
,
from
6
,
and
there
remains
5
,
place
5
over
6
,
and
cancel
0
and
6
,
lastly
,
7
from
7
there
remains
0
,
which
you
need
not
set
down
,
but
cancel
the
two
sevens
,
then
will
the
work
stand
as
above
,
and
the
remainder
will
be
597.
Secondly
,
make
a
prick
under
the
next
figure
of
the
dividend
,
namely
5
,
and
say
how
many
times
2345
can
I
have
in
5975
,
answer
two
times
,
place
2
in
the
quotient
,
by
which
multiply
the
divisor
,
saying
2
times
5
is
10
,
place
0
under
5
,
and
carry
1
,
then
2
times
4
is
8
and
1
is
9
,
place
9
under
7
,
then
2
times
3
is
6
,
place
6
under
9
,
lastly
,
2
times
2
is
4
,
place
4
under
5
,
so
is
the
product
of
this
multiplication
4690
,
which
you
must
subtract
from
5975
,
saying
0
from
5
and
there
remains
5
,
place
5
over
5
,
and
cancel
0
and
5
,
then
9
out
of
17
,
there
remains
8
,
place
8
over
7
,
and
cancel
9
and
7
,
then
1
carryed
and
6
is
7
,
from
9
there
remains
2
,
place
2
over
9
and
cancel
6
and
9
,
lastly
,
4
from
5
rests
1
,
place
1
over
5
,
and
cancel
4
and
5
,
so
have
you
finished
your
second
figure
,
and
your
work
will
stand
thus
,
and
your
remainder
will
be
128.
Thirdly
,
make
a
prick
under
the
next
figure
of
your
dividend
(
namely
under
8
,
)
and
aske
how
many
times
2345
can
I
have
in
12858
,
(
or
how
many
times
2
can
I
have
in
12
,
)
say
5
times
,
place
5
in
the
quotient
,
by
which
multiply
the
divisor
,
saying
5
times
5
is
25
,
place
5
under
8
and
carry
2
,
then
5
times
4
is
20
,
and
2
is
22
,
place
2
under
0
,
and
carry
2
,
then
5
times
3
is
15
,
and
2
is
17
,
place
7
under
9
,
and
carry
1
,
then
5
times
2
is
10
,
and
1
is
11
,
place
11
under
4
and
6
,
so
is
the
product
of
this
multiplication
11725
to
be
subtracted
from
12858
,
saying
5
from
8
rests
3
,
place
3
over
8
,
and
cancel
5
and
8
,
then
2
from
5
rests
3
,
place
3
over
5
,
and
cancel
2
and
5
,
then
7
from
8
rests
1
,
place
1
over
8
and
cancel
7
and
8
,
then
1
from
2
rests
1
,
place
1
over
2
,
and
cancel
1
and
2
,
lastly
,
1
from
1
rests
nothing
,
so
is
your
work
ended
,
which
you
shall
find
to
stand
as
in
the
margent
,
the
remainder
being
1133.
The
Proof
of
this
Division
.
This
kind
of
division
is
also
proved
by
Addition
,
for
,
if
you
draw
a
line
under
the
work
,
and
adde
all
the
figures
between
the
two
lines
together
,
(
in
order
as
they
there
stand
)
taking
in
the
remainder
(
if
any
be
)
the
Total
of
this
addition
will
be
equal
to
the
Dividend
,
if
the
work
be
true
.
Other
Examples
for
Practice
proved
.
Questions
performed
by
Division
only
.
Question
1.
If
a
square
piece
of
Land
contained
42952
square
perches
,
and
one
of
the
sides
thereof
be
236
perches
long
,
how
long
must
the
other
side
be
?
Divide
42952
by
236
,
the
quotient
will
be
182
,
and
so
many
perches
long
must
the
other
side
be
.
Question
2.
In
a
yeaar
there
are
8760
houres
,
and
in
every
natural
day
there
are
24
houres
.
I
demand
how
many
dayes
be
there
in
a
year
?
Divide
8760
by
24
the
Quotient
will
be
365
,
and
so
many
dayes
be
there
in
a
year
.
Question
3.
The
distance
from
London
to
Coventry
is
133760
yards
,
and
in
one
mile
there
is
contained
1760
yards
,
now
I
would
know
how
many
miles
it
is
from
London
to
Coventry
;
Divide
133760
by
1760
,
the
quotient
will
be
76
,
and
so
many
miles
is
it
from
London
to
Coventry
.
These
Questions
performed
by
Division
only
,
are
the
converse
of
those
that
were
performed
by
Multiplication
,
which
I
the
rather
make
choice
of
,
that
the
reader
might
see
how
Multiplication
and
Division
prove
each
other
.
There
are
one
or
two
more
kinds
of
Division
,
something
like
these
last
,
but
I
shall
forbear
exemplifying
them
;
for
much
variety
helps
to
make
a
Book
rather
great
then
fit
.
¶
Here
is
to
be
noted
that
in
the
following
Rules
,
where
there
is
continual
use
of
Division
,
I
sometimes
use
one
kind
of
Division
,
and
sometimes
another
,
for
variety
sake
,
but
the
practicioner
may
use
which
he
is
best
skill'd
in
,
for
they
all
produce
the
same
effect
.
Reduction
.
IS
twofold
,
first
,
that
which
turns
greater
denominations
into
smaller
,
as
pounds
into
shillings
or
pence
,
this
is
done
by
Multiplication
as
followeth
.
Example
1.
Let
it
be
asked
how
many
pence
are
contained
in
729
li.
11
s.
7
d
?
First
,
a
shilling
is
contained
in
a
pound
20
times
,
therefore
multiply
729
by
20
,
or
(
which
is
the
same
,
but
shorter
)
by
2
,
and
put
o
to
the
product
,
as
in
the
margine
,
this
shews
that
in
729
l.
there
are
14580
shillings
.
To
which
adde
11
s.
it
makes
14591
shillings
.
Again
,
because
one
peny
is
conteined
in
one
shilling
12
times
,
multiply
14591
by
12
,
it
produceth
175092
,
to
which
adde
the
7
pence
,
so
the
summe
will
be
175099
,
and
so
many
pence
are
contained
in
729
li.
11
s.
7
d.
Example
2.
Let
it
be
asked
how
many
pintes
are
contained
in
4
Tun
,
3
Hogsheads
,
and
27
Gallons
?
First
,
1
Tun
is
equal
to
4
Hogsheads
,
therefore
4
Tun
is
equal
to
16
Hogsheads
,
to
which
adde
the
3
Hogsheads
,
so
there
is
19
intire
Hogsheads
.
Againe
,
because
one
Hogshead
containes
63
Gallons
,
multiply
19
by
63
,
it
produceth
1197
Gallons
,
to
which
add
27
,
it
gives
1224
Gallōs
.
Lastly
,
because
every
Gallon
contains
8
pintes
,
multiply
1224
by
8
,
it
produceth
9792
,
and
so
many
pintes
are
contained
in
4
Tuns
,
3
Hogsheads
and
27
Callons
.
After
the
same
sort
might
drie
Measures
be
reduced
,
as
quarters
to
bushels
,
pecks
,
or
gallons
,
and
likewise
all
weights
and
outlandish
Coins
,
of
which
the
proportion
of
the
greater
to
the
lesser
is
(
before
)
known
,
or
given
.
Secondly
,
it
is
often
requisite
to
turn
smaller
denominations
to
greater
:
this
is
done
by
Division
,
as
followeth
.
Example
1.
Let
it
be
asked
how
many
pounds
are
contained
in
80976
shillings
?
Divide
80976
by
20
,
the
quotient
is
4048
li.
and
16
s.
remaining
,
which
is
the
true
answer
.
Example
2.
Let
it
be
asked
how
many
pounds
are
in
109754
d
?
Because
a
pound
contains
a
shilling
20
times-and
a
shilling
containes
a
peny
12
times
,
there
fore
if
109754
be
divided
first
by
12
,
the
quotient
9146
shillings
,
and
2
pence
over
,
then
if
9146
be
divided
by
20
,
the
quotient
is
457
pounds
and
six
shillings
remaining
;
so
that
109754
pence
is
equal
to
457
li.
6
s.
2
d.
Or
if
109754
had
been
at
first
divided
by
12
times
20
,
that
is
by
240
,
(
which
is
the
number
of
pence
contained
in
a
pound
)
the
quotient
had
been
457
pounds
,
and
74
pence
remaining
,
which
is
all
one
with
the
former
;
for
74
pence
is
equal
to
6
shillings
2
pence
.
More
instances
shall
not
need
herein
,
because
the
thing
of
it self
is
very
clear
.
Progression
.
IS
also
of
two
sorts
,
the
first
is
of
certain
numbers
in
Arithmetical
Proportion
from
1
,
that
is
,
such
as
differ
equally
,
as
1
,
2
,
3
,
4
,
5
,
6
,
where
the
common
difference
is
1
,
(
as
is
easily
seen
,
)
or
1
,
3
,
5
,
7
,
9
,
11
,
where
the
common
difference
is
2
,
or
any
other
,
as
1
,
8
,
15
,
22
,
29
,
36
,
where
the
common
difference
is
7
,
this
is
called
Arithmetical
Progression
.
2
Secondly
,
of
certaine
numbers
in
Geometrical
proportion
from
1
,
that
is
such
as
increase
by
a
common
Multiplication
,
as
1
,
2
,
4
,
8
,
16
,
32
,
where
the
common
multiplier
is
2
,
that
is
the
first
by
2
,
produceth
the
second
,
and
the
second
multiplied
by
2
,
produceth
the
third
,
and
so
on
.
Or
as
1
,
3
,
9
,
27
,
81
,
243
,
where
the
common
Multiplier
is
3
,
this
is
called
Geometrical
Progression
.
Both
the
common
difference
(
in
the
first
)
and
the
common
Multiplication
(
in
the
later
)
shall
for
shortnesse
hereafter
be
called
the
common
excesse
.
First
,
now
of
the
first
sort
,
or
Arithmetical
Progression
,
the
principal
use
of
this
is
,
1
If
the
number
of
places
,
and
common
excesse
be
given
,
to
find
the
last
number
.
2
When
the
number
of
places
,
and
the
last
number
is
given
,
to
find
the
aggregate
,
or
total
summe
of
all
the
numbers
.
3
When
the
last
number
,
and
the
total
summe
is
given
,
to
find
the
number
of
places
.
4
The
Number
of
places
,
and
total
summe
being
given
to
find
the
last
number
.
5
The
last
number
,
and
number
of
places
given
,
to
find
the
common
excesse
.
6
The
last
number
,
and
common
excesse
being
given
;
to
find
the
number
of
places
.
I
will
instance
in
no
more
,
few
of
these
ever
hapning
to
be
used
.
For
the
first
of
these
,
let
there
be
given
the
number
of
places
|
100
|
The
common
excesse
|
1
|
To
find
the
last
number
also
;
|
100
|
THE
RVLE
.
Multiply
the
number
of
places
lesse
by
1
,
by
the
common
excesse
,
and
to
the
product
adde
the
first
number
;
the
summe
is
equal
to
the
last
number
.
So
here
multiply
99
by
1
,
the
product
is
99
,
(
for
1
neither
multiplies
nor
divides
)
to
this
adde
the
first
number
1
,
it
gives
100
for
the
last
number
.
Or
let
the
numbers
be
1
,
7
,
13
,
19
,
25
,
31
,
where
the
common
excesse
is
6.
and
the
number
of
places
also
6.
Now
(
if
the
number
of
places
lesse
by
1
,
that
is
)
5
be
multiplied
by
the
common
excesse
,
which
is
6
,
the
product
is
30
,
to
which
adding
the
first
number
which
is
1
,
the
last
number
31
,
is
thereby
composed
.
This
is
so
casie
that
it
needs
no
proof
.
2
For
the
second
,
which
is
,
The
last
number
,
and
the
number
of
places
given
,
to
find
the
total
summe
of
all
the
numbers
.
THE
RVLE
.
Adde
the
first
and
last
numbers
together
,
and
multiply
the
summe
by
half
the
number
of
places
,
the
product
is
equal
to
the
aggregate
or
summe
of
all
the
numbers
added
together
.
So
if
to
the
first
number
1
be
added
the
last
number
100
,
it
gives
101
,
which
multiplied
by
50
(
which
is
half
the
number
of
places
)
produceth
5050
,
which
is
equal
to
all
the
hundred
numbers
added
together
.
And
hereby
may
that
vulgar
question
be
answered
,
which
is
,
If
a
man
take
up
100
stones
placed
a
yard
one
from
another
all
in
a
right
line
,
by
one
at
once
,
and
bring
them
back
one
by
one
to
his
first
standing
,
how
many
yards
doth
he
goe
backwards
and
forwards
?
It
is
shewed
before
that
he
goes
forward
5050
yards
,
and
he
must
needs
come
back
just
as
much
;
that
is
,
in
all
10100
yards
.
which
is
5
miles
and
3
quarters
;
wanting
20
yards
.
Or
secondly
,
suppose
the
numbers
were
1
,
9
,
17
,
25
,
33
,
41.
Whereof
the
common
excesse
is
8
,
the
first
and
last
added
gives
42
,
which
multiplied
by
3
,
(
half
the
number
of
places
)
the
product
is
126
,
which
is
the
summe
of
them
all
.
3
For
the
third
thing
,
that
is
,
by
the
last
number
and
the
total
,
to
find
the
number
of
places
.
THE
RULE
.
Adde
the
first
and
last
numbers
,
and
by
the
summe
of
them
divide
the
total
,
the
quotient
will
be
equal
to
half
the
number
of
places
.
This
is
so
plain
it
needs
no
cleering
.
4
For
the
fourth
,
if
the
total
,
and
number
of
places
bee
given
,
to
find
the
last
number
.
THE
RULE
.
Divide
the
total
by
half
the
number
of
places
,
the
quotient
is
a
number
,
from
which
if
1
be
taken
,
the
rest
is
the
last
number
.
As
let
the
number
be
1
,
3
,
5
,
7
,
9
,
11
,
13
,
15
,
or
any
other
(
in
arithmetical
proportion
)
whatsoever
.
The
summe
of
these
is
to
be
64.
And
the
number
of
places
is
8
,
the
half
of
it
4.
Now
if
64
be
divided
by
4
,
the
quotient
is
16
,
from
which
if
1
be
taken
,
there
remains
15
for
the
last
number
.
5
Now
for
the
fist
variety
,
If
the
last
number
and
number
of
places
be
given
,
to
find
the
common
excesse
.
THE
RVLE
.
From
the
last
number
take
1
,
and
the
remaine
shall
be
the
Dividend
;
then
from
the
number
of
places
also
take
1
,
and
make
this
later
remaine
the
Divisor
▪
then
the
quotient
of
this
Division
shall
be
the
common
excesse
.
Example
,
Let
the
numbers
be
1
,
4
,
7
,
10
,
13
,
16
,
from
16
take
1.
remains
15
,
for
the
dividend
,
then
from
6
,
(
which
is
the
number
of
places
)
take
also
1
,
remains
5
for
the
Divisor
.
Now
when
15
is
divided
by
5
,
the
quotient
is
3.
And
3
is
also
the
common
excesse
,
or
difference
between
1
and
4
,
or
4
and
7
,
&c.
6
Lastly
,
let
the
last
number
,
and
the
common
excesse
be
given
,
to
find
the
number
of
places
.
THE
RULE
.
From
the
last
number
take
1
,
and
divide
the
remaine
by
the
common
excesse
;
then
to
the
quotient
adde
1
,
the
summe
is
the
number
of
places
.
As
let
the
numbers
be
1
,
5
,
9
,
13
,
17
,
21
,
25
,
29
,
from
29
take
1
,
remains
28
,
which
divide
by
4
,
(
which
is
the
excesse
)
the
quotient
is
7
,
to
which
adde
1
,
the
summe
is
8
,
which
is
the
number
of
places
,
as
the
reader
may
easily
count
.
Geometrical
Progression
.
I
shall
not
be
so
large
in
this
as
in
the
former
,
because
these
things
are
of
little
use
to
the
Arithmetician
,
except
where
a
number
is
to
be
many
times
doubled
,
tripled
,
or
the
like
,
which
cannot
be
so
easily
abridged
here
,
as
in
the
other
,
because
there
the
last
number
arising
of
many
additions
of
the
excesse
to
1
,
was
easily
found
by
one
multiplication
:
but
here
the
last
number
being
made
by
many
Multiplications
of
the
excesse
,
is
therefore
many
times
harder
then
the
other
The
varieties
here
shall
be
but
two
.
1
The
common
excesse
and
number
of
places
being
given
,
to
find
the
last
number
.
2
The
excesse
,
and
last
number
being
given
,
to
find
the
total
summe
.
The
first
of
these
may
thus
be
found
.
Let
the
numbers
be
1
,
2
,
4
,
8
,
16
,
32
,
64
,
128
,
256
,
512
,
the
exceesse
is
2
,
the
places
10
,
find
out
the
fifth
number
(
which
is
easily
done
,
for
any
one
may
reckon
so
farre
by
heart
,
)
that
is
here
16
,
and
multiply
16
by
16
,
it
produceth
256
,
which
is
the
ninth
number
,
lastly
,
multiply
256
by
the
excesse
2
,
thence
ariseth
512
,
the
number
desired
.
So
if
the
places
had
been
more
,
as
72
,
having
found
the
9
number
256
,
multiply
it
by
256
,
thence
comes
65536
for
the
17th
.
number
,
which
multiplied
by
the
excesse
2
,
gives
131072
for
the
18th
.
place
,
which
multiplied
by
131072
,
gives
17179869184
,
for
the
35
place
;
and
that
multiplied
againe
by
the
excesse
2
,
gives
34359738368
,
for
the
36
place
,
that
multiplied
by
34359738368
,
then
the
product
will
be
1180591620717411327424
,
for
the
71
place
,
which
lastly
,
multiplied
by
the
excesse
,
gives
2361183241434822654848
,
for
the
72
place
,
which
is
the
last
number
of
the
progression
required
to
be
found
.
Perhaps
this
may
seem
somewhat
tedious
but
where
things
cannot
be
performed
without
labour
,
the
Reader
must
content
himself
with
such
Rules
as
make
it
lesse
;
for
it
is
certaine
that
this
way
is
much
shorter
then
to
have
multiplied
still
by
the
excesse
71
times
,
which
else
he
must
have
done
.
All
this
notwithstanding
,
he
is
not
bound
to
use
the
same
numbers
,
much
lesse
in
other
questions
where
the
number
of
places
is
not
the
same
,
but
whereas
I
began
from
the
place
9
,
he
may
begin
at
8
,
10
,
or
12
,
or
where
he
pleases
;
so
as
he
remembers
still
where
he
is
;
for
this
is
general
,
if
the
number
belonging
to
any
place
whatsoever
,
be
multiplyed
by
it self
,
the
product
shall
be
the
number
belonging
to
twice
so
many
places
want
one
place
.
Now
for
the
second
thing
,
which
is
to
find
the
summe
of
all
the
numbers
.
THE
RULE
.
From
the
last
number
take
the
first
,
and
divide
the
remaine
by
the
excesse
want
1
,
then
multiply
the
quotient
by
the
excesse
;
and
to
the
product
adde
the
first
number
,
the
summe
of
them
is
equal
to
the
summe
of
all
the
numbers
.
So
if
from
the
last
number
,
or
72
place
be
taken
1
remain
is
2361183241434
22654847
which
should
be
divided
by
1
,
(
that
is
the
excesse
want
1
for
the
excesse
is
but
2
)
but
because
1
neither
multiplies
nor
divides
that
labour
is
saved
;
Now
multiply
this
remain
by
the
excesse
,
the
product
is
4722366482869645309696
,
to
which
adding
the
first
number
1
,
by
making
the
figure
6
next
the
right
hand
to
be
7
,
you
have
the
total
summe
of
all
the
72
numbers
.
A
Question
resolved
by
Geometrical
Progression
.
A
Londoner
sojourning
in
a
Countrey
Market
Towne
,
in
Winter
,
made
himselfe
a
new
freez
,
Suit
and
Coat
,
on
which
were
set
6
dozen
of
buttons
of
Silke
and
Silver
;
A
Baker
being
in
his
company
liked
it
so
well
he
would
buy
it
of
him
;
the
Citizen
consented
to
let
him
have
it
,
paying
for
the
first
button
a
single
barley
corn
,
for
the
second
2
,
for
the
third
4
,
and
so
on
doubling
to
the
last
.
The
bargaine
was
liked
on
both
parts
for
the
present
,
but
shortly
after
revoked
,
for
it
could
not
be
performed
,
and
no
man
can
be
holden
to
an
impossibility
.
But
why
this
could
not
be
performed
:
may
be
judged
;
first
,
by
inquiring
the
worth
of
so
much
barley
in
Money
:
And
secondly
,
the
weight
of
it
;
and
how
it
should
be
removed
.
1
For
the
first
,
allowing
10000
corns
to
a
pinte
(
which
is
more
then
enough
)
then
5120000
corns
make
a
quarter
;
and
yet
(
for
shortning
the
division
)
we
will
allow
10000000
corns
to
a
quarter
;
by
which
dividing
the
whole
number
of
corns
(
which
is
done
by
cutting
off
the
first
7
figures
toward
the
right
hand
,
)
the
quotient
will
be
472239648286964
,
and
so
many
whole
quarters
there
are
omitting
the
remaine
,
as
in
this
case
not
considerable
.
Now
allowing
barley
were
to
be
sold
at
15
d.
the
bushel
(
which
is
cheap
,
)
it
is
so
many
Angels
:
and
therefore
dividing
by
2
it
is
23611832414143482
pounds
sterling
:
which
is
in
words
,
Two
hundred
thirty
six
millions
of
millions
,
one
hundred
and
eighteen
thousand
,
three
hundred
twenty
four
millions
:
one
hundred
forty
three
thousand
;
four
hundred
eighty
two
pounds
,
whick
I
take
to
be
too
much
for
any
Trades-man
to
get
or
keep
.
And
reckoning
land
for
ever
at
twenty
yeares
purchase
,
if
this
summe
of
pounds
be
divided
by
20
,
the
quotient
is
the
yearly
rent
of
11805916207174
pounds
.
And
this
divided
again
by
365
(
the
number
of
dayes
in
a
year
)
the
quotient
32344975918
,
that
is
above
thirty
two
thousands
of
millions
a
day
for
ever
.
So
great
a
vanity
may
be
concluded
on
for
want
of
a
little
premeditation
.
2
Now
,
secondly
,
for
the
weight
of
it
,
if
we
put
8
Bushels
to
weigh
2
hundred
pound
weight
(
for
sure
it
doth
weigh
more
)
then
the
whole
number
of
quarters
multiplied
by
2
,
gives
the
weight
of
all
to
be
944473296573928
hundred
weight
,
and
if
this
be
divided
by
20
,
(
which
is
but
cutting
off
one
figure
toward
the
right
hand
,
and
dividing
the
rest
by
2
)
or
which
is
all
one
,
cut
off
one
figure
from
the
number
of
quarters
,
the
quotient
47223664828696
,
is
so
many
tuns
.
And
therefore
it
will
require
47223664828
ships
of
1000
a
piece
to
carry
it
:
And
consequently
if
every
Nation
in
the
World
had
above
10000
such
ships
,
yet
there
must
be
above
four
millions
of
such
Nations
:
which
I
suppose
are
not
to
be
found
in
this
World
.
And
here
I
will
leave
this
,
having
used
this
long
example
,
(
which
though
it
require
more
labour
as
all
great
examples
do
,
yet
the
same
skill
will
do
it
,
as
if
the
places
had
been
fewer
)
that
the
Reader
being
throughly
exercised
thereby
,
may
the
easier
leap
over
others
which
are
shorter
.
THE
GOLDEN
RULE
,
Or
Rule
of
Three
.
THis
is
the
most
useful
and
most
easie
Rule
in
Arithmetique
,
and
deserves
a
Golden
name
.
It
is
when
there
are
three
numbers
given
or
known
to
find
a
fourth
in
proportion
with
them
.
But
4
numbers
are
in
proportion
,
and
called
Proportional
,
when
as
the
first
is
to
the
third
,
so
is
the
second
to
the
fourth
.
As
if
there
were
given
,
3
,
4
,
and
6
,
to
find
a
fourth
which
may
be
to
4
,
as
6
to
3
,
that
is
double
,
and
that
fourth
number
is
8
;
And
this
is
called
Proportion
direct
:
and
the
Rule
whereby
it
is
done
,
The
Direct
Rule
.
There
is
also
another
proportion
called
Reciprocal
;
which
is
when
as
the
first
is
to
the
third
,
so
is
the
fourth
to
the
second
:
As
3
,
4
,
6
,
and
2
,
this
is
called
The
Reverse
Rule
.
In
direct
proportion
,
the
product
of
the
two
middle
numbers
multiplied
together
,
is
ever
equal
to
the
product
of
the
first
and
last
multiplied
together
;
which
serves
not
onely
for
a
Proofe
,
but
a
ground
of
the
Rule
,
which
Rule
shall
here
follow
:
the
Reverse
Rule
being
deferred
till
we
have
done
with
this
.
The
Rule
Direct
.
Multiply
the
second
terme
(
or
number
)
by
the
third
and
divide
the
product
by
the
first
;
the
quotient
shall
be
the
fourth
number
desired
.
Example
.
Let
the
three
numbers
given
be
2
,
6
3
,
multiply
6
by
3
,
the
product
is
18
,
then
divide
18
by
2
,
the
quotient
is
9
,
which
is
th
fourth
number
in
proportion
with
2
,
6
,
and
3.
For
as
2
to
3
,
so
3
times
2
,
which
is
6
,
is
to
3
times
●
▪
which
is
9.
And
so
the
product
18
divided
by
2
,
and
the
quotient
9
,
causeth
that
the
product
of
2
into
9
shall
be
also
18
,
and
consequently
if
2
be
the
first
of
the
4
proportional
numbers
,
and
6
and
3
the
two
middlemost
,
then
9
is
the
last
.
Otherwise
.
Divide
the
second
by
the
first
,
and
multiply
the
third
by
the
quotient
,
the
product
shall
be
the
fourth
.
So
if
one
divide
6
by
2
,
the
quotient
is
3
,
by
which
multiply
3
,
the
product
is
9
,
for
the
fourth
number
,
as
before
,
otherwise
still
this
Rule
might
be
expressed
;
but
where
the
first
way
is
so
short
and
clear
,
there
many
other
wayes
would
rather
trouble
then
help
the
person
that
should
use
them
.
In
the
first
way
(
which
here
we
mean
to
use
and
no
other
)
if
the
first
number
be
1
,
then
the
product
of
the
second
and
third
gives
the
fourth
,
without
any
Division
.
Or
if
the
second
,
or
third
number
be
1
,
then
there
needs
no
multiplication
,
but
dividing
the
greater
of
them
by
the
first
,
the
quotient
(
in
whole
numbers
for
yet
we
speak
of
them
)
is
the
fourth
number
,
which
was
sought
.
Note
1.
To
know
when
to
use
the
direct
,
or
the
Reverse
Rule
,
consider
,
if
more
,
require
more
;
or
if
lesse
,
require
still
lesse
:
then
use
the
direct
Rule
,
but
if
more
require
lesse
,
or
lesse
more
,
then
use
the
Reverse
Rule
,
this
will
be
easily
understood
when
we
come
to
examples
.
Note
2.
To
know
how
to
place
the
three
numbers
when
they
are
confusedly
given
,
remember
that
two
of
them
are
alwayes
of
one
denomination
,
as
both
pounds
,
or
both
sheep
,
or
both
yards
,
or
acres
;
and
the
other
number
hath
another
denomination
,
now
know
that
this
single
number
is
ever
the
second
number
in
order
.
And
one
of
the
other
two
,
namely
,
that
which
hath
some
relation
to
this
second
,
is
the
first
;
and
the
other
is
the
third
number
,
whose
relation
is
sought
for
in
the
fourth
,
whence
its
plaine
that
the
second
and
fourth
are
also
of
the
same
denomination
.
And
having
premised
these
things
,
let
us
now
exemplifie
the
Rule
in
some
questions
.
Question
1.
If
3
yards
of
cloth
cost
4
li.
what
shall
21
yards
cost
?
Set
the
numbers
in
order
as
in
the
example
,
if
3
yards
cost
4
li.
what
21
yards
?
Here
you
see
that
the
first
number
and
the
third
number
are
both
of
one
denomination
,
viz.
both
yards
,
and
the
second
number
is
of
another
denomination
,
namely
pounds
,
wherefore
the
fourth
number
which
is
sought
for
,
must
be
also
pounds
,
therefore
multiplying
(
according
to
the
Rule
before
given
)
the
second
number
by
the
third
,
and
dividing
the
product
by
the
first
,
the
quotient
shall
answer
the
question
.
First
,
21
multiplied
by
4
,
(
which
is
the
third
●●mber
multiplied
by
the
second
)
produceth
84
,
which
divided
by
3
the
first
number
,
the
quotient
is
28
li.
and
so
much
shall
21
yards
cost
:
for
28
is
to
4
,
as
21
to
3
,
seeing
each
containes
either
7
times
.
And
the
work
will
stand
thus
,
Question
2.
If
4
men
eat
2
pecks
of
corn
in
a
week
,
how
many
pecks
shall
serve
100
men
?
Place
your
numbers
as
here
you
see
,
then
multiply
100
by
2
,
(
that
is
the
third
number
by
the
second
,
and
the
product
is
200
,
which
divided
by
4
,
the
quotient
is
50
,
for
the
number
of
pecks
required
.
Question
3.
If
20
sheep
cost
13
pound
13
shillings
4
pence
,
what
is
that
for
every
sheep
?
Turn
the
shillings
and
pounds
into
pence
,
thus
,
Multiply
12
s.
by
12
the
product
is
|
156
|
And
13
li.
by
240
(
because
240
pence
make
one
pound
)
the
product
is
|
3120
|
To
which
adde
the
4
d.
|
4
|
It
makes
in
all
|
3280
|
Then
the
question
will
be
,
If
20
sheep
cost
3280
pence
,
what
shall
one
sheep
cost
?
By
the
rule
before
delivered
,
I
should
multiply
the
second
number
by
the
third
,
but
in
this
example
,
the
third
number
being
1
,
it
doth
not
multiply
;
I
therefore
divide
3280
the
second
number
,
by
20
the
first
number
,
and
the
motient
164
,
is
the
price
of
one
sheep
in
pence
,
which
divided
by
12
,
the
quotient
is
13
s.
and
8
d.
remaining
,
the
price
of
every
sheep
therefore
is
13
s.
8
d.
Question
4.
How
many
10
inch
tiles
will
pave
a
floore
that
contains
16
square
yards
?
First
,
remember
there
are
36
inches
in
one
yard
in
length
;
which
multiplied
into
36
,
gives
1296
,
for
the
square
inches
in
one
square
yard
;
multiply
1296
therefore
by
16
,
thence
comes
20736
,
the
summe
of
all
the
16
yards
in
inches
.
Secondly
,
seeing
every
tile
is
10
inches
in
length
,
and
10
in
breadth
,
multiply
10
by
10
,
it
produceth
100
for
the
square
inches
in
one
tile
;
See
the
manner
of
worke
.
Then
by
the
golden
Rule
.
If
100
inches
require
1
tile
;
what
shall
207
;
6
inches
require
?
Here
because
1
doth
neither
multiply
nor
divide
(
as
hath
been
several
times
intimated
)
I
therefore
divide
the
third
20736
,
by
the
first
100
,
the
quotient
is
207
,
and
36
remaining
.
So
it
appears
that
207
is
too
little
,
and
208
too
much
to
doe
the
worke
:
the
just
number
being
207
36/100
,
we
shall
not
trouble
the
reader
with
this
till
he
know
something
of
Fractions
.
Question
5.
If
100
li.
give
6
li.
interest
for
a
year
,
how
much
shall
750
li.
give
?
Multiply
750
by
6
,
the
product
is
4500
,
which
divided
by
100
,
the
quotient
is
45
li.
for
the
thing
required
.
Question
6.
If
750
li.
give
45
li.
interest
for
a
year
,
what
shall
100
li.
give
?
Multiply
45
by
100
,
the
product
is
4500
,
which
divided
by
750
,
the
quotient
is
6
li.
for
the
interest
of
100
li.
for
a
year
.
Many
other
questions
might
be
added
,
but
the
rule
is
so
plaine
that
it
needs
them
not
;
and
so
general
,
that
he
which
can
resolve
one
,
may
aswel
resolve
any
other
:
And
for
that
reason
;
and
because
in
all
the
Rules
which
follow
,
this
rule
will
be
constantly
made
use
of
,
I
will
say
no
more
of
it
here
.
The
Golden
Rule
Reverse
.
Question
1.
IF
12
workmen
do
any
piece
of
worke
in
8
moneths
:
how
many
workmen
shall
do
the
same
in
2
moneths
?
THE
RVLE
.
Multiply
the
first
term
by
the
second
;
and
divide
the
Product
by
the
third
,
the
quotient
is
the
number
desired
.
Here
12
is
not
the
first
number
,
though
it
be
first
named
;
but
the
three
numbers
placed
in
order
,
stand
thus
,
8
,
12
,
2
,
for
the
middle
terme
must
alwayes
be
of
the
same
denomination
with
that
which
is
required
.
Now
multiply
12
by
8
,
the
product
is
96
,
which
divided
by
2
,
the
quotient
is
48
,
which
answers
the
question
,
As
in
the
example
,
For
if
8
moneths
require
12
men
;
then
(
a
fourth
part
of
8
)
2
moneths
shall
require
(
four
times
12
,
that
is
)
48
men
.
For
here
lesse
requires
more
,
that
is
lesse
time
,
more
hands
;
and
therefore
is
it
wrought
by
the
Reverse
Rule
.
Question
2.
How
many
Ells
of
Tapestry
will
serve
to
hang
a
roome
3
yards
high
,
6
yards
long
;
and
5
yards
broad
;
not
regarding
Doors
,
Windowes
,
or
Chimney
,
but
as
if
there
were
no
such
?
First
,
multiply
6
by
3
,
the
product
is
18
,
which
doubled
(
because
there
are
2
sides
called
lengths
)
is
36
yards
for
all
the
length
.
Secondly
(
for
the
same
reason
)
multiply
3
by
twice
5
,
that
is
by
10
,
the
product
is
30
yards
,
for
all
the
breadth
;
which
added
to
36
,
gives
66
yards
equal
to
all
the
length
and
breadth
in
yards
.
But
now
because
Ells
,
that
is
,
Flemish
Ells
(
for
such
measure
are
hangings
sold
by
)
is
equal
to
3
quarters
of
a
yard
,
that
is
,
their
Ell
is
to
our
Yard
as
3
to
4.
Say
therefore
if
4
give
66
,
what
3
?
multiply
66
by
4
,
it
produceth
264
;
then
divide
264
by
3
,
the
quotient
is
88.
Againe
,
multiply
88
by
4
,
and
divide
the
product
(
which
is
352
)
by
3
,
the
quotient
is
117
,
and
1
remaining
,
to
which
the
divisor
3
being
applyed
;
the
number
justly
answering
the
question
is
117
Ells
,
and
one
third
part
of
an
Ell.
Note
1.
Because
here
we
had
to
deal
with
things
which
had
equal
length
and
breadth
,
that
is
square
yards
,
and
square
ells
;
therefore
one
multiplication
and
division
was
not
sufficient
to
proportion
this
;
but
if
instead
of
working
by
4
and
3
,
we
had
done
it
by
their
squares
which
are
16
and
9
,
it
might
have
been
performed
at
once
,
thus
,
multiply
66
by
16
,
product
is
1056
,
which
divided
by
9
,
the
quotient
is
117⅓
,
as
before
,
but
I
began
not
with
this
way
,
for
I
supposed
my
reader
ignorant
of
squares
.
Note
2.
It
might
also
have
been
done
,
by
reducing
all
the
terms
into
quarters
of
a
yard
at
the
first
,
and
after
the
number
is
found
,
reducing
them
againe
to
ells
;
but
because
it
is
more
proper
to
worke
thus
,
till
fractions
have
been
taught
,
I
leave
that
,
and
proceed
to
another
question
.
Question
3.
If
1
close
would
graze
21
horses
for
6
weeks
;
then
(
supposing
no
waste
to
be
made
)
how
many
horses
would
it
feed
for
7
weeks
?
Multiply
21
by
6
it
produceth
126
,
which
divided
by
7
,
the
quotient
is
18.
At
that
rate
therefore
it
would
keep
18
horses
for
7
weeks
.
Question
4.
If
1
close
will
feed
18
horses
for
7
weeks
,
how
long
shall
it
feed
63
horses
.
Multiply
(
according
to
the
rule
)
18
by
7
,
the
product
is
126
,
which
divided
by
63
,
the
quotient
is
2
,
therefore
2
weeks
it
shall
keep
them
.
The
like
way
serves
for
hay
,
oates
,
or
any
other
provision
for
Man
or
Beast
;
which
may
be
of
use
in
Garrisons
,
and
such
like
cases
where
scarcity
may
be
feared
,
to
proportion
either
the
mouthes
to
the
meat
,
or
meat
to
mouthes
I
will
say
no
more
of
this
Rule
;
Neither
will
I
treat
of
the
double
Rule
of
Three
,
as
a
rule
by
it selfe
;
but
come
to
the
Rule
of
five
numbers
,
which
is
held
an
abridgment
of
the
other
.
The
Golden
Rule
Compound
of
five
Numbers
.
Question
1.
IF
a
hundred
pound
weight
(
that
is
112
pound
weight
)
carryed
120
miles
cost
14
s.
how
much
shall
three
quarters
of
a
hundred
,
(
that
is
84
pound
)
cost
,
being
carryed
40
miles
?
THE
RULE
.
Multiply
the
three
last
numbers
one
into
another
,
(
that
is
)
the
third
by
the
fourth
,
and
that
product
by
the
fifth
;
the
last
product
shall
be
the
Dividend
.
Again
,
Multiply
the
two
first
numbers
together
;
the
product
shall
be
the
divisor
.
This
Division
being
made
,
the
quotient
will
be
the
number
of
shillings
desired
.
Example
of
the
former
Question
.
First
,
place
your
numbers
according
to
the
tenor
of
the
question
thus
,
Your
numbers
being
placed
in
order
,
reduce
the
13
s.
into
pence
,
and
it
is
168
d.
then
multiply
168
by
84
,
the
product
is
14112
,
which
multiplied
by
40
the
later
product
,
it
produceth
564480
,
for
the
dividend
.
Then
multiply
112
by
120
it
produceth
13440
,
for
the
Divisor
.
Divide
5644800
,
by
13440
,
the
quotient
will
be
42
pence
;
which
is
3
s.
6
d.
and
answers
the
question
.
In
this
Rule
the
first
number
and
fourth
,
also
the
second
and
fifth
;
and
also
the
third
and
sixth
,
are
of
like
denomination
and
nature
.
Question
.
2.
If
100
li.
for
6
moneths
yield
3
li.
interest
,
what
shall
625
li.
yeield
for
36
moneths
?
Place
them
,
100
,
6
,
3
,
625
,
36.
Multiply
the
three
last
,
as
before
is
shewed
,
the
later
product
is
67500
for
the
dividend
;
And
the
2
first
multiplied
make
600
the
Divisor
,
then
divide
67500
by
600
(
or
675
by
6
,
which
is
all
one
)
the
quotient
will
be
112
whole
pounds
;
and
300
(
or
3
)
remaining
,
which
because
it
is
half
the
divisor
,
fignifies
the
half
of
a
pound
;
that
is
10
shillings
.
So
the
answer
to
the
question
is
112
li.
10
s.
Which
might
have
been
given
in
one
denomination
,
namely
2250
shillings
,
if
before
the
work
the
pounds
had
been
turned
into
shillings
,
by
multiplying
them
by
20
,
as
hath
been
shewed
before
.
But
since
most
questions
,
except
such
as
are
studied
for
the
purpose
,
are
apt
to
end
in
some
fraction
,
I
shall
next
treat
of
fractions
.
Onely
first
,
having
spoken
of
the
double
Rule
of
Three
,
this
may
let
you
know
,
that
all
questions
which
are
wrought
at
once
by
the
compound
rule
of
five
;
may
be
done
at
twice
by
the
single
rule
of
three
:
and
the
doing
of
them
so
by
two
operations
is
is
called
the
double
rule
.
As
in
our
last
question
there
are
two
things
considerable
,
the
difference
of
money
;
and
the
difference
of
time
.
First
,
for
the
money
,
Say
,
if
100
li.
give
3
li.
what
625
li.
?
answer
18
75/100
li.
Secondly
,
for
the
time
;
Say
if
6
li.
give
18
75/100
li.
what
36
mo
.
answer
112
50/100
li.
But
this
will
be
better
understood
anon
;
and
then
the
Reader
may
use
that
which
he
likes
best
.
Of
Fractions
.
THe
word
Fraction
signifies
a
breaking
or
breach
of
any
intire
thing
into
parts
;
and
when
a
number
is
broken
so
,
the
parts
(
which
must
needs
be
every
one
lesse
then
the
whole
;
and
the
whole
is
accounted
but
One
,
or
Vnity
)
being
lesse
then
Unity
,
are
called
fractions
(
that
is
fragments
,
or
pieces
)
of
Unity
.
Now
the
Unite
,
or
intire
number
which
is
to
be
broken
,
may
be
any
thing
,
as
one
pound
,
in
respect
of
which
,
shillings
and
pence
,
and
farthings
are
fractions
;
or
one
shilling
,
in
respect
of
which
,
pence
,
and
farthings
are
fractions
;
or
one
peny
,
in
respect
of
which
,
farthings
are
fractions
;
and
the
like
of
weights
and
measures
,
or
any
other
thing
to
be
broken
into
parts
.
In
Fractions
,
we
shall
treat
first
of
Numeration
,
then
of
Multiplication
and
Division
,
then
of
Reduction
;
then
lastly
,
of
Addition
and
Subtraction
.
The
reason
of
this
Order
will
soon
be
seen
;
for
Multiplication
and
Division
are
here
much
easier
then
Addition
,
&c.
and
therefore
ought
to
be
learned
before
them
.
Numeration
.
NUmeration
is
nothing
else
but
the
way
of
writing
Fractions
;
and
that
this
may
be
done
,
we
must
consider
that
any
Unite
,
or
Number
representing
an
Unite
,
may
be
broken
into
two
parts
equal
;
and
then
each
of
the
parts
is
called
one
second
,
or
halfe
;
or
it
may
be
parted
into
three
equal
parts
,
and
then
each
part
is
called
one
third
;
and
two
of
them
are
called
two
thirds
;
and
the
like
may
be
understood
if
it
were
parted
into
4
,
5
,
6
,
7
,
8
,
9
,
20
,
50
,
or
100
,
or
how
many
soever
.
Now
to
write
these
;
doe
thus
,
-
Write
one
half
Thus
½
-
Write
one
third
Thus
⅓
-
Write
one
fourth
Thus
¼
-
Write
one
fifth
Thus
⅕
-
Write
one
sixth
Thus
⅙
-
Write
one
seventh
Thus
1/7
-
Write
one
eighth
Thus
⅛
-
Write
one
nine
Thus
1/9
-
Write
one
tenth
Thus
1/10
In
every
one
of
these
10
fractions
,
the
Number
below
the
line
is
called
the
Denominator
,
and
it
shews
into
how
many
parts
the
Unite
is
broken
.
The
Number
above
the
line
shews
how
many
of
those
parts
are
taken
,
or
contained
in
the
fraction
,
and
is
therefore
called
the
Numerator
:
So
in
the
fraction
⅗
:
the
Denominator
●
shewes
the
Unite
to
be
broken
into
5
parts
:
and
the
Numerator
3
signifies
3
of
such
parts
to
be
contained
in
the
fraction
;
which
fraction
therefore
is
called
three
fifths
.
And
here
it
is
plaine
,
that
as
the
Numerator
is
in
proportion
to
the
Denominator
:
so
is
the
fraction
to
1
,
for
3/3
or
5/5
:
or
any
the
like
,
is
equal
to
1.
And
therefore
all
fractions
are
quotients
of
lesser
numbers
divided
by
greater
,
as
4/7
signifies
4
to
be
divided
by
7
,
and
as
the
dividend
4
,
is
to
the
divisor
7
:
so
is
the
quotient
4/7
to
Unity
.
And
therefore
this
line
of
separation
which
is
drawn
between
the
dividend
and
divisor
,
doth
properly
signifie
division
.
Hitherto
we
have
spoken
onely
of
such
fractions
as
are
lesse
then
1
,
and
those
are
called
Proper
Fractions
:
but
there
are
also
2½
,
3¾
,
5
1/7
,
6⅗
,
and
the
like
mixed
Numbers
;
which
so
written
signifie
two
and
an
half
,
3
and
3
quarters
,
five
and
a
seventh
,
6
and
3
fifths
.
These
by
multiplying
the
whole
Numbers
,
by
the
Denominator
,
and
to
the
product
adding
the
Numerators
respectively
are
turned
to
5/2
,
15/4
,
36/7
,
33/5
,
which
are
called
Improper
fractions
,
because
every
one
of
them
contains
more
then
Unity
.
These
,
neverthelesse
may
be
multiplyed
,
divided
,
added
,
or
subtracted
in
the
same
way
as
are
proper
fractions
.
And
this
shall
serve
for
Numeration
of
Fractions
.
Multiplication
.
THE
RULE
.
MVltiply
all
the
Numerators
together
,
the
last
product
shall
be
the
Numerator
of
the
product
required
:
Likewise
multiply
all
the
Denominators
together
,
the
last
product
shall
be
the
Denominator
of
the
product
sought
.
Example
1.
If
⅗
be
multiplyed
by
4/9
,
multiply
the
Numerator
3
by
the
Denominator
4
,
the
product
is
12
,
for
the
Numerator
of
the
new
product
.
Also
multiplying
the
Denominator
5
,
by
the
Denominator
9
,
they
produce
45
,
for
the
Denominator
of
the
desired
product
,
so
that
product
which
was
required
,
is
12/45.
Example
2.
If
½
,
¾
,
⅘
,
5/9
,
and
3/11
were
to
be
multiplyed
all
together
,
begin
with
the
Numerators
,
saying
,
once
3
is
3
,
and
3
times
4
is
12
,
and
12
times
5
is
60
,
and
60
times
5
is
180
,
for
the
Numerator
:
Then
multiply
the
Denominators
:
saying
2
times
4
is
8
,
and
8
times
5
is
40
,
and
40
times
9
is
360
,
and
360
times
11
is
3960
,
for
the
new
Denominator
.
So
that
the
product
of
all
these
is
180/3960
,
that
is
equal
to
1/22
,
as
shall
be
seen
hereafter
in
Reduction
.
And
thus
it
appears
that
proper
fractions
being
lesse
then
one
,
are
still
made
lesse
by
multiplying
;
as
here
the
product
1/22
is
much
lesse
then
3/11
,
which
is
the
least
Multiplyer
,
and
the
reasons
hereof
is
plain
,
for
seeing
Multiplication
is
but
the
taking
of
a
Number
,
a
certaine
number
of
times
,
if
that
number
of
times
be
more
then
1
,
then
the
Number
to
be
taken
is
increased
by
being
taken
more
then
once
;
but
if
the
Number
of
times
be
1
,
it
is
not
increased
,
nor
diminished
,
but
is
still
the
same
;
Lastly
,
If
that
number
of
times
be
lesse
then
1
,
as
½
,
the
number
not
being
taken
once
,
but
half
of
once
,
produceth
a
number
lesse
by
half
;
that
is
the
half
of
the
number
to
be
taken
;
and
the
like
reason
is
of
all
others
.
Example
3.
Multiply
the
mixt
Numbers
3½
,
4⅓
,
and
5¾
:
First
,
(
as
hath
been
shewn
already
)
turn
them
to
improper
fractions
:
thus
,
first
,
say
2
times
3
is
6
,
and
1
is
7.
So
the
first
is
7/2.
Secondly
,
3
times
4
is
12
,
and
1
is
13
:
so
the
second
is
13/3.
Lastly
,
4
times
5
is
20
,
and
3
is
23
:
so
the
last
is
23/4
Now
the
fractions
to
be
multiplyed
are
7/2
,
13/3
,
and
23/4
;
First
,
for
a
new
Numerator
,
say
,
7
times
13
is
91
,
and
91
times
23
is
2093
,
for
a
new
Numerator
.
Then
say
,
2
times
3
is
6
,
and
6
times
4
is
24.
So
the
new
Denominator
is
23.
And
the
product
of
all
these
fractions
is
2093/24
,
that
is
,
if
real
division
be
made
,
87
5/24.
Division
.
DIvision
of
one
fraction
by
another
,
is
but
crosse
multiplication
of
them
;
that
is
,
the
Numerator
of
the
one
,
by
the
Denominator
of
the
other
,
and
hereby
the
proportion
of
one
fraction
to
another
is
seen
.
Example
1.
Divide
¾
by
6/8
,
to
do
it
,
set
them
thus
:
and
multiply
as
the
crosse
leads
,
saying
,
3
times
8
is
24
,
for
a
new
Numerator
,
and
6
times
4
is
also
24
,
for
a
new
Denominator
;
so
the
quotient
is
24/24
,
that
is
1
,
which
shewes
the
fractions
to
be
equal
one
to
another
.
Example
2.
In
division
it
is
to
be
remembred
that
the
Numerator
of
the
quotient
ever
ariseth
of
the
Numerator
of
the
Dividend
;
And
the
Denominator
of
the
quotient
comes
of
the
Denominator
of
the
Dividend
,
each
being
crosse
multiplied
,
as
before
.
If
a
Fraction
be
to
be
divided
by
a
whole
number
,
multiply
the
Denominator
by
that
number
,
the
product
gives
the
new
Denominator
,
and
the
Numerator
remaines
the
same
.
So
if
¼
be
divided
by
9
,
say
9
times
4
is
36.
So
the
quotient
is
1/36.
Or
if
¼
were
to
be
multiplyed
by
9
,
the
product
(
by
multiplying
the
Numerator
by
9
)
,
will
be
9/4
:
that
is
,
2¼
.
Example
3.
For
320/8
is
equal
to
40
,
and
45/9
equal
to
5
,
but
40
contains
5
eight
times
.
And
so
in
the
second
example
,
it
may
be
proved
,
that
as
27
to
20
,
so
is
⅗
to
4/9.
For
first
multiply
the
two
middlemost
,
then
20
times
⅗
is
60/5
,
that
is
12.
Secondly
,
multiply
the
first
and
last
,
and
then
27
times
4/9
is
108/9
:
that
is
also
12.
Wherefore
by
that
which
hath
been
said
in
the
Golden
Rule
,
the
four
Numbers
27
,
20
,
3/5
,
4/2
,
are
proportional
.
Reduction
.
REductioon
of
Fractions
is
threefold
.
1
To
reduce
one
fraction
(
which
is
not
already
in
the
least
)
to
a
lesser
Denomination
.
2
To
reduce
many
fractions
of
divers
denominations
,
to
one
denomination
.
3
To
reduce
any
fraction
from
one
denomination
(
as
neer
as
may
be
)
to
any
other
denomination
desired
.
For
the
first
of
these
,
divide
both
the
Numerator
,
and
the
Denominator
,
by
the
greatest
common
divisor
that
you
can
think
of
;
the
two
quotients
being
placed
respectively
in
a
fraction
,
that
fraction
shall
be
equal
to
the
former
fraction
,
and
in
lesser
termes
.
So
(
in
the
3
Example
of
Division
)
to
reduce
2880/360
to
8/1
,
divide
2880
by
360
,
quotient
is
8
,
then
divide
360
by
360
,
the
quotient
is
1
,
and
the
new
fraction
8/1
is
equal
to
the
former
fraction
,
2880/360
,
and
in
lesse
termes
;
as
you
may
see
.
But
to
find
the
greatest
common
divisor
.
The
Rule
is
,
Divide
the
greater
terme
by
the
lesser
(
I
mean
by
termes
,
the
Numerator
and
Denominator
)
and
by
the
remainer
(
if
any
be
)
divide
the
divisor
,
and
if
any
thing
still
remains
,
by
that
divide
the
last
divisor
,
continuing
this
course
till
nothing
remain
greater
then
Vnity
;
that
divisor
which
is
last
of
all
is
the
greatest
common
measure
of
both
termes
,
by
which
both
being
divided
,
and
the
quotients
placed
like
a
fraction
,
that
fraction
shall
be
equal
to
the
former
fraction
,
and
in
the
least
termes
.
Example
.
Reduce
148/16
to
the
least
termes
;
first
divide
148
by
16
,
the
quotient
is
9
,
and
4
remaines
:
againe
,
divide
16
by
4
,
the
quotient
is
4
,
and
nothing
remaines
;
wherefore
taking
4
,
(
the
last
divisor
)
for
the
greatest
common
divisor
,
by
it
divide
148
,
the
quotient
is
37
,
and
by
it
divide
16
,
the
quotient
is
4.
These
two
last
quotients
placed
orderly
in
a
fraction
,
make
37/4
,
which
is
equal
to
148/16
,
and
in
the
least
terms
,
for
no
number
greater
then
1
,
will
divide
evenly
both
37
and
4.
Other
wayes
there
are
of
lessening
fractions
,
as
dividing
the
termes
(
if
they
be
even
Numbers
)
by
2
,
and
the
quotients
(
if
even
)
again
by
2
,
or
else
by
3
,
o●
any
other
Number
that
will
divide
them
both
evenly
,
that
is
,
leave
nothing
remaining
,
but
the
former
Rule
being
general
and
easie
,
shall
serve
for
all
.
Now
secondly
,
to
reduce
many
Denominations
to
one
common
denominator
;
let
the
fractions
be
½
,
¾
,
⅘
,
⅞
,
9/10
,
to
be
reduced
all
to
the
denomination
3200.
THE
RULE
.
Multiply
all
the
Denominators
together
:
saying
2
times
4
is
8
,
and
8
times
5
is
40
,
and
40
times
8
is
320
,
and
320
times
10
is
3200
,
this
last
product
3200
,
shall
be
the
common
Denominator
.
Then
to
get
Numerators
for
every
one
of
them
.
Multiply
every
particular
Numerator
into
all
the
Denominators
except
his
own
:
As
first
,
for
the
first
,
say
1
time
4
is
4
,
and
4
times
5
is
20
,
and
20
times
8
is
160
,
and
160
times
10
is
1600.
For
the
first
Numerator
;
so
the
first
fraction
reduced
is
1600/3200.
Then
for
the
second
Numerator
:
say
,
3
times
2
is
6
,
and
6
times
5
is
30
,
and
30
times
8
is
240
,
and
240
times
10
is
2400.
So
the
second
fraction
reduced
,
is
2400/3200.
After
the
same
manner
may
the
other
three
be
reduced
to
2560/3200
for
the
third
:
2800/3200
for
the
fourth
:
and
2880/3200
for
the
last
;
these
are
severally
equal
to
the
other
,
the
first
to
the
first
,
&c.
as
may
be
proved
;
thus
,
Let
the
Unity
be
a
pound
Sterling
,
then
|
s.
|
|
The
½
of
it
is
|
10
|
|
and
¾
is
|
15
|
|
and
⅘
is
|
16
|
d.
|
and
⅞
is
|
17
|
6
|
and
9/10
is
|
18
|
|
|
s.
|
d.
|
In
all
|
76
|
6
|
That
is
3
whole
Unites
,
and
16
s.
6
d.
over
,
Turn
16
s.
6
d.
all
to
six
pences
,
it
is
33
,
and
because
6
d.
is
the
fortieth
part
of
a
pounds
,
therefore
all
the
fractions
are
equal
to
3
33/40.
Now
adde
the
new
Fractions
(
which
being
all
of
one
denomination
)
may
be
added
like
whole
Numbers
:
thus
,
Which
divided
by
the
Denominator
3200
,
the
quotient
is
3
2640/3200.
Now
2640/3200
,
reduced
to
the
least
termes
,
as
hath
been
shewed
how
it
may
,
will
be
33/40
,
so
the
sum
of
these
also
is
3
33/40.
which
is
equal
to
the
sum
of
the
fractions
given
to
be
reduced
,
and
therefore
they
are
equal
in
sum
,
and
might
be
thus
proved
equal
severally
,
that
is
,
the
first
of
them
propounded
,
to
the
first
reduced
.
Divide
the
Numerator
1600
by
the
Numerator
1
,
quotient
is
1600.
Also
divide
the
Denominator
3200
,
by
the
Denominator
2
,
the
quotient
is
also
1600
,
and
so
may
any
of
the
rest
be
proved
equal
by
the
equality
of
quotients
.
But
I
leave
it
as
plaine
enough
already
.
Thirdly
,
Any
fraction
being
given
,
to
change
the
denomination
to
any
other
more
requisite
,
retaining
still
(
as
neer
as
may
be
)
the
same
value
.
THE
RVLE
.
Multiply
the
Numerator
given
,
by
the
Denominator
required
,
and
divide
the
product
by
the
Denominator
given
;
the
quotient
shall
be
the
Numerator
required
.
Example
.
Let
the
Fraction
given
be
7/13
of
a
pound
Sterling
,
what
is
that
in
twentieth
parts
or
shillings
?
Multiply
7
by
20
,
the
product
is
140
,
which
divided
by
13
,
the
quotient
10
10/13
,
that
is
,
10
s.
and
10/13
of
a
shilling
:
which
may
be
brought
to
pence
thus
,
multiply
10
by
12
,
product
is
120
,
which
divided
by
13
,
quotient
9
3/13
d.
And
againe
,
multiply
3
by
4
,
the
product
is
12
,
which
divided
by
13
,
quotient
is
12/13
of
a
farthing
,
so
seven
thirteenths
of
a
pound
is
10
s.
9
d.
and
almost
a
farthing
.
But
he
which
is
resolved
to
have
it
in
the
smallest
coine
,
may
do
it
at
first
work
;
for
seeing
a
farthing
is
the
960
part
of
a
pound
,
multiply
7
by
960
,
they
produce
6720
,
which
divided
by
13
,
the
quotient
is
516
farthings
,
and
12/13
of
a
farthing
:
these
farthings
may
be
turned
to
shillings
,
dividing
by
48
,
or
to
pence
by
4.
This
Rule
though
it
be
brief
and
plain
;
is
of
great
use
in
Arithmetick
;
either
for
turning
Natural
and
surd
Fractions
into
Decimals
;
or
any
other
desired
Denomination
,
with
such
facility
and
speed
as
may
be
wished
.
Fractions
of
Fractions
.
In
reduction
of
Fractions
,
some
make
another
,
or
more
parts
,
as
fractions
of
fractions
for
one
:
that
is
,
when
there
is
a
part
of
a
fraction
,
or
a
part
of
a
part
of
a
fraction
,
&c.
to
be
valued
in
one
fraction
.
THE
RULE
.
Multiply
all
the
Numerators
together
,
the
last
product
shall
be
the
Numerator
desired
:
then
multiply
all
the
Denominators
together
,
and
this
last
product
shall
be
the
Denominator
sought
.
Example
.
Let
the
Fractions
of
Fractions
propounded
,
be
⅘
of
¾
of
½
,
for
so
they
are
usually
written
;
and
let
the
Numerator
be
multiplyed
:
saying
,
4
times
3
is
12
,
and
12
times
1
is
12
,
the
Numerator
therefore
required
is
12
:
then
for
the
Denominator
,
say
,
5
times
4
is
20
,
and
20
times
2
is
40
,
for
the
Denominator
required
;
and
12/40
is
equal
to
⅘
of
¾
of
½
.
Proof
.
Let
the
Unite
be
40
s.
one
fifth
of
40
is
8
,
and
therefore
⅘
is
32
,
of
which
one
fourth
is
8
,
and
¾
is
24
,
of
which
one
half
is
12
,
and
therefore
12/40
is
the
just
sum
of
all
the
Fractions
,
This
needs
no
further
exemplifying
.
Addition
.
TO
adde
many
fractions
into
one
sum
,
consider
whether
they
be
of
one
Denomination
or
divers
;
if
of
one
,
then
adde
all
the
Numerators
together
into
one
summe
,
that
summe
is
the
new
Numerator
:
and
the
Denominator
,
in
this
case
is
not
altered
.
Example
.
Let
the
fractions
to
be
added
be
2/4
,
4/4
,
5/4
,
¼
,
Adde
the
Numerators
:
saying
2
and
4
is
6
,
and
6
and
5
is
11
,
and
11
and
1
is
12.
So
the
summe
of
them
all
is
12/4
,
that
is
3
Unites
.
As
,
let
the
Unite
be
20
s
,
one
fourth
is
5
s.
and
2/4
is
10
s.
and
4/4
is
20
s.
which
added
to
10
s.
is
30
s.
then
5/4
is
25
s.
which
added
to
thirty
shilling
gives
55
s.
And
lastly
,
¼
is
5
s.
which
added
to
55
s.
makes
60
s.
that
is
3
times
20
s.
that
is
3
l.
or
3
Unites
.
But
if
the
fractions
to
be
added
,
be
of
divers
denominations
,
as
let
them
be
3/2
,
¾
,
⅘
,
⅞
,
then
(
by
the
reduction
afore-going
)
they
must
be
turned
all
into
one
denomination
,
and
then
they
will
be
320/480
,
360/480
,
384/480
,
and
420/480
,
and
may
be
added
like
those
before
:
thus
,
So
the
summe
of
all
is
1484/480
,
or
371/120
,
that
is
,
3
11/120
,
which
if
it
be
money
,
and
the
Unite
1
l.
it
is
then
3
l.
1
s.
and
10
d.
as
may
be
tryed
thus
:
First
,
⅔
of
a
pound
,
is
13
s.
and
4
d.
and
¾
is
15
s.
and
⅘
is
16
s.
Lastly
,
⅞
is
17
s.
6
d.
These
all
added
together
,
the
summe
if
3
l.
1
s.
10
d.
Subtraction
.
IN
Subtraction
of
one
fraction
from
another
,
if
they
be
both
of
one
denomination
;
it
is
done
by
taking
the
Numerator
of
one
from
the
Numerator
of
the
other
,
the
remaine
is
the
new
Numerator
,
and
the
Denominator
the
same
as
before
.
So
if
2/9
be
subtracted
from
8/9.
the
remaine
is
,
6/9
the
like
of
all
others
.
But
if
they
be
not
of
one
Denomination
,
they
must
first
be
reduced
to
be
so
;
then
that
which
is
said
before
is
sufficient
.
The
golden
Rule
in
fractions
is
the
same
as
in
whole
Numbers
,
I
will
give
you
but
one
instance
.
If
¾
of
a
yard
of
tape
cost
½
of
a
peny
,
what
shall
one
inch
,
that
is
,
1/36
of
a
yard
cost
?
Multiply
the
second
by
the
third
,
the
product
is
1/72
,
which
divided
by
¾
,
the
quotient
is
4/216
of
a
peny
,
for
the
price
of
1/36
of
a
yard
.
Otherwise
,
Seeing
¾
of
a
yard
may
be
turned
to
27
inches
;
Say
,
if
27
cost
½
,
what
1
?
divide
½
by
27
,
it
makes
1/54
for
the
answer
:
which
is
equal
to
4/216
,
and
in
the
least
termes
.
And
wheresoever
this
may
be
done
,
to
have
the
first
and
third
Numbers
fractions
of
one
denomination
,
the
best
way
is
to
work
with
their
Numerators
,
not
regarding
their
denominator
at
all
:
as
if
⅔
cost
¾
,
what
7/3
?
instead
thereof
write
,
if
2
cost
¾
,
what
7
?
multiply
¾
by
7
,
it
produceth
21/4
,
which
divided
by
2
,
the
quotient
is
21/8
,
and
that
is
the
answer
in
the
least
terms
.
And
all
this
while
it
should
have
been
noted
that
the
fractions
are
ever
written
in
a
smaller
figure
then
the
whole
Numbers
.
The
Rule
of
Fellowship
.
THis
Rule
is
usefull
for
Merchants
,
and
all
such
as
trade
in
Companies
,
with
a
joint
stock
;
and
must
share
a
proportional
part
of
the
gaines
,
or
losse
;
every
one
according
to
his
stock
which
he
laid
in
.
The
Rule
is
two-fold
,
with
equal
time
;
or
with
unequal
time
.
That
which
is
with
equal
time
,
is
commonly
called
,
The
Rule
of
Fellowship
without
time
.
Of
this
we
will
first
speak
.
THE
RULE
.
As
the
whole
joint
stock
,
is
to
all
the
gain
or
losse
;
So
is
each
mans
particular
stock
,
to
his
part
of
the
gaine
,
or
losse
.
Example
1.
Two
Merchants
A.
and
B.
buy
700
l.
a
yeare
Land
for
ever
,
(
when
money
is
at
8
,
per
cent
.
)
for
14000
l.
of
which
A.
paid
8000
l.
and
B.
6000
l.
after
5
yeares
(
money
being
fallen
to
6
per
Cent.
)
they
sell
it
for
18700
l.
so
there
is
gained
4700
l.
how
much
of
this
must
A.
have
?
First
for
A.
Say
,
if
14000
gaine
4700
,
what
8000
?
answer
,
2685
10000/14000.
Then
for
B.
If
14000
,
gaine
4700
,
what
6000
?
answer
2014
4000/14000.
As
by
the
following
operation
doth
appeare
.
(
I
)
For
A.
(
II
)
For
B.
Here
note
,
that
this
work
might
have
been
much
abreviated
,
if
from
each
of
the
three
numbers
you
had
cut
off
two
Cyphers
towards
the
right
hand
,
as
hath
been
formerly
shewed
in
the
Compendium
of
Multiplication
and
Division
.
Now
for
the
proof
hereof
,
If
you
adde
|
2685
10000/14000.
|
which
is
the
sum
that
A.
gained
;
|
|
|
To
—
2014
4000/14000.
|
|
The
summe
of
them
is
4700.
|
Which
is
equal
to
the
total
gain
.
And
according
to
the
proportion
of
these
two
Numbers
:
that
is
,
as
8
to
6
,
or
4
to
3.
So
they
ought
to
have
parted
the
yearly
rent
also
,
all
the
time
they
received
it
:
that
is
,
A.
ought
to
have
400
l.
yearly
;
and
B.
300
l.
Example
2.
A.
B.
and
C.
joyne
their
moneys
to
make
a
stock
of
25000
l.
of
which
A.
laid
in
10000
l.
B.
8000
l.
and
C.
put
in
7000
l.
with
this
(
after
a
certain
time
in
trading
)
they
gained
7500
l.
how
must
this
be
parted
?
First
for
A.
Say
,
if
25000
gaine
7500
,
what
10000
?
Or
shorter
,
if
25
get
½
,
what
10
?
Multiply
7½
by
10
,
it
produceth
75
,
which
divided
by
25
,
the
quotient
is
●
,
that
is
,
(
restoring
the
3
Cyphers
)
3000
l.
for
A.
Then
for
B.
Say
,
if
25000
gaine
7500
,
what
8000
?
Or
shorter
,
if
250
get
75
,
what
80.
Multiply
and
divide
as
the
Golden
Rule
requires
,
and
to
the
quotient
,
restore
the
2
Cyphers
,
then
it
will
be
2400
l.
for
B.
Lastly
,
for
C.
Say
,
if
250
give
75
,
what
70
?
answer
21
,
to
which
put
the
2
Cyphers
,
it
makes
2100
for
C.
And
these
three
3000
,
2400
,
and
2100
,
being
added
together
,
make
7500.
And
have
that
proportion
as
the
particular
stocks
had
:
and
therefore
the
work
is
right
.
(
I
)
for
A.
(
II
)
for
B.
(
III
)
for
C.
And
if
instead
of
gaining
7500
l.
whereby
every
one
is
supposed
to
have
his
stock
,
and
a
part
of
the
gaines
;
they
had
lost
7500
l.
then
their
particular
stocks
had
not
been
due
to
them
,
but
so
much
as
would
be
left
after
their
proportional
parts
of
the
losse
were
abated
.
Example
3.
A.
B
and
C.
with
a
joynt
stock
of
25000
l.
gaine
7500
:
of
which
A.
gets
3000.
B.
2400.
C.
2100
,
what
was
their
stock
?
This
is
but
the
Converse
of
the
former
,
therefore
say
,
if
5700
require
25000
,
what
doth
3000
require
?
10000
for
A.
and
so
work
for
the
other
two
.
Many
examples
are
of
little
use
,
(
except
to
load
the
Readers
memory
)
where
the
Rule
is
so
short
and
plain
,
I
will
therefore
adde
no
more
to
this
part
of
the
Rule
,
but
immediately
come
to
the
Rule
of
Fellowship
with
time
.
The
Rule
of
Fellowship
with
time
.
THis
Rule
is
to
be
used
when
the
times
of
the
continuance
of
the
particular
stocks
are
unequal
,
and
differ
;
so
that
here
the
difference
of
time
,
and
also
the
difference
of
stock
being
both
to
be
considered
;
it
can
be
done
no
better
way
then
by
taking
the
Power
of
them
both
to
be
the
particular
stock
;
and
all
those
Powers
added
,
to
be
the
whole
stock
,
that
which
I
call
the
Power
,
is
the
product
of
the
money
of
every
one
multiplyed
by
his
time
;
And
then
THE
RVLE
.
As
the
summe
of
those
Products
,
is
to
the
whole
gain
;
so
is
each
particular
product
,
to
its
part
of
the
gain
.
Question
1.
Three
Merchants
A.
B.
C.
make
a
stock
of
10000
l.
of
which
A
layes
in
4000
for
3
moneths
,
B.
3000
l.
for
6
moneths
;
and
C.
3000
l.
for
8
moneths
,
with
this
they
gain
2000
l.
what
is
each
mans
share
?
First
,
for
A.
multiply
4000
by
3
,
it
makes
12000
,
let
that
be
accounted
his
particular
stock
.
Secondly
,
for
B.
multiply
30000
by
6
,
it
makes
18000.
Lastly
for
C.
multiply
3000
by
8
,
it
produceth
24000
,
for
his
stock
,
adde
these
,
they
make
54000
l.
for
the
general
stock
;
then
say
,
For
A.
If
54000
give
2000
,
what
12000
?
answer
,
666
24000/54000.
Then
for
B.
If
54000
give
2000
,
what
18000
?
answer
444
36000/54000.
Lastly
,
for
C.
If
54000
give
2000
,
what
24000
?
answer
,
888
48000/54000.
The
three
Fractions
may
be
reduced
(
by
dividing
each
Numerator
,
and
Denominator
by
6000
)
and
then
the
three
shares
will
be
444
4/9
,
66
6/9
,
and
888
8/9
,
which
altogether
make
2000
,
as
they
ought
.
Question
2.
Three
Farmers
,
A.
B.
and
C.
lay
out
1000
l.
to
stock
their
grounds
with
Cattel
,
of
which
A.
put
in
200
l.
for
6
yeares
;
B.
had
300
l.
going
for
4
yeares
;
and
C.
500
l.
for
2
years
;
at
the
end
(
by
unseasonable
times
)
there
was
lost
200
l.
which
made
the
remain
of
their
stock
but
800
l.
what
had
each
man
left
?
Multiply
200
by
6
,
it
gives
1200
:
Likewise
,
300
by
4
,
it
gives
1200.
Lastly
,
500
by
2
,
the
product
is
1000
:
all
these
are
3400
for
the
joynt
stock
.
Then
first
for
A.
Say
,
if
3400
lose
200
,
what
1200
?
answer
,
70
2000/3400
for
A
,
to
which
B
is
equal
,
because
the
power
of
his
stock
is
so
.
Therefore
for
C.
Say
,
if
3400
lose
200
,
what
1000
?
answer
,
58
2800/3400.
So
the
3
shares
are
70
20/34
,
70
20/34
,
and
28
2●/34
,
equal
to
200.
Now
because
A
put
in
200
l.
and
lost
70
20/34
subtract
the
losse
from
the
stock
,
remains
129
14/34.
And
so
doing
for
B
,
his
remain
will
be
289
14/34.
And
for
C
,
his
remain
is
441
6/34.
Now
these
three
remains
,
129
14/34
,
229
14/34
,
and
441
6/34
,
make
up
800
l.
which
was
the
whole
remain
.
Quest.
3.
A.
rents
a
close
for
a
year
,
to
pay
80
l.
he
puts
into
it
200
sheep
:
2
months
after
B.
puts
40
sheep
in
;
and
5
moneths
after
that
C.
puts
in
100
sheep
;
how
much
must
every
one
pay
of
the
rent
?
Multiply
200
by
12
,
it
produceth
|
2400
|
And
40
by
10
,
produceth
|
400
|
Lastly
,
100
by
5
,
(
which
is
C
time
)
produceth
|
500
|
In
all
|
3300
|
Then
for
A.
If
3300
pay
80
,
what
2400
?
answer
,
58
600/3300.
Then
for
B.
If
3300
pay
80
,
what
400
?
answer
,
9
2300/3300.
And
for
C.
If
3300
pay
80
,
what
500
?
answer
,
9
400/3300.
The
whole
numbers
make
79
,
and
the
broken
numbers
make
1.
In
all
80.
Note
Whereas
,
hitherto
we
have
considered
onely
difference
of
time
and
money
;
it
may
be
noted
,
that
there
may
be
difference
of
other
kinds
,
as
persons
or
place
;
but
whatsoever
they
are
the
power
of
all
is
found
like
these
by
multiplication
;
and
are
to
be
worked
like
these
,
with
so
many
Uses
of
the
Golden
Rule
,
as
the
question
requires
.
I
will
therefore
adde
but
one
question
more
,
which
is
this
.
Quest.
4.
One
leaves
a
Legacy
of
900
l.
among
foure
Kinsfolks
.
A.
B.
C.
D
;
so
as
B
may
have
twice
as
much
as
A.
and
C
thrice
as
much
as
B.
and
D
as
much
and
half
as
much
as
C
,
what
is
every
one
to
have
.
Say
,
if
A
be
1
,
B
is
2
,
C
6
,
and
D
9
,
adde
these
Numbers
:
1
,
2
,
6
,
9
,
together
,
they
give
18
,
then
say
,
If
18
require
900
,
what
1
?
answer
is
50.
So
A
is
to
have
50
l.
B
100
l.
C
3000
l.
and
D
450
l.
which
are
their
just
parts
;
and
altogether
are
equal
to
900
l.
and
the
work
right
.
The
Rule
of
Alligation
.
THis
hath
its
name
from
binding
,
tying
,
or
uniting
many
particulars
in
one
Masse
or
Summe
,
the
nature
of
it
will
be
understood
in
working
some
Questions
or
Examples
.
Question
1.
A
Corn
Master
would
mix
4
sorts
of
graine
together
,
viz.
wheat
at
4
s.
the
bushel
,
wheat
at
4
s.
0
d.
the
bushel
;
Rie
,
at
3
s.
the
bushel
,
and
Barly
at
2
s.
8
d.
the
bushel
;
so
as
to
make
15
quarters
in
all
,
to
be
sold
at
3
s.
6
d.
the
bushel
;
How
much
must
he
take
of
each
?
THE
RULE
.
Multiply
the
whole
Masse
to
be
made
,
by
any
particular
difference
;
and
divide
the
product
by
the
summe
of
all
the
differences
,
the
quotient
shall
be
the
just
quantity
of
that
particular
kind
,
whose
price
standeth
against
the
difference
you
worked
with
.
Example
.
First
turn
the
quarters
into
bushels
,
by
saying
,
8
times
15
is
120
,
then
for
the
quantity
of
the
first
sort
at
4½
:
multiply
120
by
⅚
,
the
product
is
100
,
which
divided
by
2⅚
,
the
quotient
is
35
5/7
bushels
of
that
sort
at
4
s.
6
d.
and
working
so
for
every
of
the
other
;
they
will
be
found
to
be
thus
:
At
4
s.
6
d.
|
35
5/7
bushels
.
|
At
4
s.
|
21
3/17
bushels
.
|
At
3
s.
6
d.
|
21
3/17
bushels
.
|
At
2
s.
8
d.
|
42
6/17
bushels
.
|
In
all
|
120
|
Now
to
prove
this
right
;
first
multiply
the
whole
Masse
120
bushels
,
by
the
desired
price
3½
s.
omitting
the
denominations
,
the
summe
is
420
s.
Then
secondly
,
multiply
38
5/17
by
4½
,
it
is
,
|
158
14/17
|
And
21
3/17
by
4
,
it
produces
|
84
12/17
|
And
21
3/17
by
3
,
it
makes
|
63
2/17
|
And
42
6/17
by
2⅔
is
|
112
16/17
|
In
all
|
420
|
But
so
much
all
should
be
worth
in
shillings
.
And
therefore
the
question
in
rightly
solved
.
Quest.
2.
One
hath
6
sorts
of
Fruits
at
several
prices
,
Dates
at
2
s.
Almonds
at
1
s.
4
d.
Currance
at
10
d.
Raisons
at
5
d.
Prunes
at
4
d.
and
Figs
at
3
d.
the
pound
;
and
would
take
of
every
sort
some
to
make
a
mixed
quantity
of
30
l.
weight
to
sell
one
with
another
for
9
d.
the
pound
,
how
much
must
he
take
of
each
?
So
30
,
multiplyed
by
6
,
produceth
180
,
which
Divided
by
38
,
the
quotient
is
4
22/38
of
a
pound
weight
:
and
so
much
must
be
taken
of
Dates
,
at
24
d.
Secondly
,
5
times
30
is
150
,
which
divided
by
38
,
the
quotient
is
3
36/38
for
Almonds
.
And
working
after
the
same
manner
with
4
,
1
,
7
,
15
,
their
respective
quantities
will
be
found
to
be
these
;
|
pounds
38
parts
.
|
Dates
|
4
,
28
|
Almonds
|
3
,
36
|
Currance
|
3
,
6
|
Raisons
|
0
,
30
|
Prunes
|
5
,
20
|
Figges
|
11
,
32
|
In
all
|
26
,
152
|
That
is
26
252/32
,
and
the
reduction
of
the
fraction
will
make
it
30
,
as
it
ought
to
be
,
and
by
comparing
the
prices
of
these
particulars
added
,
with
the
price
of
30
li.
weight
,
at
9
d.
per
l.
weight
,
which
makes
470
d.
this
may
be
proved
like
the
former
.
But
that
the
Reader
may
be
perfect
in
it
,
I
will
doe
it
here
also
,
as
followeth
.
Say
first
,
24
times
4
is
96
,
and
24
times
28
is
672
;
for
the
first
,
set
them
thus
:
96
,
672.
And
16
times
3
is
48
,
and
16
times
36
is
576
|
48
,
576
|
And
10
times
3
is
30
,
and
10
times
6
is
60
|
30
,
60
|
And
5
times
30
is
150
|
00
,
150
|
And
4
times
5
is
20
,
and
4
times
20
is
80
|
20
,
80
|
Lastly
.
3
times
11
is
33
,
and
3
times
32
is
96
|
32
,
96
|
In
all
|
227
,
1634
|
Now
this
1634
being
the
summe
of
the
Numerators
of
Fractions
,
whose
common
Denominator
is
38
,
must
be
divided
by
38
,
and
the
quotient
will
be
43
,
which
added
to
the
whole
number
227
:
the
summe
is
270.
And
so
much
is
30
,
multiplyed
by
9
,
which
shewes
the
work
to
be
right
.
The
Combination
,
or
linking
of
Numbers
may
be
varied
at
pleasure
,
as
whereas
above
I
linked
24
,
and
3
,
also
16
with
4
,
and
10
with
5
,
it
might
have
been
24
with
5
,
and
16
with
4
,
and
10
with
3.
Or
14
with
4
,
and
16
with
3
,
and
10
with
4
,
of
which
diversity
of
linking
would
follow
diversity
of
solutions
,
but
all
true
,
as
the
Reader
may
easily
prove
by
himself
.
Likewise
,
if
the
Numbers
to
be
linked
were
3
,
5
,
7
,
or
any
odde
Number
,
one
of
them
may
be
linked
to
two
severally
,
to
make
the
worke
even
.
Example
3.
If
the
Numbers
were
12
,
10
,
8
,
6
,
and
4
,
and
the
mean
or
common
price
required
were
9
,
you
might
first
linke
them
as
you
see
here
,
takeing
12
twice
,
or
else
you
might
take
any
other
twice
as
you
shall
like
;
and
so
the
work
will
be
every
way
right
,
though
not
the
same
;
if
the
differences
be
rightly
set
off
and
orderly
used
,
as
is
taught
before
in
the
first
question
.
Question
3.
A
Goldsmith
would
mix
3
sorts
of
silver
,
A.
B.
C.
A
is
10
d.
weight
better
;
B
7
d.
weight
better
;
and
C
,
4
d.
weight
better
,
to
make
an
Ingot
of
50
li.
weight
,
which
should
be
in
finesse
8
d.
weight
better
:
How
much
must
be
taken
of
each
?
1.
Then
first
50
multiplyed
by
4
|
200
|
and
divided
by
9
,
quotient
is
|
9
|
2.
50
multiplyed
by
2
,
and
|
50
|
divided
by
9
,
quotient
is
|
9
|
3.
50
multiplyed
by
2
,
and
|
100
|
divided
by
9
,
quotient
is
|
9.
|
4.
And
again
,
the
same
|
100
|
|
9
|
In
all
|
450
|
|
9
|
Which
is
equal
to
50
,
the
quantity
required
,
Now
the
first
fraction
multiplyed
by
10
,
(
omitting
the
Denominator
)
is
|
2000
|
The
second
also
by
10
gives
|
500
|
The
third
by
7
makes
|
700
|
The
last
by
4
makes
|
400
|
In
all
|
3600
|
That
is
,
1600/9
,
which
is
equal
to
400
,
and
if
the
whole
Ingot
50
,
be
multiplyed
by
the
betternesse
required
,
namely
,
by
8
,
they
shall
produce
400
also
:
So
this
is
proved
.
In
every
Alligation
,
or
linking
of
two
numbers
,
this
is
evident
,
that
if
the
summe
of
the
numbers
linked
be
greater
then
the
mean
number
required
taken
so
many
times
as
there
are
numbers
to
be
linked
,
the
question
would
be
absurd
;
and
the
resolution-thereof
impossible
.
And
this
shall
serve
for
the
rule
of
Alligation
.
The
Rule
of
False
Position
.
THis
Rule
serves
to
resolve
questions
,
which
are
not
presently
fit
for
the
Golden
Rule
;
and
therefore
in
stead
of
the
true
Number
which
is
sought
:
Suppose
any
number
great
or
small
,
and
make
trial
of
it
,
whether
it
resolve
the
question
without
any
errour
;
if
so
,
it
is
the
true
Number
:
if
not
note
what
errour
is
at
the
end
of
the
worke
,
and
whether
it
be
too
much
or
too
little
:
if
too
much
,
mark
it
thus
+
but
if
too
little
,
thus
−
Then
suppose
again
another
Number
,
(
it
imports
not
whether
it
be
neerer
or
further
off
)
and
try
as
before
,
and
mark
that
errour
also
;
And
,
then
THE
RULE
.
Multiply
the
first
Position
by
the
second
Errour
,
and
the
second
Position
by
the
first
errour
,
and
(
if
the
errours
be
both
+
or
both
−
)
Subtract
the
lesser
product
from
the
greater
,
and
keep
the
remaine
for
a
Dividend
,
and
the
difference
of
the
errours
for
the
Divisor
:
the
quotient
of
that
division
is
the
true
Number
required
.
But
if
the
errours
had
been
one
+
,
the
other
−
,
then
the
summe
of
the
products
added
together
had
been
the
Dividend
:
and
the
summe
of
the
errours
,
the
Divisor
;
the
rest
of
the
work
the
same
as
before
.
Question
1.
A
man
is
to
drive
48
young
Turkies
40
miles
:
and
for
every
Turkie
which
comes
alive
to
the
end
of
the
Journey
,
he
is
to
receive
3
d.
but
for
every
one
which
dyes
by
the
way
,
he
is
to
pay
6
d.
at
the
end
,
he
received
72
d.
How
many
dyed
by
the
way
?
Let
the
first
supposition
be
That
by
the
way
there
dyed
|
20
|
For
them
he
was
to
pay
120
d.
and
for
28
which
lived
,
he
was
to
receive
84
d.
so
he
paid
more
then
he
received
|
36
d.
|
and
should
have
got
clear
|
72
d.
|
|
Adde
108
|
Wherefore
the
first
errour
is
|
−
108
|
Let
the
second
supposition
be
|
10
|
For
these
he
paid
60
d.
and
for
the
rest
he
received
114
d.
the
difference
is
54
,
and
should
be
72
:
so
the
second
errour
is
|
−
18
|
Now
20
multiplyed
by
18
,
produceth
|
360
|
and
10
by
108
produceth
|
1080
|
and
the
difference
is
720
for
the
Dividend
,
likewise
the
difference
of
the
errours
is
90
for
the
Divisor
,
and
the
quotient
is
8
,
which
is
the
true
Number
of
those
he
lost
by
the
way
.
As
may
be
proved
by
trial
.
Question
3.
If
it
was
required
to
make
up
a
pound
Sterling
of
shillings
and
groats
only
:
and
so
as
the
number
of
groats
,
may
be
to
the
Number
of
shillings
,
as
7
to
1
,
how
many
shillings
must
there
be
?
First
,
suppose
the
shillings
|
4
|
then
the
groats
must
be
equal
to
16
s.
that
is
48
groats
;
but
the
shillings
taken
7
times
are
28
,
to
which
48
should
be
equal
,
but
is
|
+20
|
Secondly
,
suppose
the
shillings
|
2
|
then
the
groats
(
making
18
s.
)
are
25
,
|
|
which
should
be
equal
to
7
times
2
,
but
is
|
+40
|
Multiply
4
by
40
product
is
160
,
then
|
|
Multiply
2
by
20
,
the
product
is
40
,
which
taken
from
160
,
rests
for
the
Dividend
|
120
|
And
the
difference
of
errours
is
|
20
|
Lastly
,
120
divided
by
29
,
quotient
is
|
6
|
The
number
of
shillings
therefore
is
|
6
|
And
the
number
of
groats
is
|
42
|
But
as
7
to
1
,
so
is
6
times
7
which
is
42
to
6
times
1
,
which
is
6
:
so
the
work
is
done
.
Question
3.
If
there
be
4
several
weights
,
A.
B.
C.
D.
of
which
D
is
24
ounces
,
and
C
is
double
to
B
,
and
triple
to
A
,
and
D
with
twice
A
is
double
to
C.
and
quadruple
to
B
:
How
much
doth
every
one
of
these
weights
weigh
?
First
,
suppose
A
to
be
then
D
with
twice
A
is
24
,
and
16
,
that
is
40
,
of
which
●
being
the
halfe
is
●0
,
and
B
10.
|
8
|
Now
thrice
A
is
24
,
to
which
C
should
be
equal
,
but
is
|
−
4
|
Secondly
,
let
A
be
supposed
|
4
|
then
D
more
,
twice
A
is
32
,
and
C
16
and
B
is
8
,
but
thrice
A
is
12
,
to
which
16
should
be
equal
,
but
is
|
+4
|
Then
8
multiplied
by
4
,
gives
32
,
and
4
by
4
,
produceth
16
:
both
these
products
gives
48
for
the
Dividend
:
and
the
summe
of
the
errours
(
because
the
first
is
−
,
the
other
+
)
gives
8
,
for
the
Divisor
;
and
the
quotient
will
be
6
,
to
which
A
is
equal
,
and
twice
A
more
D
is
36
,
of
which
C
,
(
being
halfe
is
18
,
and
B
is
9
,
and
thrice
A
is
equal
to
C
,
namely
,
18
,
and
all
right
.
Whereas
the
first
errour
is
equal
here
to
the
second
,
it
followes
that
the
Positions
were
equally
false
:
and
therefore
their
difference
which
is
4
,
being
parted
into
two
equal
parts
,
2
and
2
,
if
2
be
taken
from
8
,
the
remaine
is
the
true
number
6
,
or
if
●
be
added
to
4
,
(
which
was
the
second
position
)
the
summe
will
be
also
6.
And
further
,
whensoever
the
erronrs
be
one
+
,
the
other
−
,
though
they
be
not
equal
;
yet
then
if
the
difference
between
the
positions
be
parted
into
two
parts
,
which
are
in
proportion
one
to
another
,
as
the
two
errours
are
one
to
another
respectively
:
then
if
the
first
part
be
taken
from
the
first
position
(
if
that
be
the
greater
)
or
adde
to
it
(
if
it
be
the
lesse
,
the
same
number
required
is
thereby
had
.
As
,
let
the
last
question
be
resumed
,
And
let
the
first
position
for
A
be
|
15
|
Then
the
first
errour
will
be
|
−
18
|
Then
let
the
second
position
be
|
3
|
And
so
the
second
errour
will
be
|
+6
|
And
the
difference
of
positions
is
|
12
|
which
divided
into
two
parts
9
and
3
,
which
have
that
proportion
one
to
another
as
have
the
errours
18
and
6
,
then
if
the
first
part
9
,
be
taken
from
the
first
position
15
,
there
remains
the
true
Number
6.
Or
else
if
the
second
part
3
,
be
added
to
the
second
position
3
:
thereby
also
is
made
the
true
Number
6.
The
way
of
parting
12
,
(
or
any
other
)
into
two
parts
proportional
with
the
errours
is
easily
done
by
the
Golden
Rule
,
thus
:
-
As
the
summe
of
the
Errours
24
,
-
is
to
the
difference
of
positions
12
;
-
So
is
the
greater
errour
18
-
to
the
greater
part
required
,
namely
,
9.
Many
other
questions
are
in
other
bookes
exemplifyed
and
wrought
by
this
Rule
;
but
seeing
I
intend
not
to
write
a
great
book
;
and
also
because
some
of
those
questions
may
be
easily
resolved
without
this
Rule
,
I
will
adde
no
more
:
ondy
mention
one
of
those
questions
.
I
there
be
a
Cisterne
with
4
Cocks
,
which
holls
8
barrels
of
water
,
and
the
first
cock
will
runit
all
out
in
6
houres
,
the
second
in
4
,
the
third
in
3
,
and
the
last
in
2
houres
:
in
what
time
shal
all
of
them
runne
it
out
?
If
the
first
in
6
houres
run
|
8
|
the
second
in
the
same
time
would
●un
|
12
|
the
third
|
16
|
the
last
|
24
|
In
all
|
60
|
Then
say
,
If
60
require
6●
what
8
?
The
answer
48/60
,
that
is
4/5
o●
an
hour
;
in
which
time
all
the
4
Cocks
together
would
run
out
all
the
8
Barrels
of
water
.
The
Rule
of
Ceres
and
Virginum
.
THis
is
the
most
uncertaine
,
and
unnecessary
Rule
in
Arithmetick
;
being
seldome
used
except
in
sporting
questions
to
puzzle
young
beginners
,
with
easie
problems
;
such
as
follow
.
Question
1.
A
Caterer
bought
8
birds
of
two
sorts
,
as
Geese
and
Hens
for
20
s.
the
Geese
cost
4
s.
a
piece
,
the
Hens
2
s.
a
piece
;
How
many
did
he
buy
of
each
sort
?
This
may
be
done
by
the
Rule
of
False
;
and
also
thus
:
multiply
the
whole
number
8
,
into
the
least
price
2
,
it
produceth
16
,
which
taken
from
the
whole
price
20
,
rests
4
for
a
Dividend
;
which
divided
by
2
,
which
is
the
difference
of
the
particular
prices
,
the
quotient
is
2
,
for
the
number
of
Geese
;
and
6
must
be
for
the
Hens
:
the
proofe
is
easie
.
Question
2.
If
21
Persons
,
Men
,
Women
,
and
Children
spend
26
shillings
;
so
that
every
man
payes
2
s.
every
woman
1
s.
every
child
6
d.
How
many
is
there
of
each
sort
?
THE
RULE
.
Multiply
the
number
of
Persons
by
the
least
expence
,
and
take
the
product
of
it
from
the
whole
expence
,
the
rest
shall
be
the
Dividend
;
which
divided
by
the
difference
betwixt
the
greatest
and
least
particular
expences
:
the
quotient
is
a
Number
which
the
Number
of
men
(
or
they
which
spend
most
)
comes
neer
to
:
but
cannot
exceed
:
or
of
the
said
Dividend
be
divided
by
the
summe
of
the
greatest
and
least
expences
,
the
quotient
is
a
Number
then
which
the
Number
of
men
or
those
which
spend
most
)
cannot
be
much
lesse
.
So
here
21
multiplyed
by
6
d.
that
is
,
by
½
,
the
product
is
10½
,
which
taken
from
26
,
rests
15½
,
for
the
Dividend
:
and
then
taking
½
from
2
,
rests
1½
for
the
Divisor
,
and
the
quotient
is
3⅓
,
which
is
something
more
then
10
,
the
Number
of
men
therefore
must
be
but
9.
Then
turn
the
Dividend
,
and
the
Divisor
both
into
whole
Numbers
,
by
multiplying
them
by
the
common
Denominator
2
,
so
they
reduced
will
be
31
and
3
,
as
before
is
to
be
seen
in
the
quotient
.
Multiply
the
Divisor
3
,
by
9
,
(
which
is
the
Number
of
Men
)
the
product
is
27
,
which
taken
from
31
,
(
which
is
the
reduced
Dividend
)
the
remaine
is
4
,
for
the
Number
of
Women
;
and
the
Children
must
be
8.
Example
1.
9
men
at
2
s.
each
|
18
s.
|
4
women
at
1
s.
each
|
4
|
8
children
at
6
d.
each
|
4
|
In
all
21
|
In
all
26
|
But
the
number
of
men
may
be
also
8
,
which
multiplyed
by
the
reduced
Divisor
3
,
product
is
24
,
which
taken
from
31
,
the
remaine
is
7
for
the
women
;
and
then
the
children
must
be
6.
Example
2.
8
men
at
2
s.
each
|
16
|
7
women
at
1
s.
each
|
7
|
6
children
at
6
d.
each
|
3
|
In
all
21
|
In
all
26
|
Or
the
number
of
men
may
be
7
,
which
multiplyed
by
3
,
produceth
21
,
which
taken
from
31
,
remaines
10
for
the
women
;
and
4
children
.
Example
3.
7
men
at
2
s.
each
|
14
s.
|
10
women
at
1
s.
each
|
10
|
4
children
at
6
d.
each
|
2
|
In
all
21
|
In
all
26
|
So
here
is
already
seen
3
various
solutions
of
this
question
,
which
makes
this
Rule
the
lesse
to
be
regarded
.
But
further
,
the
number
of
men
may
be
10
,
and
not
more
,
for
if
you
put
them
11
,
that
multiplyed
by
3
,
produceth
33
,
which
is
greater
then
31
,
from
which
it
should
be
taken
,
but
I
say
it
may
be
10
,
and
then
there
is
only
one
woman
,
and
5
children
:
this
confirmes
the
former
part
of
the
Rule
.
Now
for
the
later
part
,
if
the
Dividend
31
be
divided
by
the
summe
of
the
two
extream
expences
(
reduced
by
doubling
as
the
Dividend
is
)
4
,
the
quotient
will
be
7¾
.
And
the
men
may
be
7
,
as
hath
been
shewed
;
but
they
may
be
also
but
6
,
and
fewer
they
cannot
be
:
as
6
men
,
13
women
,
and
2
children
;
for
if
you
put
them
5
,
that
multiplyed
by
3
,
produceth
15
,
which
taken
from
31
,
there
remaines
16
for
the
women
,
and
so
there
should
be
no
children
:
which
is
contrary
to
the
supposition
.
And
further
,
because
the
quotient
was
7¾
,
the
number
of
men
might
be
so
,
if
pure
Arithmetical
division
be
onely
regarded
:
and
then
the
women
also
are
in
number
7¾
,
and
the
children
5½
,
as
may
easily
be
tryed
;
I
need
not
exemplifie
it
.
Question
3.
If
there
be
an
Exhibition
of
900
l.
per
annum
to
30
persons
:
some
Clerks
,
some
Messengers
,
and
some
Door-keepers
,
at
60
l.
each
Clerk
,
40
l.
each
messenger
,
and
20
l.
each
door-keeper
,
how
many
must
there
be
of
each
sort
?
Multiply
(
according
to
the
Rule
)
30
by
20
,
the
product
is
600
,
which
taken
from
900
,
remains
300
for
the
Dividend
;
and
60
want
20
,
that
is
40
,
for
the
Divisor
:
and
the
quotient
is
7½
,
and
more
the
Clerks
cannot
be
;
Also
divide
by
60
more
20
,
that
is
80
,
quotient
is
3¾
,
and
much
fewer
the
Clerks
cannot
be
.
Not
to
stand
upon
the
Fractions
(
in
this
case
of
dividing
men
)
the
Clerks
may
be
7
,
6
,
5
,
4
,
or
3
:
and
the
messengers
1
,
3
,
5
,
7
,
or
9
,
and
the
Door-keepers
22
,
21
,
20
,
19
,
or
18
,
that
the
Clerks
cannot
(
in
whole
Numbers
)
be
more
then
7
,
or
lesse
then
3
,
may
thus
be
proved
;
First
,
let
them
be
8
,
then
8
times
40
is
320
,
which
is
more
then
300
,
out
of
which
it
should
be
taken
:
Secondly
,
let
them
be
2
,
then
2
times
40
is
80
,
out
of
300
remains
220
,
which
divided
by
20
,
gives
the
quotient
11
,
for
the
messengers
,
so
the
Clerks
and
messengers
being
13
,
the
remaine
thereof
to
30
,
namely
,
17
,
must
be
Door-keepers
.
but
,
2
Clerks
at
60
l.
each
|
120
l.
|
11
Messengers
at
40
l.
each
|
480
|
17
Door-keepers
at
20
l.
each
|
340
|
In
all
30
|
In
all
940
|
Which
is
40
l.
too
much
,
therefore
the
Clerks
cannot
be
two
.
Note
.
It
may
be
asked
,
why
the
remaine
220
should
be
divided
by
20
:
whereas
the
like
remaine
in
the
former
Example
,
namely
16
,
was
taken
(
without
any
division
)
absolutely
for
the
number
of
Women
,
or
middle
number
?
I
answer
,
although
the
greatest
or
first
Number
being
found
,
(
as
here
to
be
2
)
the
residue
of
2
to
30
,
might
be
rightly
parted
into
two
fit
parts
in
the
same
manner
as
the
first
question
of
this
Rule
was
resolved
,
or
else
by
the
Rule
of
False
:
yet
to
give
further
satisfaction
,
the
cause
of
this
is
,
the
difference
betwixt
the
two
lesser
expences
,
was
there
½
,
which
(
before
the
division
)
was
reduced
to
1
,
which
neither
multiplies
nor
divides
any
Number
,
but
leaves
it
the
same
:
whereas
,
in
this
last
,
the
middle
expence
(
or
exhibition
)
being
40
,
and
the
least
20
,
the
difference
of
them
was
20
,
by
which
dividing
the
Remaine
of
the
last
subtraction
:
the
quotient
is
ever
the
Number
of
the
middle
persons
.
Which
may
serve
as
an
addition
to
the
Rule
,
where
the
sorts
of
things
are
but
three
.
Question
4.
If
there
be
10
persons
of
four
several
Countries
.
English
,
French
,
Dutch
,
and
Spanish
,
to
pay
a
Debt
of
1000
l.
So
that
every
English
man
payes
50
l.
every
French
man
70
l.
every
Dutch
man
130
l.
and
every
Spaniard
150
l.
How
many
is
thereof
each
?
The
Dividend
,
(
according
to
the
former
Rule
)
is
500.
Now
to
make
the
Divisor
,
take
his
summe
that
payes
least
(
namely
50
,
)
out
of
each
of
the
other
three
150
,
130
,
and
70
,
and
the
Remaines
will
be
100
,
80
,
and
20.
Adde
the
first
and
last
for
the
Divisor
,
it
is
120.
And
the
quotient
will
be
4
2/12
,
and
the
Spaniards
cannot
be
more
.
Secondly
,
adde
the
first
and
second
together
for
the
Divisor
,
it
is
180
,
and
the
quotient
is
2
14/18
,
and
the
Spaniards
cannot
be
lesse
.
I
mean
,
they
cannot
be
much
more
than
4
,
or
lesse
then
2
:
and
therefore
,
seeing
any
one
solution
will
serve
,
let
them
be
3
,
and
by
that
multiply
100
,
and
take
the
product
out
of
500
,
there
remaines
200
for
a
second
Dividend
,
which
divided
by
(
the
second
remain
)
80
,
the
quotient
is
2½
:
therefore
the
Dutchmen
are
2
,
which
multiplyed
by
80
make
160
,
take
that
out
of
200
,
there
remaines
40
for
a
third
Dividend
:
which
divided
by
(
the
third
remaine
)
20
,
the
quotient
is
2
for
the
French-men
also
;
and
consequently
the
English
must
be
3
,
because
all
of
them
are
10
:
But
the
Spaniards
may
be
also
4
or
2.
Example
.
4
Spaniards
,
at
150
l.
each
|
600
|
1
Dutchman
at
130
l.
|
130
|
1
Frenchman
,
at
70
l.
|
70
|
4
English
,
at
50
l.
each
|
200
|
10
In
all
|
1000
|
2
Spaniards
at
150
l.
each
|
300
|
3
Dutch
,
at
130
l.
each
|
390
|
3
French
,
at
70
l.
each
|
210
|
2
English
,
at
50
l.
each
|
100
|
10
In
all
|
1000
|
The
reason
why
the
Spaniards
and
English
,
as
also
the
Dutch
and
French
are
equal
in
number
,
is
because
their
payments
differ
equally
from
100
,
which
is
the
mean
summe
with
which
10
men
should
pay
1000
l.
and
making
it
so
,
this
question
,
and
many
other
of
this
nature
may
be
answered
by
the
Rule
of
Alligation
:
thus
,
And
also
,
because
the
other
two
differences
30
and
30
are
equal
:
the
Number
of
the
French
is
equal
to
the
Number
of
the
Dutch.
But
both
those
numbers
together
are
4
,
because
3
Spaniards
,
and
3
English
,
taken
out
of
10
,
the
remaine
must
be
4.
Wherefore
the
Number
of
the
French
is
2
,
and
the
Dutch
also
are
2.
Or
thus
,
If
160
require
10
,
what
30
?
Answer
is
1
160/160
,
(
that
is
,
in
this
case
2
)
for
the
Spaniards
;
and
consequently
2
English
,
and
therefore
the
French
and
Dutch
each
3.
But
where
any
one
of
the
particular
summes
,
is
equal
to
the
mean
summe
,
there
this
cannot
so
well
be
done
by
Alligation
.
Example
.
If
one
should
buy
12
loaves
of
bread
for
12
pence
,
so
that
some
might
be
two
peny
;
some
peny
,
some
half-peny
;
and
some
farthing
loaves
:
and
it
be
required
to
know
how
many
he
must
buy
of
each
;
then
because
of
12
loaves
for
12
pence
,
the
mean
price
is
1
,
but
one
of
the
particulars
being
also
1
,
there
should
be
no
peny
loaves
,
because
there
is
no
difference
betwixt
the
mean
price
,
and
a
peny
.
But
it
may
be
found
by
the
Rule
of
Ceres
and
Virginum
,
to
be
either
.
4
two
peny
loaves
|
8
pence
|
2
peny
loaves
|
2
pence
|
2
half-peny
loaves
|
1
peny
|
4
farthing
loaves
|
1
peny
|
In
all
12
loaves
.
|
In
all
12
pence
.
|
Or
else
,
3
two
peny
loaves
|
6
d.
|
4
peny
loaves
|
4
|
3
half-peny
loaves
|
1
½
|
2
farthing
loaves
|
0
½
|
In
all
12
|
In
all
12
|
Extraction
of
Roots
.
ANd
first
for
the
square
Root
,
that
is
,
a
square
Number
being
given
,
to
find
the
root
or
side
of
it
in
a
Number
:
which
root
or
side
,
being
multiplyed
into
it self
,
must
therefore
produce
that
square
Number
.
The
doing
of
this
is
shewed
in
(
almost
)
all
books
of
Arithmetick
;
and
the
reason
of
it
(
in
some
)
which
is
taken
from
the
fourth
proposition
of
the
second
book
of
Euclide
;
which
saith
,
If
a
right
line
be
divided
by
chance
;
the
squares
made
of
the
parts
,
together
with
the
Rectangle
made
of
the
parts
twice
,
is
equal
to
the
square
of
the
whole
.
Example
.
Now
let
AB
,
be
supposed
20
,
AE
15
,
and
BE
5.
Then
the
square
of
AB
,
(
which
is
made
by
multiplying
the
root
20
into
it selfe
)
is
equal
to
400.
And
the
square
AE
,
that
is
,
15
times
15
is
equal
to
|
225
|
And
the
square
of
BE
is
5
times
5
|
25
|
And
the
rectangle
AO
,
5
times
15
|
75
|
And
the
rectangle
DO
,
5
times
15
|
75
|
In
all
|
400
|
which
is
equal
to
the
square
of
AB
,
as
before
.
If
therefore
a
square
Number
,
as
334084
be
given
to
have
the
root
extracted
:
first
,
make
a
point
over
4
,
(
the
place
of
Unity
)
and
another
over
o
,
the
second
figure
from
it
;
and
so
another
over
o
,
the
second
figure
from
o
,
towards
the
left
hand
,
observing
the
like
order
still
,
if
there
were
more
secondary
figures
,
point
them
all
as
in
the
Margine
;
Then
look
at
the
figures
under
the
first
point
towards
the
left
hand
which
are
33
,
and
take
the
greatest
square
root
of
them
,
which
is
5
,
(
for
6
times
6
is
36
)
and
say
5
times
5
is
25
,
which
take
out
of
of
33
,
there
remaines
Then
multiply
the
root
5
by
20
,
it
gives
100
for
a
Divisor
;
and
the
Dividend
is
840
,
namely
,
the
figures
which
reach
to
the
next
point
,
and
so
the
quotient
might
be
8
,
but
must
be
but
7
,
because
the
square
of
the
quotient
,
being
now
49
,
must
(
together
with
7
times
the
Divisor
)
be
taken
out
of
840.
And
the
remainer
will
be
Adde
this
last
quotient
7
,
being
put
to
the
former
quotient
5
,
after
the
manner
used
in
plain
division
,
it
will
be
57
,
which
multiply
still
by
20
,
the
product
is
1140
for
a
new
Divisor
,
and
the
Dividend
(
because
there
remaines
but
one
point
)
is
all
the
figures
9184.
And
the
quotient
can
be
but
8
,
and
must
be
so
much
;
for
8
times
1140
is
|
9120
|
and
8
times
8
|
64
|
In
all
|
9184
|
Which
taken
out
of
the
remaine
9184
,
there
now
remaines
nothing
:
which
shews
the
square
334084
is
justly
resolved
,
and
putting
the
last
quotient
8
to
the
two
former
,
5
and
7
,
the
whole
will
be
578
,
which
is
the
true
root
required
.
And
the
parts
of
it
,
if
they
be
3
,
are
500
,
70
,
and
8.
Or
if
but
two
570
and
8.
And
the
truth
may
be
either
way
proved
by
adding
the
squares
and
twice
the
rectangles
of
the
parts
;
for
the
sum
of
them
shall
be
equal
to
the
whole
square
334084
,
as
hath
been
shewed
before
.
It
is
ever
certaine
that
there
shall
be
as
many
figures
,
or
places
of
figures
in
the
root
,
as
there
are
points
in
the
square
,
ordered
as
before
.
And
by
reading
this
little
seriously
,
may
any
one
be
able
to
find
the
root
of
any
other
square
whatsoever
:
the
first
operation
in
all
being
to
take
the
greatest
square
contained
under
that
point
next
the
left
hand
,
and
put
the
root
thereof
for
the
first
figure
in
a
quotient
.
The
Secondary
operation
,
must
be
repeated
so
often
as
there
are
remaining
points
;
as
hath
been
plainly
shewed
in
the
fore-going
Example
.
I
will
therefore
adde
no
more
,
but
give
you
two
other
squares
,
and
their
roots
,
leaving
the
Reader
to
extract
them
himself
.
So
,
if
there
were
given
the
square
88̇30̇
The
root
of
it
according
to
the
former
practice
may
be
found
to
be
94.
Or
let
there
be
given
the
square
1̇52̇27̇56̇
The
root
of
that
will
be
1234.
Here
followeth
a
Table
of
Roots
and
their
Squares
from
1
,
to
1000.
1
Cent.
R.
|
Square
|
1
|
1
|
2
|
4
|
3
|
9
|
4
|
16
|
5
|
25
|
6
|
36
|
7
|
49
|
8
|
64
|
9
|
81
|
10
|
100
|
11
|
121
|
12
|
144
|
13
|
169
|
14
|
196
|
15
|
225
|
16
|
256
|
17
|
289
|
18
|
324
|
19
|
361
|
20
|
400
|
21
|
441
|
22
|
484
|
23
|
529
|
24
|
576
|
25
|
625
|
26
|
676
|
27
|
729
|
28
|
784
|
29
|
841
|
30
|
900
|
31
|
961
|
32
|
1024
|
33
|
1089
|
34
|
1156
|
35
|
1225
|
36
|
1296
|
37
|
1369
|
38
|
1444
|
39
|
1521
|
40
|
1600
|
41
|
1681
|
42
|
1764
|
43
|
1849
|
44
|
1936
|
45
|
2025
|
46
|
2116
|
47
|
2209
|
48
|
2304
|
49
|
2401
|
50
|
2500
|
51
|
2601
|
52
|
2704
|
53
|
2809
|
54
|
2916
|
55
|
3025
|
56
|
3136
|
57
|
3249
|
58
|
3●64
|
59
|
3481
|
60
|
3600
|
61
|
3721
|
62
|
3844
|
63
|
3969
|
64
|
4096
|
65
|
4225
|
66
|
4356
|
67
|
4489
|
68
|
4624
|
69
|
4761
|
70
|
4900
|
71
|
5041
|
72
|
5184
|
73
|
5329
|
74
|
5476
|
75
|
5625
|
76
|
5776
|
77
|
5929
|
78
|
6084
|
79
|
6241
|
80
|
6400
|
81
|
6561
|
82
|
6724
|
83
|
6889
|
84
|
7056
|
85
|
7225
|
86
|
7396
|
87
|
7569
|
88
|
7744
|
89
|
7921
|
90
|
8100
|
91
|
8281
|
92
|
8464
|
93
|
8649
|
94
|
8836
|
95
|
9025
|
96
|
9216
|
97
|
9409
|
98
|
9604
|
99
|
9801
|
100
|
10000
|
101
|
10201
|
102
|
10404
|
103
|
10609
|
104
|
10816
|
105
|
11025
|
106
|
11236
|
107
|
11449
|
108
|
11664
|
109
|
11881
|
110
|
12100
|
111
|
12321
|
112
|
12544
|
113
|
12769
|
114
|
12996
|
115
|
13225
|
116
|
13456
|
117
|
13689
|
118
|
13924
|
119
|
14161
|
120
|
14400
|
121
|
14641
|
122
|
14884
|
123
|
15129
|
124
|
15376
|
125
|
15625
|
126
|
15876
|
127
|
16129
|
128
|
16384
|
129
|
16641
|
130
|
16900
|
131
|
17161
|
132
|
174●4
|
133
|
17689
|
134
|
17956
|
135
|
18225
|
136
|
18496
|
137
|
18769
|
138
|
19044
|
139
|
19321
|
140
|
19600
|
141
|
19881
|
142
|
20164
|
143
|
20449
|
144
|
20736
|
145
|
21025
|
146
|
21316
|
147
|
21609
|
148
|
21904
|
149
|
22201
|
150
|
22500
|
151
|
22801
|
152
|
23104
|
153
|
23409
|
154
|
23716
|
155
|
24025
|
156
|
24336
|
157
|
24649
|
158
|
24964
|
159
|
25281
|
160
|
25600
|
161
|
25921
|
162
|
26●44
|
163
|
26569
|
164
|
26896
|
165
|
27225
|
166
|
27556
|
167
|
27889
|
168
|
28224
|
169
|
28561
|
170
|
28900
|
171
|
29241
|
172
|
29584
|
173
|
29929
|
174
|
30276
|
175
|
30625
|
176
|
30976
|
177
|
31329
|
178
|
31684
|
179
|
32041
|
180
|
32400
|
2
Cent.
R.
|
Square
|
181
|
32761
|
182
|
33124
|
183
|
33489
|
184
|
33856
|
185
|
34225
|
186
|
34596
|
187
|
34969
|
188
|
35344
|
189
|
35721
|
190
|
36100
|
191
|
36481
|
192
|
36864
|
193
|
37249
|
194
|
37636
|
195
|
38025
|
196
|
38416
|
197
|
38809
|
198
|
39204
|
199
|
39601
|
200
|
40000
|
201
|
40401
|
202
|
40804
|
203
|
41209
|
204
|
41616
|
205
|
42025
|
206
|
42436
|
207
|
42849
|
208
|
43264
|
209
|
43681
|
210
|
44100
|
211
|
44521
|
212
|
44944
|
213
|
45369
|
214
|
45796
|
215
|
46225
|
216
|
46656
|
217
|
47089
|
218
|
47524
|
219
|
47961
|
220
|
48400
|
221
|
48841
|
222
|
49284
|
223
|
49729
|
224
|
50176
|
225
|
50625
|
226
|
51076
|
227
|
51529
|
228
|
51984
|
229
|
52441
|
230
|
52900
|
231
|
53361
|
232
|
53824
|
233
|
54289
|
234
|
54756
|
235
|
55225
|
236
|
55696
|
237
|
56169
|
238
|
56644
|
239
|
57121
|
240
|
57600
|
241
|
58081
|
242
|
58564
|
243
|
59049
|
244
|
59536
|
245
|
60025
|
246
|
60516
|
247
|
61009
|
248
|
61504
|
249
|
62001
|
250
|
62500
|
251
|
63001
|
252
|
63504
|
253
|
64009
|
254
|
64516
|
255
|
6●025
|
256
|
65536
|
257
|
66049
|
258
|
66564
|
259
|
67081
|
260
|
67600
|
261
|
68121
|
262
|
68644
|
263
|
69169
|
264
|
69696
|
265
|
70225
|
266
|
70756
|
267
|
71289
|
268
|
71824
|
269
|
72361
|
270
|
72900
|
271
|
73441
|
272
|
73984
|
273
|
74529
|
274
|
75076
|
275
|
75625
|
276
|
76176
|
277
|
76729
|
278
|
77284
|
279
|
77841
|
280
|
78400
|
281
|
78961
|
282
|
79524
|
283
|
80089
|
284
|
80656
|
285
|
81225
|
286
|
81796
|
287
|
82369
|
288
|
82944
|
289
|
83521
|
290
|
84100
|
291
|
84681
|
292
|
85264
|
293
|
85849
|
294
|
86436
|
295
|
87025
|
296
|
87616
|
297
|
88209
|
298
|
88804
|
299
|
89401
|
300
|
90000
|
3
Cent.
R.
|
Square
|
301
|
90601
|
302
|
91204
|
303
|
91809
|
304
|
92416
|
305
|
93025
|
306
|
93636
|
307
|
94249
|
308
|
94864
|
309
|
95481
|
310
|
96100
|
311
|
96721
|
312
|
97344
|
313
|
97969
|
314
|
98596
|
315
|
99225
|
316
|
99856
|
317
|
100489
|
318
|
101124
|
319
|
101761
|
320
|
102400
|
321
|
103041
|
322
|
103684
|
323
|
104329
|
324
|
104976
|
325
|
105625
|
326
|
106276
|
327
|
106929
|
328
|
107584
|
329
|
108241
|
330
|
108900
|
331
|
109561
|
332
|
110224
|
333
|
110889
|
334
|
111556
|
335
|
112225
|
336
|
112896
|
337
|
113569
|
338
|
114244
|
339
|
114921
|
340
|
115600
|
341
|
116281
|
342
|
116964
|
343
|
117649
|
344
|
118336
|
345
|
119025
|
346
|
119716
|
347
|
120409
|
348
|
121104
|
349
|
121801
|
350
|
122500
|
351
|
123201
|
352
|
123904
|
353
|
124609
|
354
|
125316
|
355
|
126025
|
356
|
126736
|
357
|
127449
|
358
|
128164
|
359
|
128881
|
360
|
129600
|
361
|
130321
|
362
|
131044
|
363
|
131769
|
364
|
132496
|
365
|
133225
|
366
|
133956
|
367
|
134689
|
368
|
135424
|
369
|
136161
|
370
|
136900
|
371
|
137641
|
372
|
138384
|
373
|
139129
|
374
|
139876
|
375
|
140625
|
376
|
141376
|
377
|
142129
|
378
|
142884
|
379
|
143641
|
380
|
144400
|
381
|
145161
|
382
|
145924
|
383
|
146689
|
384
|
147456
|
385
|
148225
|
386
|
148996
|
387
|
149769
|
388
|
150544
|
389
|
151321
|
390
|
152100
|
4
Cent.
R.
|
Square
|
391
|
152881
|
392
|
153664
|
393
|
154449
|
394
|
155236
|
395
|
156025
|
396
|
156816
|
397
|
157609
|
398
|
158404
|
399
|
159201
|
400
|
160000
|
401
|
160801
|
402
|
161604
|
403
|
162409
|
404
|
163216
|
405
|
164025
|
406
|
164836
|
407
|
165649
|
408
|
166464
|
409
|
167281
|
410
|
168100
|
411
|
168921
|
412
|
169744
|
413
|
170569
|
414
|
171396
|
415
|
172225
|
416
|
173056
|
417
|
173889
|
418
|
174724
|
419
|
175561
|
420
|
176400
|
421
|
177241
|
422
|
178084
|
423
|
178929
|
424
|
179776
|
425
|
180625
|
426
|
181476
|
427
|
182329
|
428
|
183184
|
429
|
184041
|
430
|
184900
|
431
|
185761
|
432
|
186624
|
433
|
187489
|
434
|
188356
|
435
|
189225
|
436
|
190096
|
437
|
190969
|
438
|
191844
|
439
|
192721
|
440
|
193600
|
441
|
194481
|
442
|
195364
|
443
|
196249
|
444
|
197136
|
445
|
198025
|
446
|
198916
|
447
|
199809
|
448
|
200704
|
449
|
201601
|
450
|
202500
|
451
|
203401
|
452
|
204304
|
453
|
205209
|
454
|
206116
|
455
|
207025
|
456
|
207936
|
457
|
208849
|
458
|
209764
|
459
|
210681
|
460
|
211600
|
461
|
212521
|
462
|
213444
|
463
|
214369
|
464
|
215296
|
465
|
216225
|
466
|
217156
|
467
|
218089
|
468
|
219024
|
469
|
219961
|
470
|
220900
|
471
|
221841
|
472
|
222784
|
473
|
223729
|
474
|
224676
|
475
|
225625
|
476
|
226576
|
477
|
227529
|
478
|
228484
|
479
|
229441
|
480
|
230400
|
5
Cent.
R.
|
Square
|
481
|
231361
|
482
|
232324
|
483
|
233289
|
484
|
234256
|
485
|
235225
|
486
|
236196
|
487
|
237169
|
488
|
238144
|
489
|
239121
|
490
|
240100
|
491
|
241081
|
492
|
242064
|
493
|
243049
|
494
|
244036
|
495
|
245025
|
496
|
246016
|
497
|
247009
|
498
|
248004
|
499
|
249001
|
500
|
250000
|
501
|
251001
|
502
|
252004
|
503
|
253009
|
504
|
254016
|
505
|
255025
|
506
|
256036
|
507
|
257049
|
508
|
258064
|
509
|
259081
|
510
|
260100
|
6
Cent.
R.
|
Square
|
511
|
261121
|
512
|
262144
|
513
|
263169
|
514
|
264196
|
515
|
265225
|
516
|
266256
|
517
|
267289
|
518
|
268324
|
519
|
266361
|
520
|
270400
|
521
|
271441
|
522
|
272484
|
523
|
273529
|
524
|
274576
|
525
|
275625
|
526
|
276676
|
527
|
277729
|
528
|
278784
|
529
|
279841
|
530
|
280900
|
531
|
281961
|
532
|
283024
|
533
|
284089
|
534
|
285156
|
535
|
286225
|
536
|
287296
|
537
|
288369
|
538
|
289444
|
539
|
290521
|
540
|
291600
|
5
Cent.
R.
|
Square
|
541
|
292681
|
542
|
293764
|
543
|
294849
|
544
|
295936
|
545
|
297025
|
546
|
298116
|
547
|
299209
|
548
|
300304
|
549
|
301401
|
550
|
302500
|
551
|
303601
|
552
|
304704
|
553
|
305809
|
554
|
306916
|
555
|
308025
|
556
|
309136
|
557
|
310249
|
558
|
311364
|
559
|
312481
|
560
|
313600
|
561
|
314721
|
562
|
315844
|
563
|
316969
|
564
|
318096
|
565
|
319225
|
566
|
320356
|
567
|
321489
|
568
|
322624
|
569
|
323761
|
570
|
324900
|
571
|
326041
|
572
|
327184
|
573
|
328329
|
574
|
329476
|
575
|
330625
|
576
|
331776
|
577
|
332929
|
578
|
334084
|
579
|
335241
|
580
|
336400
|
581
|
337561
|
582
|
338724
|
583
|
339889
|
584
|
341056
|
585
|
342225
|
586
|
343396
|
587
|
344569
|
588
|
345744
|
589
|
346921
|
590
|
348100
|
591
|
349281
|
592
|
350464
|
593
|
351649
|
594
|
352836
|
595
|
354025
|
596
|
355216
|
597
|
356409
|
598
|
357604
|
599
|
358801
|
600
|
360000
|
6
Cent.
R.
|
Square
|
601
|
361201
|
602
|
362404
|
603
|
363609
|
604
|
364816
|
605
|
366025
|
606
|
367236
|
607
|
368449
|
608
|
369964
|
609
|
370881
|
610
|
372100
|
611
|
373321
|
612
|
374544
|
613
|
375769
|
614
|
376996
|
615
|
378225
|
616
|
379456
|
617
|
380689
|
618
|
381924
|
619
|
383161
|
620
|
384400
|
621
|
385641
|
622
|
386884
|
623
|
388129
|
624
|
389376
|
625
|
390625
|
626
|
391876
|
627
|
393129
|
628
|
394384
|
629
|
395641
|
630
|
396900
|
7
Cent.
R.
|
Square
|
631
|
398161
|
632
|
399424
|
633
|
400689
|
634
|
401956
|
635
|
403225
|
636
|
404496
|
637
|
405769
|
638
|
407044
|
639
|
408321
|
640
|
409600
|
641
|
410881
|
642
|
412164
|
643
|
413449
|
644
|
414736
|
645
|
416025
|
646
|
417316
|
647
|
418609
|
648
|
419904
|
649
|
421201
|
650
|
422500
|
651
|
423801
|
652
|
425104
|
653
|
426409
|
654
|
427716
|
655
|
429025
|
656
|
430336
|
657
|
431649
|
658
|
432964
|
659
|
434281
|
660
|
435600
|
661
|
436921
|
662
|
438244
|
663
|
439569
|
664
|
440896
|
665
|
442225
|
666
|
443556
|
667
|
444889
|
668
|
446224
|
669
|
447561
|
670
|
448900
|
671
|
450241
|
672
|
451584
|
673
|
452929
|
674
|
454276
|
675
|
455625
|
676
|
456976
|
677
|
458329
|
678
|
459684
|
679
|
461041
|
680
|
462400
|
681
|
463761
|
682
|
465124
|
683
|
466489
|
684
|
467856
|
685
|
469225
|
686
|
470596
|
687
|
471969
|
688
|
473344
|
689
|
474721
|
690
|
476100
|
8
Cent.
R.
|
Square
|
691
|
477481
|
692
|
478864
|
693
|
480249
|
694
|
881636
|
695
|
483025
|
696
|
484416
|
697
|
485809
|
698
|
487204
|
699
|
488601
|
700
|
490000
|
701
|
491401
|
702
|
492804
|
703
|
494209
|
704
|
495616
|
705
|
497025
|
706
|
498436
|
707
|
499849
|
708
|
501264
|
709
|
502681
|
710
|
504100
|
711
|
505521
|
712
|
506944
|
713
|
508369
|
714
|
509796
|
715
|
511225
|
716
|
512656
|
717
|
514089
|
718
|
515524
|
719
|
516961
|
720
|
518400
|
7
Cent.
R.
|
Square
|
721
|
519841
|
722
|
521284
|
723
|
522729
|
724
|
524176
|
725
|
525625
|
726
|
527076
|
727
|
528529
|
728
|
529984
|
729
|
531441
|
730
|
532900
|
731
|
534361
|
732
|
535824
|
733
|
537289
|
734
|
538756
|
735
|
540225
|
736
|
541696
|
737
|
543169
|
738
|
544644
|
739
|
546121
|
740
|
547600
|
741
|
549081
|
742
|
550564
|
743
|
552049
|
744
|
553536
|
745
|
555025
|
746
|
556516
|
747
|
558009
|
748
|
559504
|
749
|
561001
|
750
|
562500
|
751
|
564001
|
752
|
565504
|
753
|
567009
|
754
|
568516
|
755
|
570025
|
756
|
571536
|
757
|
573049
|
758
|
574564
|
759
|
576081
|
760
|
577600
|
761
|
579121
|
762
|
580644
|
763
|
582169
|
764
|
583696
|
765
|
585225
|
766
|
586756
|
767
|
588289
|
768
|
589824
|
769
|
591361
|
770
|
592900
|
771
|
594441
|
772
|
595984
|
773
|
597529
|
774
|
599076
|
775
|
600625
|
776
|
602176
|
777
|
603729
|
778
|
605284
|
779
|
606841
|
780
|
608400
|
8
Cent.
R.
|
Square
|
781
|
609961
|
782
|
611524
|
783
|
613089
|
784
|
614656
|
785
|
619225
|
786
|
617796
|
787
|
619369
|
788
|
620944
|
789
|
622521
|
790
|
624100
|
791
|
625681
|
792
|
627264
|
793
|
628849
|
794
|
630436
|
795
|
632025
|
796
|
633616
|
797
|
635209
|
798
|
636804
|
799
|
638401
|
800
|
640000
|
801
|
641601
|
802
|
643204
|
803
|
644809
|
804
|
646416
|
805
|
648025
|
806
|
649636
|
807
|
651249
|
808
|
652864
|
809
|
654481
|
810
|
656100
|
811
|
657721
|
812
|
659344
|
813
|
660969
|
814
|
662596
|
815
|
664225
|
816
|
665856
|
817
|
667489
|
818
|
669124
|
819
|
670761
|
820
|
672400
|
821
|
674041
|
822
|
675684
|
823
|
677329
|
824
|
678976
|
825
|
680625
|
826
|
682276
|
827
|
683929
|
828
|
685584
|
829
|
687241
|
830
|
688900
|
831
|
690561
|
832
|
692224
|
833
|
693889
|
834
|
695556
|
835
|
697225
|
836
|
698896
|
837
|
700569
|
838
|
702244
|
839
|
703921
|
840
|
705600
|
841
|
707281
|
842
|
708964
|
843
|
710649
|
844
|
712336
|
845
|
714025
|
846
|
715716
|
847
|
717409
|
848
|
719104
|
849
|
720801
|
850
|
722500
|
851
|
724201
|
852
|
725904
|
853
|
727609
|
854
|
729316
|
855
|
731025
|
856
|
732736
|
857
|
734449
|
858
|
736164
|
859
|
737881
|
860
|
739600
|
861
|
741321
|
862
|
743044
|
863
|
744769
|
864
|
746496
|
865
|
748225
|
866
|
749956
|
867
|
751689
|
868
|
753424
|
869
|
755161
|
870
|
756900
|
871
|
758641
|
872
|
760384
|
873
|
762129
|
874
|
763876
|
875
|
765625
|
876
|
767376
|
877
|
769129
|
878
|
770884
|
879
|
772641
|
880
|
774400
|
881
|
776161
|
882
|
777924
|
883
|
779689
|
884
|
781456
|
885
|
783225
|
886
|
784996
|
887
|
786769
|
888
|
788544
|
889
|
790321
|
890
|
792100
|
891
|
793881
|
892
|
795664
|
893
|
797449
|
894
|
799236
|
895
|
801025
|
896
|
802816
|
897
|
804609
|
898
|
806404
|
899
|
808201
|
900
|
810000
|
9
Cent.
R.
|
Square
|
901
|
811801
|
902
|
813604
|
903
|
815409
|
904
|
817216
|
905
|
819025
|
906
|
820836
|
907
|
822649
|
908
|
824464
|
909
|
826281
|
910
|
828100
|
911
|
829921
|
912
|
831744
|
913
|
833569
|
914
|
835396
|
915
|
837225
|
916
|
839056
|
917
|
840889
|
918
|
842724
|
919
|
844561
|
920
|
846400
|
921
|
848241
|
922
|
850084
|
923
|
851929
|
924
|
853776
|
925
|
855625
|
926
|
857476
|
927
|
859329
|
928
|
861184
|
929
|
863041
|
930
|
864900
|
931
|
866761
|
932
|
868624
|
933
|
870489
|
934
|
872356
|
935
|
874225
|
936
|
876096
|
937
|
877969
|
938
|
879844
|
939
|
881721
|
940
|
883600
|
941
|
885481
|
942
|
887364
|
943
|
889249
|
944
|
891136
|
945
|
893025
|
946
|
894916
|
947
|
896809
|
948
|
898704
|
949
|
900601
|
950
|
902500
|
951
|
904401
|
952
|
906304
|
953
|
908209
|
954
|
910116
|
955
|
912025
|
956
|
913936
|
957
|
915849
|
958
|
917764
|
959
|
919681
|
960
|
921600
|
961
|
923521
|
962
|
925444
|
963
|
927369
|
964
|
929296
|
965
|
931225
|
966
|
933156
|
967
|
935089
|
968
|
937024
|
969
|
938961
|
970
|
940900
|
971
|
942841
|
972
|
944784
|
973
|
946729
|
974
|
948676
|
975
|
950625
|
976
|
952576
|
977
|
954529
|
978
|
956484
|
979
|
958441
|
980
|
960400
|
981
|
962361
|
982
|
964324
|
983
|
966289
|
984
|
968256
|
985
|
970225
|
986
|
972196
|
987
|
974169
|
988
|
976144
|
989
|
978121
|
990
|
980100
|
991
|
982081
|
992
|
984064
|
993
|
986049
|
994
|
988036
|
995
|
990025
|
996
|
992016
|
997
|
994009
|
998
|
996004
|
999
|
998001
|
1000
|
1000000
|
Extraction
of
the
Cube
Root
.
ACube
is
a
solid
,
or
Body
contained
within
six
equal
squares
;
and
may
be
fitly
represented
by
a
Die
.
When
a
Cube
is
given
,
as
is
435̇519̇512̇
;
point
the
Number
as
you
see
in
the
place
of
Unity
,
and
every
third
figure
after
;
then
see
what
is
the
root
of
the
greatest
Cube
contained
under
the
first
point
toward
the
left
hand
;
that
is
,
in
435
,
it
will
be
found
7
,
(
for
8
times
8
,
taken
8
times
is
5
12
,
which
is
too
much
)
put
this
7
for
the
first
figure
in
the
quotient
,
having
taken
the
Cube
thereof
,
which
is
343
,
out
of
435
:
thus
,
And
so
the
first
work
is
done
.
For
the
second
,
take
the
square
of
the
quotient
,
that
is
49
,
which
multiply
by
300
,
the
product
is
|
14700
|
to
which
adde
30
times
7
,
that
is
|
∷
∷
210
|
which
makes
the
Divisor
|
14910
|
And
the
Dividend
is
|
92519
|
So
the
quotient
might
be
6
,
but
must
be
but
5
,
because
the
Cube
of
the
new
quotient
,
and
210
times
the
square
of
the
said
quotient
,
must
be
allowed
in
this
worke
,
as
followeth
.
The
remaine
is
|
92519̇512̇
|
multiply
the
first
product
14700
by
|
78875
|
The
second
quotient
5
,
and
they
produce
|
73500
|
then
multiply
210
by
the
square
of
5
,
it
makes
|
:
5250
|
to
which
adde
the
cube
of
5.
|
..
:
:
125
|
It
makes
in
all
|
78875
|
Which
taken
from
92519
there
remaines
|
13644512
|
and
the
quotient
is
75
,
whose
square
5625
,
multiplyed
by
300
the
product
is
|
1687500
|
secondly
,
30
times
75
is
|
:
:
2250
|
And
the
new
Divisor
is
|
1710000
|
And
the
third
quotient
figure
is
8
,
by
which
multiply
1687500
,
it
is
|
13500000
|
Likewise
,
2250
,
multiplyed
by
the
square
of
8
,
(
64
)
is
|
144000
|
And
the
Cube
of
8
is
|
:
:
:
512
|
Altogether
are
|
13644512
|
Which
taken
from
the
second
remaine
,
there
remaines
now
thirdly
nothing
.
And
therefore
the
last
quotient
8
,
put
to
the
former
two
,
it
is
75.8
for
the
whole
Root
,
as
may
be
tryed
by
multiplying
758
into
it selfe
;
and
the
product
againe
by
758
,
then
the
last
product
shall
be
equal
to
the
whole
Cube
,
which
was
given
at
first
to
be
resolved
.
R.
|
Cube
|
1
|
1
|
2
|
8
|
3
|
27
|
4
|
64
|
5
|
125
|
6
|
216
|
7
|
343
|
8
|
512
|
9
|
729
|
10
|
1000
|
11
|
1331
|
12
|
1728
|
13
|
2197
|
14
|
2744
|
15
|
3375
|
16
|
4096
|
17
|
4913
|
18
|
5832
|
19
|
6859
|
20
|
8000
|
21
|
9261
|
22
|
10648
|
23
|
12167
|
24
|
13824
|
25
|
15625
|
26
|
17576
|
27
|
19683
|
28
|
21972
|
29
|
24389
|
30
|
27000
|
31
|
29791
|
32
|
32768
|
33
|
35937
|
34
|
39304
|
35
|
42825
|
36
|
48656
|
37
|
50653
|
38
|
54872
|
39
|
55419
|
40
|
64000
|
41
|
68921
|
42
|
74088
|
43
|
79507
|
44
|
85184
|
45
|
91125
|
46
|
97336
|
47
|
103823
|
48
|
110592
|
49
|
117649
|
50
|
125000
|
51
|
135651
|
52
|
140608
|
53
|
148877
|
54
|
157464
|
55
|
167375
|
56
|
175616
|
57
|
185193
|
58
|
195112
|
59
|
205379
|
60
|
216000
|
61
|
216981
|
62
|
238328
|
63
|
293047
|
64
|
262244
|
65
|
274625
|
66
|
287496
|
67
|
300753
|
68
|
314432
|
69
|
329199
|
70
|
333000
|
71
|
357911
|
72
|
373348
|
73
|
389017
|
74
|
405224
|
75
|
411875
|
76
|
438976
|
77
|
456533
|
78
|
474522
|
79
|
493039
|
80
|
512000
|
81
|
531441
|
82
|
550408
|
83
|
571787
|
84
|
592604
|
85
|
614125
|
86
|
636056
|
87
|
648303
|
88
|
681472
|
89
|
705669
|
90
|
729000
|
91
|
753571
|
92
|
778688
|
93
|
804357
|
94
|
830584
|
95
|
857375
|
96
|
884736
|
97
|
915673
|
98
|
941192
|
99
|
970299
|
100
|
1000000
|
101
|
1030301
|
102
|
1061208
|
103
|
1092727
|
104
|
1124856
|
105
|
1157625
|
106
|
1191016
|
107
|
1225043
|
108
|
1259712
|
109
|
1295029
|
110
|
1331000
|
111
|
1367631
|
112
|
1404928
|
113
|
1442897
|
114
|
1481544
|
115
|
1520875
|
116
|
1560896
|
117
|
1601613
|
118
|
1643032
|
119
|
1685159
|
120
|
1728000
|
121
|
1771561
|
122
|
1815848
|
123
|
1860867
|
124
|
1906624
|
125
|
1953125
|
126
|
2000376
|
127
|
2048383
|
128
|
2097172
|
129
|
2146689
|
130
|
2197000
|
131
|
2248091
|
132
|
2299968
|
133
|
2352637
|
134
|
2406104
|
135
|
2460375
|
136
|
2515856
|
137
|
2570353
|
138
|
2628072
|
139
|
2685619
|
140
|
2744000
|
141
|
2803221
|
142
|
2864288
|
143
|
2924207
|
144
|
2985984
|
145
|
3027525
|
146
|
3112136
|
147
|
3176523
|
148
|
3241792
|
149
|
3307949
|
150
|
3375000
|
151
|
3442951
|
152
|
3511808
|
153
|
3581577
|
154
|
3652264
|
155
|
3723875
|
156
|
3796416
|
157
|
3869893
|
158
|
3944312
|
159
|
4019679
|
160
|
4096000
|
161
|
4173281
|
162
|
4251528
|
163
|
4230744
|
164
|
4410944
|
165
|
4492125
|
166
|
4574296
|
167
|
4657463
|
168
|
4741632
|
169
|
4826809
|
170
|
4913000
|
171
|
5000211
|
172
|
5088448
|
173
|
5177717
|
174
|
5268024
|
175
|
5359375
|
176
|
5451776
|
177
|
5545233
|
178
|
5639752
|
179
|
5735339
|
180
|
5832000
|
2
Cent.
R.
|
Cube
|
181
|
5929741
|
182
|
6028568
|
183
|
6128487
|
184
|
6229504
|
185
|
6331625
|
186
|
6434856
|
187
|
6539203
|
188
|
6644672
|
189
|
6751269
|
190
|
6859000
|
191
|
6967871
|
192
|
7077888
|
193
|
7189057
|
194
|
7301384
|
195
|
7414875
|
196
|
7529536
|
197
|
7645373
|
198
|
7762392
|
199
|
7880599
|
200
|
8000000
|
201
|
8120601
|
202
|
8242408
|
203
|
8365427
|
204
|
8489664
|
205
|
8615125
|
206
|
8741816
|
207
|
8869743
|
208
|
899●912
|
209
|
9129329
|
210
|
9261000
|
211
|
9393931
|
212
|
9528128
|
213
|
9663597
|
214
|
9800344
|
215
|
9938375
|
216
|
10077696
|
217
|
10218313
|
218
|
10360232
|
219
|
10503459
|
220
|
10648000
|
221
|
10793861
|
222
|
10941048
|
223
|
11089567
|
224
|
113●9424
|
225
|
11390625
|
226
|
11543176
|
227
|
11697083
|
228
|
11852452
|
229
|
12008989
|
230
|
12167000
|
231
|
12326391
|
232
|
12487168
|
233
|
12649337
|
234
|
12812904
|
235
|
12977875
|
236
|
13144256
|
237
|
13312053
|
238
|
13481272
|
239
|
13651919
|
240
|
13824000
|
241
|
13997521
|
242
|
14172448
|
243
|
14●48907
|
244
|
14526784
|
245
|
14705125
|
246
|
14886936
|
247
|
15069223
|
248
|
15252992
|
249
|
15438249
|
250
|
15625000
|
251
|
15813251
|
252
|
16003008
|
253
|
16194277
|
254
|
16387064
|
255
|
16581375
|
256
|
16777216
|
257
|
16974593
|
258
|
17173512
|
259
|
17473979
|
260
|
175760●0
|
261
|
17779581
|
262
|
17984728
|
263
|
18191447
|
264
|
18399744
|
265
|
18609625
|
266
|
18821096
|
267
|
19034163
|
268
|
19248832
|
269
|
19465109
|
270
|
19683000
|
271
|
19902511
|
272
|
20123648
|
273
|
20346417
|
274
|
20571024
|
275
|
20796875
|
276
|
21024576
|
277
|
21253933
|
278
|
21484952
|
279
|
21717639
|
280
|
21952000
|
281
|
22188041
|
282
|
22425768
|
283
|
22665187
|
284
|
22906304
|
285
|
23149125
|
286
|
23393656
|
287
|
23539903
|
288
|
23897872
|
289
|
24137569
|
290
|
24389000
|
291
|
24642171
|
292
|
24897088
|
293
|
25153757
|
294
|
25412184
|
295
|
25672375
|
296
|
25934336
|
297
|
26198073
|
298
|
26353592
|
299
|
26730899
|
300
|
27000000
|
3
Cent.
R.
|
Cube
|
301
|
27270901
|
302
|
27543608
|
303
|
27818127
|
304
|
28094464
|
305
|
28372625
|
306
|
28652616
|
307
|
28934443
|
308
|
29218112
|
309
|
29503629
|
310
|
29791000
|
311
|
30080231
|
312
|
30371328
|
313
|
30664●97
|
314
|
30959144
|
315
|
31255875
|
316
|
31554496
|
317
|
31855013
|
318
|
32157432
|
319
|
32461759
|
320
|
32768000
|
321
|
33076161
|
322
|
33386248
|
323
|
33698267
|
324
|
34012224
|
325
|
34328125
|
326
|
34645976
|
327
|
34965783
|
328
|
35287552
|
329
|
35611289
|
330
|
35937000
|
331
|
36264691
|
332
|
36594368
|
333
|
36926037
|
334
|
37259704
|
335
|
37595375
|
336
|
37933076
|
337
|
38272753
|
338
|
38614472
|
339
|
38958219
|
340
|
39304000
|
341
|
39651821
|
342
|
40001688
|
343
|
40353607
|
344
|
40707584
|
345
|
41063625
|
346
|
41421736
|
347
|
41781923
|
348
|
42144192
|
349
|
42508549
|
350
|
42875000
|
351
|
43243551
|
352
|
43614208
|
353
|
43986977
|
354
|
44361864
|
355
|
44738875
|
356
|
45118016
|
357
|
45499293
|
358
|
45882712
|
359
|
46268279
|
360
|
46656000
|
361
|
47045881
|
362
|
47439928
|
363
|
47832147
|
364
|
48228544
|
365
|
48627125
|
366
|
49027896
|
367
|
49430863
|
368
|
49836032
|
369
|
50243409
|
370
|
50653000
|
371
|
51064811
|
372
|
51478848
|
373
|
51895117
|
374
|
52313624
|
375
|
52734375
|
376
|
53157376
|
377
|
53582633
|
378
|
54010152
|
379
|
54439939
|
380
|
54872000
|
381
|
55306341
|
382
|
55742968
|
383
|
56181887
|
384
|
56623104
|
385
|
57066625
|
386
|
57512456
|
387
|
57960603
|
388
|
58411072
|
389
|
58863869
|
390
|
59319000
|
4
Cent.
R.
|
Cube
|
391
|
59776471
|
392
|
60236288
|
393
|
60698457
|
394
|
61162984
|
395
|
61629875
|
396
|
62099136
|
397
|
62570773
|
398
|
63044792
|
399
|
63521199
|
400
|
64000000
|
401
|
64481201
|
402
|
64964808
|
403
|
65450827
|
404
|
65939264
|
405
|
66430125
|
406
|
66923416
|
407
|
67419143
|
408
|
67917312
|
409
|
68417929
|
410
|
68921000
|
411
|
69426531
|
412
|
69934528
|
413
|
70444997
|
414
|
70957944
|
415
|
71473375
|
416
|
71991296
|
417
|
72511713
|
418
|
73034632
|
419
|
73560059
|
420
|
74088000
|
421
|
74618461
|
422
|
75151448
|
423
|
75686967
|
424
|
76225024
|
425
|
76765625
|
426
|
77308776
|
427
|
77854483
|
428
|
78402752
|
429
|
78953589
|
430
|
79507000
|
431
|
80062991
|
432
|
80621568
|
433
|
81182737
|
434
|
81746504
|
435
|
82312875
|
436
|
82881856
|
437
|
83453453
|
438
|
84027672
|
439
|
84604519
|
440
|
85184000
|
441
|
85766121
|
442
|
86350888
|
443
|
86938307
|
444
|
87528384
|
445
|
88121125
|
446
|
88716536
|
447
|
89314623
|
448
|
89915392
|
449
|
90518849
|
450
|
91125000
|
451
|
91733851
|
452
|
92345408
|
453
|
92959677
|
454
|
93576664
|
455
|
94196375
|
456
|
94818816
|
457
|
95443993
|
458
|
96071912
|
459
|
96702579
|
460
|
97336000
|
461
|
97972181
|
462
|
98611128
|
463
|
99252847
|
464
|
99897344
|
465
|
100544625
|
466
|
101194696
|
467
|
101847563
|
468
|
102503232
|
469
|
103161709
|
470
|
103823000
|
471
|
104487111
|
472
|
105154048
|
473
|
105823817
|
474
|
106496424
|
475
|
107171875
|
476
|
107850176
|
477
|
108531333
|
478
|
109215352
|
479
|
109902239
|
480
|
110592000
|
5
Cent.
R.
|
Cube
|
481
|
111284641
|
482
|
111980168
|
483
|
112678587
|
484
|
113379904
|
485
|
114084125
|
486
|
114791256
|
487
|
115501303
|
488
|
116214272
|
489
|
116930169
|
490
|
117649000
|
491
|
118370771
|
492
|
119095488
|
493
|
119823157
|
494
|
120553784
|
495
|
121287375
|
496
|
122023936
|
497
|
122763473
|
498
|
123505992
|
499
|
124251499
|
500
|
125000000
|
501
|
125751501
|
502
|
126506008
|
503
|
127263527
|
504
|
128024064
|
505
|
128787625
|
506
|
129554216
|
507
|
130323843
|
508
|
131096512
|
509
|
131872229
|
510
|
132651000
|
511
|
133432831
|
512
|
134217728
|
513
|
135005697
|
514
|
135796744
|
515
|
136590875
|
516
|
137388096
|
517
|
138188413
|
518
|
138991832
|
519
|
139798359
|
520
|
140608000
|
521
|
141420761
|
522
|
142246648
|
523
|
143055667
|
524
|
143877824
|
525
|
144703125
|
526
|
145531576
|
527
|
146363183
|
528
|
147197952
|
529
|
148035889
|
530
|
148877000
|
531
|
149721291
|
532
|
150568768
|
533
|
151419437
|
534
|
152273304
|
535
|
153130375
|
536
|
153990656
|
537
|
154854153
|
538
|
155720872
|
539
|
156590819
|
540
|
157464000
|
541
|
158340421
|
542
|
159220088
|
543
|
160103007
|
544
|
160989184
|
545
|
161878625
|
546
|
162771336
|
547
|
163667323
|
548
|
164566592
|
549
|
165469149
|
550
|
166375000
|
551
|
167284151
|
552
|
168196608
|
553
|
169112377
|
554
|
170031464
|
555
|
170953875
|
556
|
171879616
|
557
|
172808683
|
558
|
173741112
|
559
|
174676879
|
560
|
175616000
|
561
|
176558481
|
562
|
177504328
|
563
|
178453547
|
564
|
179306144
|
565
|
188262125
|
566
|
181221496
|
567
|
182154263
|
568
|
183150432
|
569
|
184220009
|
570
|
185193000
|
571
|
186169411
|
572
|
187149248
|
573
|
188132517
|
574
|
189119224
|
575
|
190109375
|
576
|
191102976
|
577
|
192100033
|
578
|
193100552
|
579
|
194104539
|
580
|
195112000
|
581
|
196122941
|
582
|
197137368
|
583
|
198155287
|
584
|
199176704
|
585
|
200202625
|
586
|
201230056
|
587
|
202262003
|
588
|
203297472
|
589
|
204336469
|
590
|
205379000
|
591
|
206425071
|
592
|
207474688
|
593
|
208527857
|
594
|
209584584
|
595
|
210644875
|
596
|
211708736
|
597
|
212776173
|
598
|
113847192
|
599
|
214921799
|
600
|
216000000
|
6
Cent.
R.
|
Cube
|
601
|
217081801
|
602
|
218167208
|
603
|
219256227
|
604
|
220348864
|
605
|
221445125
|
606
|
222545016
|
607
|
223648543
|
608
|
224755712
|
609
|
225866529
|
610
|
226981000
|
611
|
228099131
|
612
|
229220928
|
613
|
230346397
|
614
|
231475544
|
615
|
232608375
|
616
|
233744896
|
617
|
234885113
|
618
|
236029032
|
619
|
237176659
|
620
|
238328000
|
621
|
239483061
|
622
|
240641848
|
623
|
241804367
|
624
|
242970624
|
625
|
244140625
|
626
|
245314376
|
627
|
246491883
|
628
|
247673152
|
629
|
248858189
|
630
|
250047000
|
631
|
251239591
|
632
|
252435968
|
633
|
253636137
|
634
|
254840104
|
635
|
256047875
|
636
|
257259456
|
637
|
258474853
|
638
|
259694072
|
639
|
260917119
|
640
|
262144000
|
641
|
263374721
|
642
|
264609288
|
643
|
265847707
|
644
|
267089984
|
645
|
268336125
|
646
|
269586136
|
647
|
270840023
|
648
|
272097792
|
649
|
273359449
|
650
|
274625000
|
651
|
275894451
|
652
|
277167808
|
653
|
278445077
|
654
|
279726264
|
655
|
281011375
|
656
|
282300416
|
657
|
283593393
|
658
|
284890312
|
659
|
286191179
|
660
|
287496000
|
661
|
288804781
|
662
|
290117528
|
663
|
291434247
|
664
|
292754944
|
665
|
294079625
|
666
|
295408296
|
667
|
296740963
|
668
|
298077632
|
669
|
299418309
|
670
|
300763000
|
671
|
302111711
|
672
|
303464448
|
673
|
304821217
|
674
|
306182024
|
675
|
307546875
|
676
|
308915776
|
677
|
310288733
|
678
|
311665752
|
679
|
313046839
|
680
|
314432000
|
681
|
315821241
|
682
|
317214568
|
683
|
318611987
|
684
|
320013504
|
685
|
321419125
|
686
|
322828856
|
687
|
324242703
|
688
|
325660672
|
689
|
327082769
|
690
|
328509000
|
7
Cent.
R.
|
Cube
|
691
|
329939371
|
692
|
331373888
|
693
|
332812557
|
694
|
334255384
|
695
|
335702375
|
696
|
3371535●6
|
697
|
338608873
|
698
|
340068392
|
699
|
341532099
|
700
|
343000000
|
701
|
344472101
|
702
|
345948408
|
703
|
347428927
|
704
|
348913664
|
705
|
350402625
|
706
|
351895816
|
707
|
353393243
|
708
|
354894912
|
709
|
356400829
|
710
|
357911000
|
711
|
359425431
|
712
|
360944128
|
713
|
362467097
|
714
|
363994344
|
715
|
365525875
|
716
|
367061696
|
717
|
368601813
|
718
|
370146232
|
719
|
371694959
|
720
|
373248000
|
721
|
374805361
|
722
|
376367048
|
723
|
377933067
|
724
|
379503424
|
725
|
381078125
|
726
|
382657176
|
727
|
384240583
|
728
|
385828352
|
729
|
387420489
|
730
|
389017000
|
731
|
390617891
|
732
|
392223168
|
733
|
393832837
|
734
|
395446904
|
735
|
397065375
|
736
|
398688256
|
737
|
400●15553
|
738
|
401947272
|
739
|
403583419
|
740
|
405224000
|
741
|
406869021
|
742
|
408518488
|
743
|
410172407
|
744
|
411830784
|
745
|
413493625
|
746
|
415160936
|
747
|
416832723
|
748
|
418508992
|
749
|
420189749
|
750
|
421875000
|
751
|
423564751
|
752
|
425259008
|
753
|
426957777
|
754
|
428661064
|
755
|
430368875
|
756
|
432081216
|
757
|
433798093
|
758
|
435519512
|
759
|
437245479
|
760
|
438976000
|
761
|
440711081
|
762
|
442450728
|
763
|
444194947
|
764
|
445943744
|
765
|
447697125
|
766
|
449455096
|
767
|
451217663
|
768
|
452984832
|
769
|
454756609
|
770
|
456533000
|
771
|
458314011
|
772
|
460099648
|
773
|
461889917
|
774
|
463684824
|
775
|
465484375
|
776
|
467288576
|
777
|
469097433
|
778
|
470910952
|
779
|
472729139
|
780
|
474552000
|
8
Cent.
R.
|
Cube
|
781
|
476379541
|
782
|
478211768
|
783
|
480048687
|
784
|
481890304
|
785
|
483736625
|
786
|
485587656
|
787
|
487443403
|
788
|
489303872
|
789
|
491169069
|
790
|
493039000
|
791
|
494913671
|
792
|
496793088
|
793
|
498677257
|
794
|
500566184
|
795
|
502459875
|
796
|
504358336
|
797
|
506261573
|
798
|
508169592
|
799
|
510082399
|
800
|
512000000
|
801
|
513922401
|
802
|
515849608
|
803
|
517781627
|
804
|
519718464
|
805
|
521660125
|
806
|
523606616
|
807
|
525557943
|
808
|
527514112
|
809
|
529475129
|
810
|
531441000
|
811
|
533411731
|
812
|
535387328
|
813
|
537367797
|
814
|
539353144
|
815
|
541343375
|
816
|
543338496
|
817
|
545338513
|
818
|
547343432
|
819
|
549353259
|
820
|
551368000
|
821
|
553387661
|
822
|
555412248
|
823
|
557441767
|
824
|
559476224
|
825
|
561515625
|
826
|
563559976
|
827
|
565609283
|
828
|
567663552
|
829
|
569722789
|
830
|
571787000
|
831
|
573856191
|
832
|
575930368
|
833
|
578009537
|
834
|
580093704
|
835
|
582182875
|
836
|
584277056
|
837
|
586376253
|
838
|
588480472
|
839
|
590589719
|
840
|
592704000
|
841
|
594823321
|
842
|
596947688
|
843
|
599077107
|
844
|
601211584
|
845
|
603351125
|
846
|
605495736
|
847
|
607645423
|
848
|
609800192
|
849
|
611960049
|
850
|
614125000
|
851
|
616295051
|
852
|
618470208
|
853
|
620650477
|
854
|
622835864
|
855
|
625026375
|
856
|
627222016
|
857
|
629422793
|
858
|
631628712
|
859
|
633839779
|
860
|
636056000
|
861
|
638277381
|
862
|
640503928
|
863
|
642735647
|
864
|
644972544
|
865
|
647214625
|
866
|
649461896
|
867
|
651714363
|
868
|
653972032
|
869
|
656234909
|
870
|
658503000
|
871
|
660776311
|
872
|
663054848
|
873
|
665338617
|
874
|
667627624
|
875
|
669921875
|
876
|
672221376
|
877
|
674526133
|
878
|
676836152
|
879
|
679151435
|
880
|
681472000
|
881
|
683797841
|
882
|
686128968
|
883
|
688465387
|
884
|
690807104
|
885
|
693154125
|
886
|
695506456
|
887
|
697864103
|
888
|
700227072
|
889
|
702595369
|
890
|
704969000
|
891
|
707347971
|
892
|
709732288
|
893
|
712121957
|
894
|
714516984
|
895
|
716917375
|
896
|
719323136
|
897
|
721734273
|
898
|
724150792
|
899
|
726572699
|
900
|
729000000
|
9
Cent.
R.
|
Cube
|
901
|
731432701
|
902
|
733870808
|
903
|
736314327
|
904
|
738763264
|
905
|
741217625
|
906
|
743677416
|
907
|
746142643
|
908
|
748613312
|
909
|
751089429
|
910
|
753571000
|
911
|
756058031
|
912
|
758550528
|
913
|
761048497
|
914
|
763551944
|
915
|
766060875
|
916
|
768575296
|
917
|
771095213
|
918
|
773620632
|
919
|
776151559
|
920
|
778688000
|
921
|
781229961
|
922
|
783777448
|
923
|
786330467
|
924
|
788889024
|
925
|
791453125
|
926
|
794022776
|
927
|
796597983
|
928
|
799178752
|
929
|
801765089
|
930
|
804357000
|
931
|
806954491
|
932
|
800557569
|
933
|
812166237
|
934
|
814780504
|
935
|
817400375
|
936
|
820025856
|
937
|
822656953
|
938
|
825293672
|
939
|
827936019
|
940
|
830584000
|
941
|
833237621
|
942
|
835896888
|
943
|
838561807
|
944
|
841232384
|
945
|
843908625
|
946
|
846590536
|
947
|
849278123
|
948
|
851971392
|
949
|
854670349
|
950
|
857375000
|
951
|
860085351
|
952
|
862801408
|
953
|
865523177
|
954
|
868250664
|
955
|
870983875
|
956
|
873722816
|
957
|
876467493
|
958
|
879217912
|
959
|
881974079
|
960
|
384736000
|
961
|
887503681
|
962
|
890277128
|
963
|
893056347
|
964
|
895841344
|
965
|
898632125
|
966
|
901428696
|
967
|
904231063
|
968
|
907039232
|
969
|
909853209
|
970
|
912673000
|
971
|
915498611
|
972
|
918330048
|
973
|
921167317
|
974
|
924010424
|
975
|
926859375
|
976
|
929714176
|
977
|
932574833
|
978
|
935441352
|
979
|
938313739
|
980
|
941192000
|
981
|
944076141
|
982
|
946966168
|
983
|
949862087
|
984
|
952763904
|
985
|
955671625
|
986
|
958585256
|
987
|
961504803
|
988
|
964430272
|
989
|
967361669
|
990
|
970299000
|
991
|
973242271
|
992
|
976191488
|
993
|
979146657
|
994
|
982107784
|
995
|
985074875
|
996
|
988047936
|
997
|
991026973
|
998
|
994011992
|
999
|
997002999
|
1000
|
1000000000
|
Some
uses
of
the
Square
and
Cube
Root
.
Uses
of
the
Square
Root
.
WHat
the
Square
and
Cube
Roots
are
,
and
how
to
extract
them
,
hath
already
been
taght
,
and
for
more
ease
and
expedition
,
there
are
Tables
ready
calculated
,
both
of
the
Square
and
Cube
Roots
,
from
1
to
1000
,
we
come
now
to
shew
some
uses
thereof
,
which
in
some
measure
will
appear
in
the
Propositions
following
.
PROPOSITION
.
I.
Admit
the
height
of
the
Wall
of
a
Fort
or
Castle
to
be
scaled
,
be
30
foot
,
and
the
breadth
of
the
Trench
about
the
Fort
be
40
foot
,
I
demand
of
what
length
a
Scaling
Ladder
should
be
,
justly
to
reach
from
the
edge
of
the
Trench
,
to
the
top
of
the
Wall
.
By
the
47th
.
of
the
first
Book
of
Euclides
Elements
,
it
is
demonstrated
that
the
square
of
the
Hypotenusal
of
all
right
angled
plain
Triangles
is
equal
to
the
squares
of
the
2
other
sides
;
I
therefore
to
resolve
this
Proposition
,
square
the
height
of
the
Wall
which
is
30
,
facit
900
,
also
I
square
the
breadth
of
the
Trench
which
is
40
,
facit
1600
,
these
two
added
together
make
2500
,
the
square
root
whereof
is
50
,
and
so
long
must
Scaling
Ladders
be
made
to
reach
from
the
edge
of
the
Trench
to
the
top
of
the
Wall
.
PROPOSITION
II.
There
be
two
Towns
,
as
Chichester
and
York
,
which
lie
North
and
South
one
from
another
,
and
their
distance
is
220
miles
,
and
Excester
lieth
directly
West
from
Chichester
120
miles
;
I
desire
to
know
the
distance
of
York
from
Excester
.
Square
120
,
Excester
120
Chichester
220
York
the
distance
of
Excester
and
Chichester
,
it
maketh
14400
,
likewise
square
220
,
the
distance
of
York
and
Chichester
,
facit
48400
,
these
two
numbers
added
together
make
62800
,
whose
square
root
extracted
(
or
found
in
the
Tables
)
will
be
250
3/5
;
neer
,
and
so
many
miles
is
Excester
distant
from
York
.
Vse
of
the
Cube
Root
.
ONe
chief
use
of
the
Cube
Root
,
is
to
find
out
a
proportion
between
,
like
Solids
,
such
are
Spheres
,
Cubes
,
and
such
like
;
as
in
the
Proposion
following
.
PROPOSITION
.
If
a
Bullet
of
Brasse
of
4
inches
Diameter
,
weigh
9
pound
,
what
shall
a
Bullet
of
Brasse
weigh
,
whose
Diameter
is
8
inches
?
Cube
4
the
Diameter
of
the
lesser
Bullet
,
makes
64
,
likewise
Cube
the
Diameter
of
the
greater
Bullet
8
,
makes
4608.
This
done
,
say
by
the
Rule
of
proportion
;
If
the
Cube
number
6●
,
give
9
li.
weight
,
what
shall
the
Cube
number
4608
give
?
Multiply
and
divide
,
you
shall
find
72
,
and
so
many
pounds
will
a
Bullet
of
Brasse
weigh
,
whose
Diameter
is
8
inches
.
THE
THIRD
PART
Conteining
INSTRUMENTAL
ARITHMETICK
.
THe
Arithmetick
,
of
which
we
now
come
to
treat
,
and
which
I
call
Instrumental
Arithmetick
,
is
not
any
new
kind
of
Arithmetick
,
but
is
indeed
the
same
with
Decimal
Arithmetick
before
taught
;
onely
,
whereas
in
Decimal
Arithmetick
,
there
were
certain
Tables
made
of
Money
,
Weight
,
and
Measure
,
by
help
of
which
the
Decimal
of
any
fraction
of
Money
,
Weight
or
Measure
might
be
set
down
(
as
it
were
)
in
whole
numbers
,
here
in
this
Instrumental
part
,
we
have
contrived
certain
Scales
of
Money
,
Weight
,
and
Measure
,
equally
divided
into
the
several
Denominations
into
which
the
several
Weights
and
Measures
,
can
be
equally
divided
,
unto
every
of
which
Scales
there
is
joyned
a
Scale
of
100
,
1000
,
or
10000
,
equal
parts
,
according
to
the
length
of
the
Scale
,
so
that
by
inspection
onely
you
may
readily
and
exactly
without
addition
(
as
in
using
the
fore
mentioned
Tables
you
must
necessarily
do
)
set
down
the
Decimal
fraction
of
any
part
of
Money
,
Weight
or
Measure
,
with
great
celerity
and
exactnesse
,
if
the
Scale
be
any
thing
well
divided
,
though
the
Scale
be
but
of
a
reasonable
length
.
Now
the
Scale
,
which
I
have
chiefly
made
choice
of
in
this
Work
,
as
being
of
most
use
with
English
men
;
(
though
other
Scales
may
be
made
for
the
Coyns
,
Weights
or
Measures
of
any
other
Countrey
as
well
,
and
upon
the
same
ground
)
are
chiefly
these
,
viz.
-
1
Of
Money
.
-
2
Of
Troy
Weight
.
-
3
Of
Averdupois
great
Weight
.
-
4
Of
Averdupois
little
Weight
.
-
5
Of
Liquid
measures
.
-
6
Of
dry
measures
.
-
7
Of
Long
measures
.
-
8
Of
Dozens
.
Unto
every
of
these
Scales
,
is
joyned
another
Scale
of
100
or
1000
equal
parts
,
these
Scales
are
made
to
face
one
another
,
so
that
if
you
look
upon
any
one
Division
,
in
the
one
,
you
shall
also
discern
plainly
what
Division
or
part
of
a
Division
answereth
thereunto
in
the
other
.
English
Coyn
,
Two
shillings
being
the
Integer
Troy
weight
,
Two
peny
weight
being
the
Integer
Averdupois
great
weight
,
28
lib:
or
one
quarter
of
an
hundred
being
the
Integer
Averdupois
little
weight
,
16
ounces
,
or
one
pound
being
the
Integer
Place
this
betweene
pag
246
&
247
Deliniavit
Antonius
Thompson
Dry
measure
,
S
Bushels
being
the
Integer
Liquid
measure
,
36
Gallons
,
or
one
Barrell
being
the
Integer
Long
measure
,
one
Ell
or
one
Yard
being
the
Integer
These
Scales
being
thus
disposed
,
as
they
may
easily
be
upon
any
Ruler
of
Silver
,
Brass
,
or
Wood
;
but
best
of
all
upon
a
square
Ruler
,
made
in
form
of
a
Parallelepipedon
,
will
by
inspection
onely
give
you
any
Decimal
fraction
required
without
Addition
,
or
(
on
the
contrary
)
reduce
the
fraction
into
the
know
parts
of
the
Integer
,
by
inspection
also
,
without
subtraction
.
Let
thus
much
suffice
for
a
general
description
of
what
I
mean
by
Scales
,
the
particular
description
of
them
,
will
more
plainly
appear
,
when
we
treat
of
Numeration
upon
the
Scales
,
unto
which
we
shall
now
proceed
.
But
first
take
a
view
of
the
Scales
as
they
are
here
disposed
,
and
as
they
may
be
set
upon
such
a
Ruler
as
I
have
here
mentioned
.
Numeration
upon
the
Scales
.
THe
Scales
here
to
be
described
are
in
number
eight
,
as
hath
been
already
shewed
,
and
as
by
the
figure
of
them
appear
.
Now
Numeration
upon
a
Scale
is
to
find
upon
what
part
of
the
Scale
,
any
number
upon
the
same
Scale
will
fall
.
We
will
begin
with
the
first
,
and
so
proceed
till
we
have
given
an
Example
in
every
one
.
1
The
first
Scale
is
of
English
money
,
and
is
divided
into
24
equal
parts
,
which
represent
24
pence
or
2
shillings
,
these
parts
are
numbred
with
Arithmetical
figures
,
from
the
beginning
thereof
,
by
1
,
2
,
3
,
4
,
5
,
&c.
to
24
,
each
division
representing
one
peny
,
and
the
whole
24
divisions
representeth
24
pence
or
2
shillings
,
So
that
where
the
figure
1
standeth
,
that
part
of
the
Scale
representeth
one
peny
,
where
the
figure
2
standeth
,
it
representeth
two
pence
,
where
the
figure
18
standeth
,
it
representeth
18
pence
,
or
one
shilling
six
pence
,
and
so
of
any
other
figure
of
the
same
Seale
.
Then
because
there
are
four
farthings
contained
in
a
peny
,
each
of
these
pence
(
or
divisions
)
is
sub-divided
into
4
other
equal
parts
by
short
lines
,
every
one
of
these
representing
one
farthing
,
so
is
the
whole
Scale
divided
in
all
in
96
equal
parts
,
which
are
the
number
of
farthings
conteined
in
two
shillings
.
This
if
you
look
into
the
Scale
of
Money
for
8
pence
3
farthings
,
you
shall
find
it
at
the
letter
a
,
which
letter
is
here
put
onely
for
example
sake
,
Also
if
you
would
find
in
the
Scale
the
place
of
18
d.
half
peny
,
you
shall
find
it
at
c
,
and
thus
may
you
find
the
place
of
any
number
of
pence
and
farthings
under
two
shillings
upon
the
Scale
.
Unto
this
Scale
of
money
(
as
to
all
the
rest
of
the
Scales
,
)
there
is
joyned
another
Scale
which
I
shall
alwayes
hereafter
call
the
Scale
of
1000
,
the
use
of
which
Scale
is
this
.
When
you
have
found
any
Number
of
pence
or
farthings
upon
the
Scale
of
Money
,
you
shall
find
upon
the
Scale
of
1000
,
what
parts
of
a
thousand
is
the
Decimal
of
those
pence
and
farthings
:
Thus
when
in
the
Scale
of
money
you
find
at
the
letter
a
8
pence
3
farthings
,
if
you
cast
your
eye
directly
crosse
to
the
Scale
of
1000
,
you
shall
find
364
to
stand
directly
against
8
d.
3
q.
which
364
is
the
Decimal
of
8
d.
3
q.
Also
,
if
you
find
upon
the
Scale
of
money
18
d.
half
penny
,
which
is
at
the
letter
c
,
you
shall
find
against
it
in
the
Scale
of
1000
,
this
number
771
,
which
is
the
Decimal
of
18
d.
half
penny
.
And
in
this
manner
may
the
Decimal
of
any
number
of
pence
or
farthings
under
two
shillings
be
most
easily
and
exactly
obteined
.
Now
on
the
contrary
,
suppose
a
Decimal
fraction
were
given
,
representing
some
part
of
English
Coin
if
you
look
in
the
Scale
of
1000
for
your
Number
given
,
right
against
it
in
the
Scale
of
Money
you
shall
find
what
Number
of
pence
and
farthings
is
representeth
thereby
.
As
for
Example
.
Suppose
.364
were
a
Decimal
given
,
and
it
were
required
to
find
what
part
of
Coin
it
doth
represent
.
Look
in
the
Scale
of
1000
for
the
Number
364
,
and
right
against
it
you
shall
find
8
pence
3
farthings
.
Also
if
.771
were
a
Decimal
given
,
if
you
look
in
the
Scale
of
1000
for
.771
,
you
shall
find
against
that
number
771
,
18
pence
2
farthings
.
And
thus
of
any
other
.
By
what
hath
been
already
said
,
it
may
be
easily
discerned
of
what
exceeding
expedition
these
Scales
thus
disposed
are
of
,
for
I
dare
affirme
,
that
I
will
set
down
2
(
if
not
3
)
numbers
by
the
Scale
,
as
soon
as
one
by
the
Tables
,
and
if
the
Scale
be
but
of
any
reasonable
length
,
altogether
with
as
much
exactnesse
,
but
if
I
should
vary
an
unite
in
the
last
place
,
in
my
estimation
in
the
Scale
of
1000
,
it
is
not
any
thing
material
.
I
have
been
very
tedious
in
shewing
the
use
of
these
Scales
to
finde
the
fraction
parts
of
money
,
but
the
reason
is
because
I
intend
to
be
the
briefer
in
the
rest
,
for
weight
and
measure
,
the
manner
of
working
(
when
the
division
of
the
Scale
is
known
,
)
being
the
same
in
all
respects
without
the
least
alteration
.
2
The
second
Scale
is
of
Troy
weight
,
two
Peny
weight
being
the
Integer
,
which
Scale
is
divided
into
48
equal
parts
or
divisions
,
each
of
which
divisions
contains
one
grain
,
and
are
numbered
by
Arithmetical
figures
at
every
three
grains
by
3
,
6
,
9
,
12
,
&c.
to
24
,
and
at
the
place
where
24
should
stand
,
there
standeth
PW
,
which
signifieth
one
peny
weight
,
or
24
grains
,
this
PW
standeth
in
the
middle
of
the
line
.
Then
is
the
same
Scale
continued
farther
by
Arithmetical
figures
3
,
6
,
9
,
12
,
&c.
as
before
to
24
,
and
there
is
written
PW
again
,
representing
two
peny
weight
,
or
48
grains
.
The
Scale
being
thus
divided
,
it
is
easie
to
finde
the
place
where
any
number
of
Grains
under
48
shall
be
upon
the
Scale
;
As
for
example
,
if
it
be
required
to
finde
where
8
peny
weight
shall
fall
,
look
upon
the
Scale
of
Troy
weight
,
from
the
beginning
thereof
,
and
count
the
figures
3
and
6
,
then
count
also
two
of
the
smaller
Divisions
,
and
that
makes
8
grains
,
which
you
shall
finde
to
stand
at
the
letter
d
,
which
is
the
place
of
8
grains
;
Also
if
upon
the
Scale
you
would
finde
the
place
of
one
peny
weight
10
grains
,
you
shall
finde
it
at
the
letter
e
,
and
so
of
any
other
number
of
grains
under
48
,
or
two
peny
weight
.
But
if
you
had
a
Decimal
given
,
and
would
know
what
number
of
grains
it
representeth
,
if
you
seek
your
Decimal
given
in
the
Scale
of
1000
,
right
against
it
in
the
Scale
of
Troy
weight
,
you
shall
finde
the
number
of
grains
represented
thereby
.
Example
.
Let
.167
be
a
Decimal
fraction
given
,
If
you
look
in
the
Scale
of
1000
,
for
167
,
right
against
it
in
the
scale
of
Troy
weight
,
you
shall
finde
8
peny
weight
.
Also
if
.708
were
a
Decimal
given
,
if
you
seek
708
in
the
scale
of
1000
,
right
against
it
you
shall
finde
1
peny
weight
10
grains
.
3
The
third
scale
is
of
Avoirdupois
great
weight
,
28
pounds
,
or
one
quarter
of
an
hundred
,
being
the
Integer
,
this
scale
is
numbered
by
1
,
2
,
3
,
4
,
&c.
to
●8
,
which
28
representeth
28
l.
or
a
quarter
of
a
hundred
,
and
each
of
those
is
subdivided
into
four
small
parts
,
each
representing
one
quarter
of
a
pound
.
Now
if
you
would
know
what
is
the
Decimal
of
any
number
of
pounds
or
quarters
under
28
,
if
you
seek
the
number
of
pounds
in
the
scale
of
Avoirdupois
great
weight
,
right
against
it
in
the
scale
of
1000
,
you
shall
finde
the
decimal
thereof
.
Thus
if
it
were
required
to
find
the
Decimal
of
8
pound
and
an
half
,
if
you
look
upon
the
Scale
for
8
pound
and
an
half
you
shall
find
it
at
the
letter
g
,
and
right
against
it
in
the
Scale
of
1000
you
shall
find
304
,
which
is
the
Decimal
of
8
pounds
and
an
half
.
On
the
contrary
,
suppose
304
were
a
Decimal
given
,
and
it
were
required
to
find
what
part
of
Avoirdupois
great
weight
were
represented
thereby
,
if
you
look
in
the
Scale
of
1000
for
304
,
right
against
it
in
the
Scale
of
Avoirdupois
great
weight
,
you
shall
find
8
pound
and
an
half
.
4
The
fourth
Scale
is
of
Avoirdupois
little
Weight
,
16
ounces
or
one
pound
being
the
Integer
,
This
Scale
is
first
divided
unto
16
equal
parts
,
and
numbred
by
1
,
2
,
3
,
4
,
&c.
to
16
,
each
Division
representing
one
ounce
.
Then
again
,
each
of
these
ounces
is
sub-divided
into
8
other
smaller
parts
or
divisions
,
each
of
which
divisions
representeth
two
Drams
;
but
if
your
Scale
be
large
enough
,
you
may
have
each
ounce
divided
into
16
equal
parts
,
or
divisions
,
each
division
representing
one
Dram.
Now
to
find
the
Decimal
belonging
to
any
number
of
Ounces
and
Drams
,
repaire
to
the
Scale
of
Avoirdupois
little
weight
,
and
on
it
find
the
quantity
of
ounces
and
drams
required
,
and
right
against
it
in
the
Scale
of
1000
,
you
shall
have
the
Decimal
thereof
.
Thus
if
it
were
required
to
find
the
Decimal
of
6
ounces
,
and
6
drams
,
if
you
look
this
in
the
Scale
of
Avoirdupois
little
weight
,
you
shall
find
it
at
the
letter
h
,
and
right
against
it
in
the
Scale
of
1000
,
you
shall
find
398
,
which
is
the
Decimal
of
6
ounces
and
6
drams
.
But
if
398
were
a
Decimal
given
,
and
it
were
required
to
find
the
value
thereof
in
Avoirdupois
little
weight
,
if
you
look
for
398
in
the
Scale
of
1000
,
right
against
it
in
the
other
scale
you
shall
find
6
ounces
6
drams
.
5
The
fifth
Scale
is
of
Dry
Measures
,
one
Quarter
or
8
Bushels
being
the
Integer
,
This
Scale
is
first
divided
into
8
equal
parts
,
and
numbred
by
1
,
2
,
3
,
&c.
to
8
,
each
of
which
divisions
representeth
a
Bushel
,
&
each
of
those
parts
is
again
sub-divided
,
first
into
4
equal
parts
or
divisions
each
representing
one
peck
,
and
then
those
again
sub-divided
into
4
other
smaller
parts
,
representing
quarters
,
halves
,
and
three
quarters
of
a
peck
.
Now
if
you
would
know
the
Decimal
belonging
to
any
number
of
Bushels
(
under
8
Bushels
or
one
quarter
)
Pecks
and
parts
of
a
Peck
,
if
you
seek
the
number
of
Bushels
,
Pecks
,
and
parts
of
a
Peck
in
the
Scale
of
Dry
Measures
,
right
against
it
in
the
Scale
of
1000
,
you
shall
have
the
Decimal
required
.
As
for
Example
,
if
it
were
required
to
find
the
Decimal
belonging
to
5
Bushels
2
Pecks
,
and
half
a
Peck
,
if
you
look
into
the
Scale
of
dry
Measures
you
shall
find
5
bushels
,
2
pecks
,
and
an
half
,
to
stand
at
the
letter
k
,
and
right
against
it
in
the
Scale
of
1000
,
you
shall
find
702
,
which
is
the
Decimal
answering
to
5
bushels
,
2
pecks
and
an
half
.
But
if
702
were
a
decimal
given
,
and
it
were
required
to
find
what
number
of
bushels
,
pecks
and
parts
of
a
peck
it
representeth
,
if
you
look
in
the
scale
of
1000
for
702
,
you
shall
find
against
it
in
the
scale
of
Dry
measures
,
5
bushels
2
pecks
and
an
half
.
6
The
sixth
Scale
is
of
Liquid
Measures
,
the
Integer
being
36
Gallons
or
one
Barrel
,
this
Scale
is
divided
first
into
36
equal
parts
or
divisions
,
and
numbered
by
5
,
10
,
15
,
&c.
to
36
,
then
every
of
these
divisions
is
again
sub-divided
into
4
other
smal
divisions
,
each
representing
a
quart
,
but
(
if
the
Scale
be
large
enough
)
you
may
sub-divide
each
Gallon
into
8
parts
,
so
will
every
part
represent
one
pinte
.
Now
to
find
the
Decimal
belonging
to
any
number
of
Gallons
(
under
36
Gallons
or
one
barrel
)
quarts
or
pintes
,
repaire
to
the
Scale
of
Liquid
measures
,
and
seek
there
upon
the
Scale
,
the
number
of
gallons
,
quarts
,
or
pintes
,
and
against
it
in
the
Scale
of
1000
,
you
shall
find
the
Decimal
thereunto
belonging
.
So
if
it
were
required
to
find
a
Decimal
representing
10
gallons
and
two
quarts
,
or
4
pintes
,
which
is
all
one
,
if
you
seek
in
the
scale
of
Liquid
measures
for
10
gallons
2
quarts
,
you
shall
find
it
at
the
letter
m
,
against
which
in
the
scale
of
1000
,
you
shall
find
292
,
which
is
the
Decimal
of
10
gallons
4
pintes
.
Likewise
if
292
were
a
Decimal
given
,
and
it
were
required
to
find
what
number
of
gallons
,
quarts
or
pintes
were
represented
thereby
,
if
you
look
in
the
scale
of
1000
for
292
,
right
against
it
in
the
scale
of
Liquid
measures
,
you
shall
find
10
gallons
,
2
quarts
,
or
4
pintes
.
7
The
seventh
scale
is
of
Long
Measures
,
the
Integer
being
Yards
or
Ells
,
this
scale
is
divided
into
4
equal
parts
,
and
numbred
by
1
,
2
,
3
,
4
,
representing
1
quarter
,
2
quarters
,
3
quarters
or
4
quarters
of
a
Yard
or
Ell
,
these
are
again
sub-divided
first
into
4
other
equal
parts
,
representing
nails
,
and
those
may
be
again
sub-divided
at
pleasure
if
need
be
.
Now
if
you
would
know
what
decimal
belongeth
to
any
number
of
quarters
or
nailes
of
a
yard
if
you
seek
the
number
of
quarters
and
nailes
in
the
scale
of
Long
measure
,
the
scale
of
1000
will
give
you
the
Decimal
thereof
.
Thus
if
it
be
required
to
find
the
Decimal
belonging
to
1
quarter
and
3
nails
,
if
you
seek
this
in
the
scale
of
long
measure
,
you
shall
find
it
to
stand
at
the
letter
o
,
against
which
in
the
scale
of
1000
you
shall
find
437
,
which
is
the
decimal
answering
to
1
quarter
&
3
nails
of
a
Yard
or
Ell.
Also
if
437
were
a
decimal
given
,
&
it
were
required
to
find
what
quantity
of
yards
or
ells
were
represented
thereby
,
if
you
look
in
the
scale
of
1000
,
for
437
,
you
shall
in
the
scale
of
long
measure
find
against
it
one
quarter
and
3
nails
.
8
The
eighth
and
last
scale
is
of
representative
inches
,
the
whole
scale
being
divided
into
12
equal
parts
and
numbred
by
1
,
2
,
3
,
&c.
to
12
,
and
those
parts
are
again
subdivided
into
halves
,
quarters
,
and
half
quarters
,
as
Carpenters-Rules
are
usually
divided
.
Unto
this
Scale
,
(
as
unto
all
the
other
)
there
is
joyned
a
Scale
of
1000
,
this
Scale
will
readily
discover
what
is
the
Decimal
belonging
to
any
number
of
Inches
,
halves
or
quarters
,
and
the
use
is
the
same
with
the
Scales
before
mentioned
Thus
have
I
given
you
a
briefe
description
of
these
Scales
and
the
uses
of
them
,
and
do
now
suppose
my
Reader
to
be
perfectly
acquainted
with
the
way
of
numbring
or
counting
upon
them
,
wherefore
I
intend
onely
to
give
you
a
question
or
two
in
the
most
usual
Rules
of
Arithmetick
,
and
so
conclude
;
for
Decimal
Arithmetick
being
already
sufficiently
explained
,
I
shall
not
need
to
repeat
the
Rules
(
or
the
manner
of
working
them
)
again
,
but
give
you
one
Example
,
by
which
the
exactnesse
and
expedition
of
these
Scales
may
the
more
evidently
appear
,
for
when
we
work
by
Scales
,
it
is
supposed
that
we
do
not
use
Vulgar
,
but
Decimal
Arithmetick
,
and
Addition
,
Subtraction
,
Multiplication
,
Division
,
the
Rule
of
Three
,
and
indeed
,
all
the
other
Rules
of
Arithmetick
,
are
to
be
performed
,
as
is
before
taught
,
the
Scale
serving
only
to
avoid
Reduction
.
Addition
.
WHat
Addition
is
,
and
the
manner
of
working
of
it
hath
been
already
taught
,
both
in
the
first
and
second
Parts
,
we
will
now
come
to
an
Example
,
which
let
be
in
Addition
of
English
Coin
,
and
let
the
sums
to
be
added
be
36
l.
8
s.
8
d.
29
l.
0
s.
2
d.
31
l.
16
s.
9
d.
and
6
l.
2
s.
5
d.
First
,
set
down
36
l.
29
l.
31
l.
and
6
l.
one
under
another
,
in
such
order
as
you
see
here
in
the
margent
,
drawing
a
line
by
the
side
of
them
as
you
see
here
done
,
and
also
a
line
under
them
.
This
done
,
seeing
that
your
first
number
to
be
set
down
to
36
l.
is
8
s.
8
d.
you
must
for
the
8
s.
(
because
two
shillings
,
which
we
called
a
Decade
,
or
the
tenth
part
of
a
pound
,
is
made
the
Integer
,
in
the
Scale
of
Money
)
set
down
4
,
which
is
done
by
memorie
,
and
after
it
make
a
comma
.
Then
your
next
number
to
be
set
by
29
l.
being
0
s.
2
d.
for
the
0
s.
set
down
a
Cypher
,
thirdly
,
for
your
number
to
be
set
by
31
l.
being
16
s.
9
d.
for
the
16
s.
set
down
8
decades
,
with
a
comma
after
it
,
and
lastly
,
the
number
to
be
set
by
6
l.
being
2
s.
5
d.
for
the
2
s.
I
set
down
1
decade
with
a
comma
after
it
,
and
then
will
your
work
stand
,
as
here
you
see
.
Then
take
your
scale
in
hand
,
and
seeing
your
first
number
of
pence
are
8
d.
look
in
your
scale
of
money
for
8
d.
and
against
it
in
the
scale
of
1000
,
you
shall
finde
333
,
which
set
to
36
l.
8
s.
behinde
the
comma
,
then
your
next
number
of
pence
being
2
d.
look
in
your
scale
for
2
d.
and
against
it
in
the
scale
of
1000
,
you
shall
finde
082
,
which
set
to
29
l.
0
s.
behinde
the
comma
.
36
|
4
,
|
333
|
29
|
0
,
|
083
|
31
|
8
,
|
376
|
6
|
1
,
|
208
|
Then
your
third
number
of
pence
being
9
d.
look
in
your
scale
for
9
d.
and
against
it
in
the
scale
of
1000
,
you
shall
finde
376
,
which
set
to
31
l.
16
s.
and
lastly
,
your
last
number
of
pence
being
3
d.
look
in
your
scale
for
5
d.
and
against
it
you
shall
finde
208
,
which
set
to
6
l.
2
s.
and
then
will
your
whole
work
stand
,
as
here
you
see
.
Your
sums
being
thus
set
down
,
which
is
done
with
more
facilitie
then
you
can
imagine
,
till
you
make
trial
and
be
something
perfect
therein
,
you
must
then
adde
all
the
numbers
together
,
as
in
Addition
of
Decimals
,
and
you
shall
finde
the
sum
of
them
to
be
1034,000
,
Now
to
know
what
this
is
in
money
,
is
as
easie
as
it
was
to
set
the
several
sums
down
,
for
the
figures
103
,
which
stand
behinde
the
down
right
line
,
are
103
l.
and
the
figure
4
which
stands
between
the
down
right
line
and
the
comma
,
l.
|
|
|
36
|
4
,
|
333
|
29
|
0
,
|
083
|
31
|
8
,
|
376
|
6
|
1
,
|
208
|
103
|
4
,
|
000
|
are
4
decades
or
8
s.
and
being
the
rest
to
the
right
hand
are
all
Cyphers
,
they
signifie
neither
pence
nor
farthings
,
so
is
the
total
of
this
addition
103
l.
8
s.
0
d.
That
the
manner
of
working
may
appear
more
plain
.
I
will
give
you
another
short
Example
as
difficult
as
I
can
invent
,
which
I
performed
by
a
Scale
of
Wood
but
of
8
inches
long
.
Let
the
sums
to
be
added
together
,
be
these
following
.
li.
|
t.
|
d.
|
q
|
332
|
17
|
4
|
1
|
159
|
6
|
8
|
1
|
217
|
5
|
3
|
3
|
709
|
9
|
4
|
1
|
First
,
set
down
your
several
sums
of
pounds
one
under
another
as
before
,
and
draw
a
line
by
the
side
of
them
,
and
another
under
them
.
So
will
they
stand
as
here
you
see
.
1
Your
sums
of
pounds
being
thus
orderly
placed
and
lines
drawn
,
repair
to
your
Scale
and
seeing
your
first
number
of
shillings
,
pence
and
farthings
is
17
s.
4
d.
1
q.
for
your
17
s.
set
down
8
Decads
,
which
is
16
s.
with
a
comma
after
it
,
then
will
there
rest
to
be
set
down
1
s.
4
d.
1
q.
or
16
d.
1
q.
which
if
you
seek
in
your
scale
of
money
,
you
shall
find
to
stand
against
it
in
the
scale
of
1000
this
number
677
,
which
is
the
Decimal
of
1
s.
4
d.
1
q.
2
Your
second
number
of
shillings
pence
and
farthings
is
6
s.
8
d.
1
q.
for
your
6
s.
set
down
3
decads
,
which
is
6
s.
and
then
there
will
remain
8
d.
1
q.
which
if
you
seek
in
your
scale
of
money
,
you
shall
find
to
stand
against
it
in
the
scale
of
1000
this
number
344
,
which
is
the
decimal
of
8
d.
1
q.
3
Your
third
number
of
shillings
,
pence
,
and
farthings
,
is
5
s.
3
d.
3
q.
for
your
5
s.
set
down
2
decads
,
which
is
4
s.
with
a
comma
after
it
,
then
will
there
rest
to
be
set
down
1
s.
3
d.
3
q.
or
15
d.
3
q.
which
if
you
seek
in
your
scale
of
money
,
you
shall
finde
to
stand
against
it
in
the
scale
of
1000
,
l.
|
|
332
|
8,677
|
159
|
3,344
|
217
|
2,656
|
709
|
4,677
|
this
number
656
,
which
is
the
Decimal
of
15
d.
3
q.
or
1
s.
3
d.
3
q.
and
the
three
sums
to
be
added
together
will
stand
as
here
you
see
.
These
sums
being
added
together
,
according
to
the
rule
for
Addition
of
Decimals
,
you
shall
finde
the
sum
of
them
to
be
7094,677
,
now
to
know
what
this
is
in
money
,
take
notice
that
the
709
which
stands
to
the
left
hand
of
the
downright
line
are
709
pounds
,
and
the
figure
4
,
which
stands
between
the
downright
line
and
the
comma
,
are
4
decads
or
8
s.
but
(
because
the
first
figure
next
after
the
comma
is
above
5
,
viz.
6
)
you
must
add
1
s.
to
the
4
decads
,
making
them
9
s.
;
then
will
there
remain
177
,
wherefore
if
you
look
in
the
Scale
of
1000
for
177
,
you
shall
find
against
it
in
the
Scale
of
Money
4
d.
1
q.
So
is
the
whole
sum
of
this
Addition
709
l.
9
s.
4
d.
1
q.
as
by
the
preceding
work
doth
appear
.
¶
Here
note
,
that
when
you
had
set
down
your
709
l.
4
decads
or
8
s.
there
remained
beyond
the
comma
677
,
which
if
you
had
sought
in
your
Scale
of
1000
,
you
should
have
found
against
it
in
the
Scale
of
money
15
d.
1
q.
or
1
s.
4
d.
1
q.
(
which
is
all
one
)
as
before
,
for
it
appeareth
plainly
by
the
Scale
that
500
in
the
line
of
1000
is
equal
to
one
shilling
.
I
might
proceed
farther
in
giving
you
Examples
in
Weight
and
Measure
answerable
to
the
Scales
,
but
that
would
only
make
the
Reader
spend
his
time
to
little
purpose
,
for
being
before
acquainted
with
Decimal
Arithmetick
,
and
(
as
by
this
time
I
suppose
he
is
)
with
Numeration
upon
the
Scales
,
he
cannot
be
deficient
in
the
applying
of
the
other
Scales
of
Weight
and
Measure
to
the
same
purpose
for
which
they
were
contrived
,
I
having
so
largely
exemplified
the
use
of
the
Scale
of
money
.
Subtraction
.
SUbtraction
(
as
hath
been
before
said
)
is
the
taking
one
or
more
smaller
sums
out
of
one
greater
,
I
shall
onely
give
you
an
Example
or
two
,
as
I
have
taken
the
numbers
from
a
Scale
.
Example
.
Delivered
to
a
Gold-smith
of
old
plate
297
ounces
13
peny
weight
,
19
grains
.
Received
of
the
same
Gold-smith
first
165
ounces
,
11
peny
weight
and
7
grains
,
and
after
that
received
of
the
same
Gold-smith
32
onnces
19
penny
weight
,
and
23
grains
,
what
plate
remains
in
the
Gold-smiths
bands
?
Take
your
numbers
out
the
scale
of
Troy
weight
,
and
set
them
down
,
as
here
you
see
.
|
ounces
|
|
Delivered
|
165
|
5,646
|
Received
more
|
32
|
9,979
|
Received
in
all
|
198
|
5,625
|
Reses
in
the
Goldsmiths
hands
|
99
|
1,271
|
or
99
ounces
—
2
peny
w.
—
13
gr.
Then
adde
the
several
weights
of
plate
received
,
together
,
and
they
make
1985
,
625
,
or
198
ounces
,
11
peny
weight
,
6
grains
,
which
if
you
subtract
from
2976,896
,
or
257
ounces
,
13
peny
weight
19
grains
,
which
was
the
quantity
of
plate
delivered
,
there
will
remain
991271
,
or
99
ounces
2
peny
weight
,
13
grains
,
and
so
much
plate
is
still
in
the
Gold-smiths
hand
.
And
let
thus
much
suffice
for
subtraction
.
Now
we
should
proceed
to
Multiplication
and
Division
,
but
being
when
the
numbers
are
taken
from
the
scale
and
set
down
,
the
manner
of
working
doth
not
at
all
differ
from
Multiplication
and
Division
of
Decimals
before
taught
.
But
that
we
are
now
a
treating
of
Instrumental
Arithmetick
(
because
I
will
not
,
alter
the
method
before
used
,
in
the
first
and
second
parts
of
this
Book
;
)
I
will
therefore
here
shew
you
how
Multiplication
and
Division
,
may
be
Instrumentally
performed
by
Nepers
Bones
.
Multiplication
by
Nepers
Bones
.
MUltiplication
and
Division
are
accounted
the
hardest
parts
of
Vulgar
Arithmetick
;
but
by
the
invention
of
the
Right
Honorable
John
Lord
Nepeir
Baron
of
Merciston
,
they
are
both
of
them
made
very
easie
,
and
more
certain
then
in
the
ordinary
way
,
the
memory
not
being
charged
at
all
,
but
as
in
Addition
and
Subtraction
,
what
this
Artifice
is
,
is
shewed
by
himself
in
his
Treatise
of
Rabdologia
,
as
also
in
that
of
Mr.
Seth
Patridge
and
others
,
and
therefore
it
will
not
be
necessary
to
make
any
large
discourse
of
it
now
,
it
will
be
sufficient
here
,
onely
to
give
you
a
description
of
them
,
and
how
to
use
them
.
These
Rods
or
Bones
are
nothing
else
,
but
the
ordinary
Multiplication
Table
,
commonly
called
Pythagor
as
his
Table
,
cut
into
pieces
,
with
Diagonal
lines
between
the
figures
of
every
product
in
single
multiplication
;
as
by
the
figure
following
it
doth
appear
.
A
Figure
of
the
Bones
.
The
Bones
are
usually
made
of
small
pieces
of
Box
,
containing
in
length
about
1
inch
and
8/10
of
an
inch
,
and
in
breadth
about
2/10
of
an
inch
,
and
in
thicknesse
neer
1/10
of
an
inch
.
One
set
of
these
Bones
consisteth
of
five
pieces
of
Box
,
and
are
graduated
with
Diagonal
lines
and
figures
,
as
you
see
here
in
the
figure
of
them
.
On
one
of
the
five
pieces
there
is
0
and
9
,
the
0
being
on
one
side
of
the
Wood
,
and
the
9
on
the
other
side
,
another
hath
1
and
8.
the
third
2
and
7
,
the
fourth
3
and
6
,
the
fifth
4
and
5.
Now
these
five
pieces
of
Box
contein
one
Table
of
Multiplication
cut
in
pieces
,
but
so
few
are
not
sufficient
for
use
,
for
you
cannot
well
have
lesse
then
six
sets
of
them
,
that
is
30
pieces
,
and
so
many
are
usually
made
.
For
the
placing
of
these
Bones
,
when
you
use
them
,
there
belongeth
a
piece
of
Wood
about
four
inches
in
length
,
and
two
inches
in
breadth
,
having
2
ledges
of
the
same
thicknesse
with
the
Bones
,
on
the
thin
legde
which
is
at
the
end
of
this
board
,
there
is
figured
the
nine
Digits
,
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9.
This
board
is
called
a
Tabulat
,
because
we
usually
say
when
the
bones
are
laid
thereon
for
any
the
Bones
are
Tabulated
.
See
the
figure
thereof
.
Now
these
Bones
thus
ordered
and
Tabulated
doth
not
onely
give
you
the
product
of
every
single
Digit
,
multiplyed
by
it self
,
or
any
one
of
the
Digits
,
but
the
several
products
of
all
the
9
Digits
together
by
every
of
the
9
Digits
,
as
if
it
were
demanded
how
much
123456789
,
will
amount
unto
,
being
multiplyed
by
6
,
in
this
Table
it
will
be
found
to
be
740740734
,
for
the
product
of
9
multiplyed
by
6
is
54
,
the
first
figure
whereof
towards
the
right
hand
,
viz.
4
,
is
placed
under
the
Diagonal
line
,
and
5
being
in
the
place
of
tens
is
placed
above
it
,
again
8
multiplyed
by
6
is
48
,
the
first
figure
whereof
towards
the
right
hand
,
viz.
8
is
placed
under
the
Diagonal
line
as
in
the
other
product
,
and
these
2
figures
5
in
the
upper
part
of
the
former
product
,
and
8
in
the
lower
part
of
this
must
be
added
together
,
and
being
so
they
make
13
,
therefore
next
unto
4
in
the
first
product
,
I
set
down
3
,
and
carry
1
to
4
the
figure
placed
above
the
Diagonal
in
this
second
product
,
which
together
with
the
lower
figure
in
the
next
product
,
viz.
2
,
do
make
7
,
for
4
and
1
that
I
carried
is
5
and
2
the
figure
under
the
Diagonal
in
the
next
product
are
7
,
and
so
the
3
first
figures
of
this
product
towards
the
right
hand
are
734
,
in
like
manner
must
the
figures
remaining
between
the
Diagonals
be
added
together
,
and
their
sum
if
it
amount
not
to
ten
must
be
set
down
in
his
proper
place
,
but
if
they
make
or
exceed
ten
,
set
down
the
overplus
,
and
carry
the
ten
in
mind
to
the
next
figures
between
the
Diagonals
as
in
the
last
product
;
So
likewise
in
that
which
followes
4
and
6
the
figures
between
the
Diagonals
being
10
,
I
set
down
a
Cypher
,
and
carry
one
to
the
next
figures
,
which
are
3
and
0
,
now
4
and
1
that
I
carried
is
4
,
and
4
and
0
is
4
still
,
and
therefore
next
to
the
Cypher
I
set
down
4
in
the
paper
;
the
figures
in
the
next
Diagonal
are
3
and
3.
that
is
7
,
wherefore
I
set
down
7
in
the
paper
,
the
next
are
2
and
8
that
is
10
,
therefore
I
set
down
0
in
the
paper
and
carry
1
,
the
next
are
1
and
2
which
with
1
that
I
carried
make
4
,
therefore
I
set
down
4
in
the
paper
,
the
last
are
1
and
6
which
make
7
and
so
at
last
the
whole
product
of
123456789
multiplyed
by
6
is
740740734.
Now
as
the
multiplication
Table
thus
ordered
doth
give
the
product
of
all
the
9
Digits
placed
orderly
by
every
of
the
9
digits
,
so
being
cut
in
pieces
or
slips
,
and
having
many
of
them
made
in
Past-board
of
Wood
,
any
other
sum
may
be
placed
on
the
Tabulat
,
and
the
several
products
thereof
by
every
of
the
9
digits
will
be
found
in
the
like
manner
;
thus
if
it
were
required
to
multiply
this
number
5767
by
every
of
the
9
digits
,
placing
the
slips
with
these
figureson
the
top
orderly
as
,
they
are
in
the
following
Figure
,
product
of
5767
by
3
will
be
found
to
be
17301
,
the
product
thereof
by
8
is
46136
,
and
so
for
any
other
.
Also
if
it
were
required
to
multiply
5767
by
749
,
the
product
of
5767
by
9
is
51903
,
and
the
product
thereof
by
4
is
23068
,
and
the
product
by
7
is
40369
,
and
these
products
being
set
down
and
added
together
,
as
in
ordinary
Multiplication
,
the
product
of
5767
multiplyed
by
749
will
be
4319483
,
as
by
the
operation
in
the
margine
it
doth
appear
.
Of
Division
by
the
Rods.
LEt
it
be
required
to
divide
4319483
by
5767
placing
this
Divisor
on
the
Tabulat
,
as
was
directed
for
Multiplication
,
the
products
of
your
Divisor
by
every
of
the
9
Digits
is
given
you
,
and
may
be
deducted
from
the
Dividend
,
as
hath
been
already
shewed
Pag.
55
,
and
according
to
the
Rules
there
delivered
,
I
ask
how
often
is
5767
in
43194
,
and
the
answer
by
my
Table
is
7
times
,
and
therefore
I
subtract
40369
the
product
of
5767
by
7
from
43194
,
and
there
rests
2825
,
which
with
the
next
figure
in
the
Dividend
,
viz.
8
,
is
28258
,
in
which
I
find
5767
to
be
conteined
4
times
,
and
therefore
I
subtract
23068
the
product
of
5767
by
4
from
28258
,
and
there
rests
5190
,
which
with
the
next
figure
in
the
Dividend
,
viz.
3
,
is
51903
,
in
which
I
find
5767
to
be
conteined
nine
times
,
and
therefore
I
subtract
51903
the
product
of
5767
by
9
,
from
51903
the
remaining
figures
of
the
Dividend
,
and
there
resteth
nothing
,
and
therefore
the
quotient
is
749
,
as
by
the
operation
it
doth
appear
.
And
thus
may
you
doe
with
any
other
sum
(
having
a
competent
number
of
Bones
or
Rods
ready
for
the
work
)
without
any
charge
to
the
memory
,
which
in
other
wayes
cannot
be
avoided
,
and
by
reason
of
which
many
mistakes
may
happen
,
which
are
in
this
prevented
.
Also
by
the
Bones
there
is
this
commodity
,
that
when
it
is
doubtfull
what
figure
to
set
in
the
Quotient
,
the
Bones
will
certainly
direct
you
,
so
that
you
cannot
possibly
mistake
.
These
Rods
,
or
Bones
(
as
they
are
commonly
called
)
are
of
excellent
use
in
extracting
the
Square
and
Cube
Roots
,
but
that
being
already
taught
in
the
first
Part
,
I
shall
not
here
insert
it
again
,
for
he
that
can
extract
the
Square
and
Cube
Roots
any
way
,
cannot
be
ignorant
to
effect
the
same
by
the
Bones
,
if
he
can
multiply
and
divide
by
them
.
Having
done
with
the
four
Species
of
Arithmetick
in
this
instrumental
way
,
I
should
now
proceed
to
the
Golden
Rule
of
three
,
and
consequently
to
the
other
Rules
of
Arithmetick
,
but
having
in
the
former
parts
of
this
Book
performed
that
already
,
both
in
whole
numbers
,
Fractions
,
and
Decimals
,
I
shall
onely
give
you
an
Example
in
the
most
necessary
Rules
,
and
so
conclude
this
third
part
.
Example
in
the
Rule
of
Three
Direct
.
IF
37
Ells
and
a
half
of
linnen
cloth
,
cost
24
l.
7
s.
9
d.
what
shall
283
Ells
and
an
half
cost
?
First
,
set
down
your
37
Ells
,
then
if
you
look
in
your
Scale
for
your
half
Ell
,
you
shall
find
it
to
stand
against
500
in
the
Scale
of
1000
,
which
500
may
be
called
5
onely
,
for
the
two
Cyphers
may
be
omitted
.
Then
set
down
your
24
l.
and
for
your
7
set
down
3
decads
,
which
is
6
s.
and
look
in
your
Scale
for
the
Decimal
of
1
s.
9
d.
which
you
shall
find
to
be
875
;
Lastly
,
set
down
your
183
Ells
,
and
for
your
half
Ell
set
down
5
as
before
,
so
will
your
numbers
stand
thus
,
Your
numbers
thus
taken
out
of
your
Scale
,
and
placed
as
here
you
see
,
if
you
multiply
the
second
and
third
together
,
you
shall
finde
the
product
of
that
multiplication
to
be
4475.10625
,
which
divided
by
the
first
number
37.5
,
giveth
in
the
quotient
119.3361
,
which
is
119
pounds
,
3
decads
,
or
6
shillings
,
and
361
,
which
reduced
by
the
Scale
giveth
8
pence
2
farthings
,
and
something
more
.
Example
in
the
Rule
of
Three
Reverse
.
IF
when
the
price
of
a
Quarter
of
Wheat
is
1
li.
5
s.
6
d.
the
penny
white
loaf
shall
weigh
12
ounces
16
penny
weight
,
I
demand
what
the
penny
white
loaf
shall
weigh
,
when
the
price
of
the
Quarter
of
Wheat
is
7
s.
6
d
?
If
you
place
your
numbers
according
to
the
tenor
of
the
Question
,
they
will
stand
as
followeth
.
1
li.
—
5
s.
—
6
d.
—
12
où
.
—
16
pw
.
—
7
s.
—
6
d.
But
being
taken
out
of
the
Scales
of
Money
an
Troy
Weight
,
they
will
stand
thus
,
Here
if
you
multiply
1.275
which
is
the
Decimal
of
1
l.
5
s.
6
d.
by
12.8
.
which
is
the
Decimal
of
12
ounces
16
peny
weight
,
you
shall
find
the
product
of
that
multiplication
to
be
16.3200
,
which
being
divided
by
.375
,
which
is
the
Decimal
of
7
s.
6
d.
the
quotient
will
be
43.5
,
which
is
the
Decimal
of
43
ounces
,
10
peny
weight
3
grains
,
and
so
much
ought
the
peny
white
loafe
to
weigh
,
when
the
quarter
of
Wheat
is
sold
for
7
s.
6
d.
Example
in
the
Double
Rule
of
Three
.
IF
24
yards
of
stuffe
of
three
quarters
broad
cost
4
l.
14
s.
what
shall
328
yards
of
the
same
stuffe
cost
being
5
quarters
broad
.
If
you
place
your
numbers
according
to
the
directions
of
this
Rule
,
they
will
stand
thus
,
If
24
yards
of
3
quarters
cost
4
li.
14
s.
what
shall
328
yards
cost
of
5
q.
But
if
you
take
your
fraction
numbers
out
of
their
proper
Scales
,
they
will
stand
thus
,
First
,
multiply
the
two
first
numbers
,
as
24
and
3
together
,
they
make
72
for
Divisor
,
then
multiply
4.7
,
which
is
the
Decimal
of
4
l.
14
s.
by
328
,
and
the
Product
is
15416
,
which
again
multiplied
by
5
the
last
number
,
giveth
77080
,
unto
this
Product
,
(
that
there
may
be
a
competent
number
of
figures
in
the
quotient
,
)
I
adde
two
Cyphers
,
making
it
7708000
,
which
I
divide
by
72
,
and
the
quotient
is
107.055
,
which
is
107
l.
1
s.
1
d.
and
so
much
is
328
yards
of
stuffe
worth
,
being
5
quarters
broad
.
Example
in
Barter
.
TWo
Merchants
having
two
several
commodities
,
are
willing
to
barter
,
or
exchange
the
one
with
the
other
.
The
one
hath
Indigo
,
which
he
will
sell
at
4
s.
the
pound
,
for
ready
money
,
but
in
Barter
he
will
have
4
s.
9
d.
the
pound
;
the
other
Merchant
hath
Kersies
,
which
for
ready
money
he
will
sell
for
3
s.
6
d.
the
yard
.
Now
the
question
is
,
at
what
price
he
must
rate
his
Kersies
in
Barter
,
to
equalize
the
9
d.
advance
upon
the
pound
of
Indigo
.
The
tenor
of
the
Question
is
this
.
If
4
s.
in
Barter
require
9
d.
what
shall
3
s.
6
d.
require
?
Your
numbers
placed
will
stand
thus
,
4
s.
—
9
d.
3
s.
6
d.
But
being
taken
out
of
your
Scale
they
will
stand
thus
,
Say
then
by
the
Rule
of
three
Direct
,
If
2
decads
or
4
s.
in
Barter
require
.
357
,
which
is
the
Decimal
of
9
d.
what
shall
1.57
require
?
which
is
the
Decimal
of
3
s.
6
d.
First
,
multiply
.375
by
1.57
,
the
product
is
.65625
,
(
but
being
it
is
a
fraction
I
cut
off
the
two
last
figures
because
we
require
onely
three
figures
in
the
quotient
,
which
divided
by
2
giveth
in
the
quotient
.328
,
which
is
the
Decimal
of
7
d.
3
q.
this
7
d.
3
q.
added
to
his
3
s.
6
d.
maketh
4
shillings
1
peny
3
farthings
,
and
so
much
ought
he
to
rate
his
Kersies
at
by
the
yard
in
Barter
to
save
himself
harmlesse
.
Example
in
Fellowship
.
THree
Persons
A
,
B
,
and
C
bought
4000
Sheep
,
which
cost
48●
li.
6
sh.
8
d.
of
which
money
A
paid
203
li.
B
paid
165
li.
15
sh.
8
d.
and
C
paid
114
l.
11
s.
First
,
say
by
the
Rule
of
three
Direct
.
1
If
483
l.
6
s.
8
d.
buy
4000
sheep
,
how
many
sheep
shall
203
l.
(
which
is
A
share
)
buy
?
Answer
1680.
2
Say
,
if
483
l.
6
s.
8
d.
buy
4000
sheep
,
how
many
sheep
shall
165
l.
15
s.
8
d.
(
which
is
B
share
)
buy
?
Answer
1372.
3
Say
again
,
if
483
l.
6
s.
8
d.
buy
4000
sheep
how
many
sheep
shall
114
li.
11
s.
(
which
is
C
share
)
buy
?
Answer
948.
Your
numbers
being
taken
out
of
your
Scale
,
proceed
as
followeth
.
First
for
A.
Secondly
for
B.
Thirdly
for
C.
The
manner
of
Work
.
1
For
A
multiply
203
l.
(
which
is
A
share
)
by
4000
,
(
which
is
the
number
of
sheep
bought
)
and
the
product
is
812000
,
which
number
should
be
divided
by
483.3333
,
but
being
it
is
greater
then
812000
,
I
therefore
add
four
Cyphers
thereto
,
that
I
may
have
four
figures
in
the
quotient
,
and
it
makes
8120000000
,
which
divided
by
483.3333
giveth
in
the
quotient
1680
,
and
so
many
sheep
belong
to
A.
2
For
B
,
multiply
165.7833
(
which
is
the
Decimal
of
B
share
)
by
4000
,
(
the
number
of
sheep
bought
)
and
it
produceth
6631332000
,
which
divided
by
483.3333
,
giveth
in
the
quotient
1372
,
and
so
many
sheep
belong
to
B.
3
For
C
,
multiply
114.55
,
(
which
is
the
Deeimal
of
C
share
,
)
by
4000
,
(
the
number
of
sheep
bought
,
)
it
produceth
45820000
,
which
number
should
be
divided
by
483.3333
,
but
being
it
is
not
large
enough
to
give
figures
enough
in
the
quotient
,
I
therefore
adde
two
Cyphers
making
it
4582000000
,
which
divided
by
483.3333
,
giveth
in
the
quotient
948
,
and
so
many
sheep
ought
Cto
have
.
Now
for
proof
,
if
you
add
the
number
of
sheep
that
A
,
B
and
C
should
severally
have
,
you
shall
find
them
in
all
to
make
4000
which
demonstrates
the
work
to
be
true
.
Examples
in
Losse
and
Grain
.
IF
one
yard
of
Stuffe
cost
6
s.
8
d.
and
I
sell
the
same
again
for
8
s.
6
d.
what
shall
I
gain
in
laying
out
100
l.
upon
such
commodity
?
Take
the
difference
between
the
price
that
your
commodity
cost
,
and
the
price
for
which
you
sell
it
,
that
is
,
in
this
Example
,
the
difference
between
6
s.
8
d.
and
8
s.
6
d.
which
is
1
s.
10
d
,
then
say
by
the
Rule
of
Three
Direct
.
If
6
s.
8
d.
gain
1
s.
10
d.
what
will
100
li.
gain
?
If
you
place
your
numbers
according
to
the
Rule
of
Three
Direct
,
as
they
are
here
given
,
they
will
stand
thus
,
6
s.
8
d.
—
1
s.
10
d.
100
li.
But
being
taken
out
of
your
Scale
and
placed
,
they
will
stand
as
followeth
.
Your
numbers
being
placed
,
multiply
425
which
is
the
Decimal
of
1
s.
10
d.
by
100
li.
and
the
Product
is
42500
,
to
which
I
adde
two
Cyphers
(
that
I
may
have
a
competent
number
of
figures
in
the
quotient
)
and
it
makes
42500●0
,
which
divided
by
.3333
the
decimal
of
6
s.
8
d.
giveth
in
quotient
127.5
,
which
is
127
li.
5
livers
or
10
s.
so
there
is
27
li.
10
s.
gained
in
laying
out
of
100
l.
I
will
here
prove
this
question
by
the
converse
.
If
one
yard
of
Stuffe
which
is
sold
for
8
s.
6
d.
there
was
gained
27
li.
10
s.
in
laying
out
of
100
li.
I
demand
what
the
said
stuffe
cost
a
yard
at
the
first
hand
?
Adde
100
li.
and
27
li.
10
s.
together
,
and
they
make
127
li.
10
s.
Then
say
by
the
Rule
of
Three
Direct
,
If
1
27
li.
10
s.
give
100
l.
what
shall
8
s.
6
d.
give
?
Take
your
numbers
out
of
your
Scale
and
place
them
as
here
you
see
.
Here
if
you
multiply
425
,
which
is
the
Decimal
of
8
s.
6
d.
by
100
,
you
shall
have
42500
,
to
which
if
you
adde
a
Cypher
,
you
make
it
425000
,
this
number
being
divided
by
127.5
,
which
is
the
Decimal
of
127
l.
10
s
,
giveth
in
the
quotient
3333
,
and
if
you
had
added
more
Cyphers
to
the
Dividend
,
you
should
have
had
more
threes
in
the
quotient
,
and
no
other
figures
,
but
these
four
threes
are
enough
,
and
are
a
Decimal
fraction
representing
6
s.
8
d.
and
so
much
did
the
yard
of
Stuffe
cost
at
the
first
hand
.
Examples
in
Losse
and
Gain
upon
Time
wrought
by
the
Double
Rule
of
Three
.
IF
one
Ell
of
Lockeram
,
cost
me
2
s.
8
d.
ready
money
,
and
I
sell
the
same
again
for
2
s.
10
d.
the
Ell
to
be
paid
at
the
expiration
of
three
moneths
,
I
demand
what
I
shall
gain
in
12
moneths
,
laying
out
100
l.
upon
that
commodity
.
This
and
such
like
questions
,
although
they
may
be
wrought
by
the
Rule
of
Three
Direct
,
at
two
operations
,
yet
they
are
best
performed
by
the
Double
Rule
of
Three
compounded
of
five
numbers
,
wherefore
the
question
may
be
thus
stated
.
If
2
s.
8
d.
in
three
moneths
,
gain
2
d.
what
shall
100
l.
gain
in
11
moneths
?
If
you
take
your
numbers
out
of
your
scale
▪
and
place
them
according
as
was
directed
in
the
first
part
of
this
Book
,
you
shall
finde
them
to
stand
thus
,
Your
numbers
being
placed
according
to
the
tenor
of
the
question
,
if
you
multiply
1.333
,
which
is
the
Decimal
of
2
s.
8
d.
by
3
moneths
,
the
product
will
be
3999
,
which
must
be
your
Divisor
,
then
multiply
.83
which
is
the
Decimal
of
2
d.
by
100
l.
and
it
makes
8300
,
that
again
multiplied
by
12
moneths
,
giveth
for
the
product
99600
for
your
Dividend
,
wherefore
if
you
divide
99600
by
3999
,
it
will
give
you
in
the
quotient
25
almost
,
which
is
25
li.
for
the
Decimal
fraction
remaining
is
so
small
,
that
it
wanteth
not
neer
a
farthing
of
25
li.
and
therefore
we
call
it
25
li
,
and
so
it
is
exactly
,
as
you
may
try
,
if
you
reduce
all
the
numbers
to
their
least
denominations
,
and
work
as
is
before
taught
in
Vulgar
Arithmetick
.
I
will
prove
this
question
by
the
converse
.
If
one
Ell
of
Lockerum
cost
me
2
s.
8
d.
ready
money
,
for
what
price
shall
I
sell
the
same
again
to
be
paid
at
the
end
of
three
moneths
.
So
that
I
may
gain
25
li.
in
100
li.
for
12
moneths
?
Say
by
the
Rule
of
Three
Direct
.
If
100
li.
in
12
moneths
gain
25
li.
what
shall
2
s.
8
d.
gain
in
3
moneths
?
If
you
take
your
numbers
out
of
your
Scale
,
and
place
them
according
to
the
Double
Rule
of
Three
,
they
will
stand
as
followeth
.
Your
numbers
being
thus
placed
,
if
you
multiply
100
li.
by
12
moneths
,
you
shall
finde
the
product
to
be
1200
,
which
is
your
Divisor
.
Then
multiply
25
li.
by
1.333
,
which
is
the
Decimal
of
2
s.
8
d.
and
the
product
thereof
will
be
33325
,
which
multiply
again
by
3
,
and
the
product
will
be
99975
for
your
Dividend
,
this
99975
divided
by
1200
,
giveth
in
the
quotient
.
83
,
which
is
the
Decimal
of
2
d
,
which
2
d.
added
to
2
s.
8
d.
the
price
which
the
Ell
of
Lockeram
cost
,
giveth
2
s.
10
d.
and
at
that
price
must
you
sell
the
same
at
3
moneths
time
so
that
you
may
gain
25
l.
in
the
100
li.
in
12
moneth●
.
I
might
further
proceed
to
shew
you
Examples
in
divers
other
Rules
;
As
in
Alligation
,
Position
,
&c.
but
those
Rules
being
already
handled
in
the
First
Part
of
this
Book
,
it
will
be
easie
to
apply
them
to
the
scales
,
And
(
as
I
intimated
at
the
beginning
of
this
Third
Part
,
)
that
although
I
have
onely
made
choice
of
the
eight
scales
there
mentioned
and
described
,
yet
it
was
easie
to
contrive
scales
for
the
Coins
,
Weights
,
and
Measures
of
other
Countries
,
and
not
onely
so
,
but
when
the
value
of
the
Money
,
Weight
,
or
Measure
of
one
Countrey
,
and
the
Money
,
Weight
,
or
Measure
of
another
Countrey
is
known
,
it
is
easie
to
contrive
two
scales
,
which
facing
one
another
,
shall
immediately
tell
you
how
many
pounds
in
one
place
shall
be
equal
to
so
many
Crowns
,
or
other
Coin
in
another
place
,
but
this
I
do
onely
intimate
,
that
such
as
are
desirous
,
may
fit
themselves
with
scales
answerable
to
their
most
necessary
occasions
.
And
thus
shall
I
conclude
this
third
part
,
referring
some
things
necessary
to
such
a
work
,
to
the
Appendix
following
.