Parag.
1.
ARITHMETICK
is
the
Art
,
or
Doctrine
of
Numbring
;
deriv'd
from
the
Greeke
,
and
signifies
Number
;
the
subject
of
Arithmetick
,
as
Magnitude
is
of
Geometry
.
Of
this
Art
there
are
two
kindes
,
viz:
Naturall
and
Artificiall
:
the
first
of
these
is
so
denominated
from
having
proper
Figures
,
and
Numbers
significant
of
themselves
;
whereas
the
other
is
Figurative
,
or
founded
upon
art
:
of
this
Arithmetick
there
be
severall
kindes
,
of
which
(
God
willing
)
I
will
treat
of
in
my
third
Book
,
and
now
proceed
.
Number
is
defin'd
to
be
a
discrete
quantity
,
being
divisible
in
its
parts
,
consisting
of
Vnites
,
by
which
every
thing
is
numbred
;
and
in
this
hath
proper
and
peculiar
characters
.
Annotation
.
This
describes
the
Characters
,
as
the
Elements
to
this
Art
;
by
which
Number
is
exprest
,
and
are
in
all
but
ten
,
which
in
order
are
these
,
viz:
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
0
|
one
|
two
|
three
|
four
|
five
|
six
|
seven
|
eight
|
nine
|
a
cypher
|
Nine
of
these
are
called
significant
Figures
of
themselves
;
the
little
circle
or
Cypher
represented
by
the
letter
O
,
signifies
nothing
of
it self
,
yet
encreases
the
value
of
number
according
to
its
place
or
position
.
The
first
of
these
figures
is
called
1
A.
a
Vnite
,
which
simply
of
it selfe
is
no
number
,
but
the
beginning
of
all
number
or
quantity
;
as
a
point
in
Geometrie
is
the
first
of
magnitude
represented
by
the
letter
A.
All
the
other
eight
significant
figures
A
—
B
are
composed
of
Vnites
,
as
a
Line
in
Geometrie
is
the
continuation
of
Points
,
represented
by
A.
B
so
that
2
is
composed
of
two
Unites
;
3
made
of
three
Unites
,
and
4
contains
four
Unites
,
and
so
of
all
the
rest
.
NVMERATION
.
THis
shews
the
value
or
quantity
of
any
number
given
;
or
how
to
expresse
in
proper
figures
any
number
assign'd
or
propounded
,
and
to
declare
the
quantity
of
each
figure
,
according
to
its
place
or
position
,
their
contents
increasing
by
degrees
ascending
from
the
right
hand
towards
the
left
,
as
doe
the
letters
and
characters
of
the
Hebrews
,
and
all
the
Eastern
people
,
from
whence
this
Art
of
Number
extracts
its
originall
.
In
this
amplification
of
Numbers
you
must
observe
the
figure
on
the
right
hand
denotes
simply
it self
,
as
2
two
;
4
four
;
6
six
;
9
nine
;
0
nothing
&c.
the
next
degree
,
or
second
figure
towards
the
left
hand
is
ten
times
its
own
quantity
;
the
third
place
a
hundred
times
;
the
fourth
degree
a
thousand
times
;
the
fift
place
ten
thousand
times
;
the
sixt
figure
in
order
is
a
hundred
thousand
times
its
own
number
;
the
seventh
place
a
million
;
the
eighth
ten
millions
;
the
ninth
place
a
hundred
millions
;
and
so
proceeding
without
end
,
encreasing
every
degree
in
this
manner
following
,
reiterating
Unites
,
Tens
,
and
Hundreds
.
A
Table
of
Numeration
,
continued
.
Degrees
.
|
1
|
1
|
One
|
1
|
1
|
Degrees
.
|
2
|
10
|
Tennes
|
21
|
2
|
3
|
100
|
Hundreds
|
321
|
3
|
4
|
1000
|
Thousands
|
4321
|
4
|
5
|
10000
|
Ten
thousands
|
54321
|
5
|
6
|
100000
|
Hundred
thousands
|
654321
|
6
|
7
|
1000000
|
Million
|
7654321
|
7
|
8
|
10000000
|
Ten
millions
|
87654321
|
8
|
9
|
100000000
|
Hundred
millions
|
987654321
|
9
|
By
this
Table
of
Numeration
you
may
observe
how
the
figures
by
degrees
doe
encrease
by
tennes
,
proceeding
from
the
right
hand
towards
the
left
,
in
value
according
to
their
places
;
as
for
example
:
the
sinister
Table
begins
with
a
Unite
,
unto
which
annexing
a
Cypher
,
and
then
the
second
degree
will
be
10
,
that
is
in
value
tenne
times
the
first
:
unto
that
annex
another
Cypher
,
and
then
in
the
third
place
it
is
100
a
hundred
,
that
is
tenne
times
greater
than
the
second
degree
,
the
fourth
place
1000
a
thousand
,
that
is
tenne
times
100
,
&c.
The
second
Table
I
also
here
begin
with
a
Unite
,
before
which
towards
the
left
hand
1
place
2
,
and
then
in
the
second
degree
it
makes
twenty
,
so
both
the
figures
21
,
and
2
in
that
place
10
times
its
own
number
:
then
put
3
before
them
,
and
it
will
be
the
hundreth
place
;
and
the
character
being
3
,
denotes
three
hundred
;
the
three
figures
making
in
all
321
three
hundred
twenty
and
one
:
the
fourth
degree
obtains
the
thousand
place
,
being
tenne
times
greater
than
the
third
degree
;
and
having
the
first
figure
before
it
4
,
the
value
of
the
four
figures
will
be
4321
,
that
is
,
four
thousand
three
hundred
twenty
and
one
:
the
next
degree
is
tenne
thousand
,
and
the
character
being
5
,
it
must
be
fifty
thousand
,
and
the
whole
value
of
those
five
figures
54321
,
and
so
all
others
proceeding
in
order
by
tennes
.
Yet
the
better
for
to
effect
the
computation
of
great
numbers
,
when
their
degrees
cannot
easily
be
contain'd
in
memory
;
then
it
is
necessary
for
to
put
periods
unto
every
centurie
,
or
three
figures
,
and
so
numbring
them
in
unites
,
tennes
,
hundreds
;
and
then
in
expressing
of
their
values
,
observe
their
places
,
and
the
quantity
of
each
figure
as
it
is
simply
of
it self
,
and
note
that
after
the
first
period
,
their
places
are
Thousands
,
and
after
the
second
period
Millions
;
and
after
the
next
period
Millions
of
millions
:
and
so
proceeding
with
reiteration
of
unites
,
tennes
,
and
hundreds
,
untill
their
periods
are
put
unto
them
;
by
this
means
avoiding
errours
,
arising
in
numbring
of
great
quantities
,
and
mistaking
their
degrees
,
as
in
this
example
shall
be
illustrated
,
and
suppose
this
number
given
,
whose
quantity
is
required
,
viz:
1
,
234
,
567
,
890
which
having
pointed
,
read
thus
:
one
Million
of
millions
two
hundred
thirty
four
millions
,
five
hundred
sixty
seven
thousand
,
eight
hundred
and
ninety
.
If
this
had
been
the
number
given
1
0
,
0
0
0
,
0
0
0
,
0
0
0
it
would
have
been
thus
exprest
,
ten
Millions
of
millions
:
or
thus
,
ten
th●usand
Millions
.
Observe
this
order
well
,
and
then
the
computation
of
any
Numbers
will
prove
as
easie
as
they
be
necessary
for
to
be
known
or
exprest
,
according
to
their
places
in
degrees
.
Number
is
also
for
to
be
consider●d
,
whether
it
be
of
one
,
or
severall
den●minations
;
those
are
said
of
one
denomination
,
when
they
doe
consist
of
one
number
,
weight
,
or
measure
,
whether
they
be
all
of
one
intire
number
,
or
in
the
same
parts
.
All
numbers
of
severall
denominations
are
said
to
be
mixt
,
when
they
doe
consist
both
of
Integers
and
Fractions
collected
in
a
summe
,
viz:
Pounds
sterling
,
Marks
,
Nobles
,
Shillings
,
Pence
;
or
in
respect
of
weight
in
Tunnes
,
Hundreds
,
Stones
,
Pounds
,
Ounces
,
Drams
,
and
Grains
;
or
in
long
measures
,
viz:
Leagues
,
Miles
,
Furlongs
,
Perches
,
Yards
,
Feet
,
Inches
;
or
any
other
parts
or
fractions
.
Every
number
is
subdivided
into
three
parts
,
and
is
either
Simple
,
Decimall
,
or
Compound
,
and
are
thus
distinguished
.
1.
Simple
,
are
onely
those
,
who
doe
consist
of
themselves
,
as
doe
the
9
significant
figures
,
viz:
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9.
2.
Decimalls
or
Tennes
,
are
those
who
doe
consist
of
any
significant
figures
,
with
cyphers
annexed
unto
them
,
as
10
,
20
,
30
,
40
,
50
,
60
,
70
,
80
,
90
,
100
,
&c.
3.
Compound
numbers
are
all
those
which
are
made
by
Simple
and
Decimall
numbers
commixt
together
,
viz:
12
,
23
,
34
,
45
,
56
,
67
,
78
,
89
,
91
,
and
so
are
all
others
which
are
composed
of
significant
figures
.
From
Numeration
there
proceeds
four
species
,
viz:
Addition
,
Substraction
,
Multiplication
,
and
Division
,
which
in
order
are
treated
of
in
the
follow
-
Paragraphs
or
Chapters
.
ADDITION
.
Parag:
2.
The
first
species
.
A
Species
is
here
defin'd
to
be
a
certain
form
of
working
by
numbers
,
in
which
four
this
Art
depends
,
the
Rules
are
fram'd
,
and
all
questions
in
Arithmetick
perform'd
.
Addition
teacheth
the
contracting
of
many
severall
quantities
or
numbers
into
one
totall
summe
.
To
effect
this
;
place
the
greater
number
first
,
and
right
under
that
the
next
,
and
so
in
order
place
the
severall
summes
,
this
is
convenient
in
the
respect
of
form
,
but
of
no
necessity
;
but
this
is
,
that
you
alwayes
observe
in
Addition
for
to
put
unites
under
unites
,
tennes
under
tennes
,
hundreds
under
hundreds
,
&c.
that
is
,
every
figure
in
order
underneath
one
another
according
to
their
degrees
or
places
,
from
the
right
hand
;
this
done
,
underneath
them
all
you
must
draw
a
line
,
as
by
the
following
Paradigma
.
Example
1.
Let
the
three
numbers
given
be
A
320
B
241
&
C
231
,
of
any
one
denomination
,
but
here
suppose
them
grains
of
Barley
,
the
least
denomination
of
English
measure
,
four
of
them
making
an
Inch
,
and
all
these
a
statute
Perch
,
whose
summe
is
required
;
and
having
plac'd
them
under
one
another
,
as
by
the
letters
A
,
B
,
C
:
begin
with
the
figures
first
on
the
right
hand
,
and
say
1
and
1
makes
2
,
which
write
down
beneath
the
line
under
unites
,
the
cypher
of
it selfe
being
nothing
:
then
goe
to
the
next
place
,
and
say
,
3
&
4
makes
7
,
and
2
is
9
;
which
place
in
the
second
rank
from
the
right
hand
:
and
proceed
to
the
next
place
,
where
say
2
&
2
makes
4
and
3
will
be
7
,
which
place
likewise
beneath
the
line
under
hundreds
:
and
so
continue
if
there
be
any
more
numbers
,
the
totall
summe
of
these
is
792
,
as
in
the
example
:
and
so
many
grains
of
Barly
or
quarters
of
inches
there
be
in
16
feet
and
a
half
.
An
illustration
Arithmeticall
.
Number
consists
in
the
addition
of
unites
,
as
a
Line
does
in
Geometrie
by
the
continuation
of
points
,
and
is
call'd
the
first
quantitie
,
and
so
is
number
discreet
,
whose
summe
is
nothing
else
,
but
the
putting
of
severall
numbers
together
,
as
is
the
addition
of
Lines
:
as
in
the
last
example
,
admit
three
lines
given
for
to
be
added
together
,
each
line
consisting
of
so
many
equall
parts
respectively
,
as
were
the
numbers
propounded
,
viz:
the
line
A
divided
into
320
parts
or
quarters
of
inches
,
each
equal
to
a
grain
of
Barly
:
B
containing
241
,
&
C
231
of
those
equal
parts
,
all
which
added
together
will
make
the
line
A
B
C
D
,
the
true
summe
of
the
three
lines
given
,
being
792
,
as
in
the
last
Scheme
appears
,
the
quarters
of
inches
contained
in
a
Statute
Pole
:
in
the
same
manner
if
12340
were
to
be
added
unto
87659
,
the
totall
summe
would
be
99999.
But
it
will
most
commonly
happen
,
when
great
numbers
are
to
be
added
,
that
the
summe
in
most
ranks
will
exceed
the
quantity
of
any
one
significant
figure
;
if
it
be
a
Decimal
,
place
a
Cypher
under
that
ranke
,
if
a
compound
number
write
the
simple
significant
figure
,
and
adde
in
the
decimall
to
the
next
place
as
an
unite
;
if
two
tens
as
2
,
three
tens
as
3
&c.
according
to
the
order
and
degrees
of
Numeration
,
for
every
figure
is
no
more
than
its
own
quantity
,
but
in
respect
of
the
degree
and
place
,
as
by
the
following
examples
will
be
made
evident
.
Example
2.
There
was
an
old
man
,
whose
|
44
|
Yeares
|
12
|
|
44
|
The
totall
|
100
|
age
was
required
,
to
which
he
replied
:
I
have
7
sonnes
,
and
there
was
2
years
between
every
one
of
them
;
in
my
44
yeare
,
my
eldest
sonne
was
born
,
which
is
just
now
the
age
of
my
youngest
.
First
set
down
the
Fathers
age
;
after
that
he
had
6
sonnes
which
made
12
yeares
more
,
write
that
down
,
and
the
age
of
his
youngest
,
which
was
equall
to
his
Fathers
age
at
the
birth
of
his
first
sonne
,
all
which
together
makes
the
old
mans
age
:
so
begin
with
the
first
ranke
,
and
say
4
&
2
makes
6
and
4
more
is
10
,
and
being
there
are
more
numbers
to
come
,
set
down
a
cypher
in
the
unite
place
,
the
decimal
being
but
1
;
in
the
next
adde
it
to
4
which
makes
5
and
1
will
be
6
and
4
makes
10
;
under
the
second
rank
place
another
cypher
,
and
being
there
are
no
more
figures
,
place
the
decimall
,
or
unite
in
the
third
place
:
the
totall
summe
is
100
,
the
old
mans
age
required
,
as
in
the
Paradigma
appears
.
Example
3.
There
was
a
Traveller
at
the
From
Barwick
to
London
.
Miles
—
|
60
|
48
|
55
|
47
|
48
|
The
total
is
|
258
|
Town
of
Barwick
inquired
how
farre
it
was
to
the
City
of
London
by
computation
,
and
was
thus
answered
:
from
hence
to
Durham
it
is
60
miles
,
from
thence
unto
the
City
of
Yorke
48
miles
,
then
to
Newarke
55
miles
,
from
thence
to
the
Town
of
Huntington
47
miles
,
and
from
thence
to
the
City
of
London
48
miles
:
having
plac'd
these
numbers
in
order
,
their
summe
will
contain
the
whole
distance
,
therefore
say
8
,
7
,
5
&
8
makes
28
,
and
according
to
the
former
rules
,
set
down
8
the
simple
figure
of
this
compound
number
,
under
the
rank
of
unites
,
and
adde
the
two
decimalls
into
the
next
place
,
saying
,
2
&
4
makes
6
and
4
is
10
&
5
will
be
15
&
4
,
19
&
6
will
be
25
;
write
under
that
rank
the
simple
number
5
,
and
goe
the
two
decimalls
,
but
being
there
are
no
more
rooms
or
places
,
write
down
the
decimall
in
the
next
degree
or
place
to
the
last
figure
,
and
so
you
will
finde
the
totall
summe
258
miles
,
the
true
distance
by
computation
from
Barwick
to
London
,
as
in
the
example
.
Yet
note
when
numbers
given
are
of
divers
denominations
,
in
adding
of
those
summes
together
,
you
must
alwayes
observe
for
to
begin
with
the
least
first
,
on
the
right
hand
as
before
,
and
when
their
summes
doe
amount
unto
any
compound
number
consisting
of
integers
and
parts
,
subscribe
the
fraction
,
or
part
,
underneath
that
columne
,
whereon
the
head
is
writ
the
denomination
,
which
is
various
both
in
number
,
weight
,
and
measure
;
the
integer
or
integers
must
be
added
as
unites
,
to
the
next
place
towards
the
left
hand
:
of
this
I
will
shew
you
some
examples
;
but
to
know
the
parts
,
I
must
referre
you
to
practice
,
they
being
over-numerous
to
be
subscrib'd
,
as
in
the
numeration
of
mixt
numbers
was
already
intimated
.
Example
4.
Addition
of
mixt
numbers
in
Coine
.
This
Table
is
divided
into
Li.
|
So.
|
De.
|
14567
|
8
|
9
|
3289
|
16
|
3
|
987
|
13
|
4
|
81
|
6
|
8
|
9
|
14
|
11
|
18935
|
19
|
11
|
3
columnes
,
for
so
many
severall
denominations
:
upon
the
head
of
the
columne
to
the
left
hand
there
is
writ
Li.
denoting
Libra
signifying
a
pound
,
not
here
in
respect
of
common
weight
but
money
,
and
for
definition
is
called
Pound
sterling
;
and
is
an
integer
,
according
to
the
English
account
in
Coines
:
the
next
columne
is
noted
with
So.
for
Solidum
,
a
coine
of
brasse
used
by
the
Romans
,
but
with
us
of
silver
,
and
signifies
a
Shilling
,
20
of
those
pieces
making
1
Pound
sterling
:
the
third
columne
,
or
first
on
the
right
hand
is
noted
with
De.
for
Denarius
,
which
signifies
10
for
so
many
pieces
it
contain'd
of
the
Romans
least
coyne
;
it
hath
had
a
various
estimate
in
our
English
coynes
,
as
it
signifies
a
Penny
,
the
twelfth
part
of
a
shilling
:
for
untill
the
Reigne
of
Henry
the
sixt
a
Penny
was
the
twentieth
part
of
an
ounce
,
and
in
his
Reigne
made
the
thirtieth
:
by
Edward
the
fourth
40
pence
in
an
ounce
:
by
Henry
the
eighth
there
were
allowed
45
pence
to
an
ounce
:
and
by
Queen
Elizabeth
an
ounce
of
silver
was
divided
into
sixty
parts
,
called
pence
,
as
at
this
day
;
so
much
for
the
Coynes
,
and
now
for
the
adding
of
them
together
,
begin
with
the
least
denomination
first
,
where
you
will
finde
●5
pence
or
2
shillings
and
a
11
pence
,
which
write
down
,
and
adde
the
2
shillings
to
the
next
columne
,
saying
,
2
&
4
is
6
&
6
is
12
&
3
is
15
,
and
6
is
21
&
8
makes
29
,
and
then
reckon
down
the
tennes
,
saying
,
29
&
10
is
39
&
10
will
be
49
and
10
more
makes
59
,
that
is
2
pound
and
19
shillings
which
subscribe
in
the
proper
columne
,
and
adde
2
unto
the
pounds
,
saying
,
2
,
9
,
1
,
7
,
9
,
7
,
makes
35
,
that
is
5
and
goe
3
,
then
3
,
8
,
8
,
8
,
6
,
makes
33
,
that
is
3
and
goe
3
,
which
adde
to
the
next
rank
,
as
3
,
9
,
2
,
5
,
that
is
19
,
subscribe
the
simple
number
,
and
adde
the
decimall
to
the
next
place
,
as
1
,
3
,
4
makes
8
,
which
as
a
simple
number
set
down
,
and
lastly
,
0
&
1
is
1
,
which
prescribe
according
to
its
place
,
and
the
whole
summe
is
18935
L.
—
19
S.
—
11
D.
as
by
the
example
in
the
Table
.
Example
5.
Addition
of
mixt
numbers
in
Weight
.
Of
Weights
there
be
two
kindes
Lb.
|
Ou.
|
Dr.
|
99
|
15
|
7
|
110
|
14
|
6
|
87
|
12
|
5
|
108
|
13
|
4
|
56
|
9
|
3
|
84
|
6
|
1
|
548
|
8
|
2
|
chiefly
u●ed
in
England
,
viz:
Averdupois
,
and
Troy
weight
;
the
first
of
these
,
which
is
also
called
the
Civil
or
Merchants
weight
,
is
divided
into
two
kindes
,
the
greater
,
and
the
lesse
:
the
integer
of
the
greater
,
is
112
pound
to
the
hundred
weight
,
and
those
subdivided
into
Quarters
,
Stones
,
and
Pounds
&c.
by
these
are
weighed
the
most
grosse
commodities
;
or
used
by
Merchants
and
Whole-sale
men
.
The
lesser
weight
by
those
who
sell
by
Retaile
,
whose
integer
is
a
pound
,
and
usually
marked
as
is
the
head
of
the
first
columne
,
or
thus
lb
this
integer
is
subdivided
into
ounces
and
sign'd
as
the
second
columne
,
and
often
thus
℥
16
making
a
pound
,
every
Ounce
is
subdivided
into
8
Drams
or
Dragmes
,
noted
as
in
the
table
,
and
often
thus
ʒ
,
every
Dragme
into
3
Scruples
,
which
usually
hath
this
marke
℈
,
&c.
But
leaving
these
parts
of
intigers
for
some
other
time
,
and
summe
up
these
in
the
Table
according
to
our
prescribed
method
,
and
begin
first
with
the
least
denomination
of
this
mixt
number
,
which
here
are
Drams
,
which
columne
added
together
makes
26
,
that
is
2
Drams
,
and
3
integers
for
to
be
added
unto
the
next
columne
,
which
according
to
the
former
order
of
Addition
,
will
make
the
next
columne
72
Ounces
,
that
is
8
ounces
,
and
4
integers
in
the
next
columne
being
pounds
,
which
being
added
into
the
pounds
according
to
the
rules
of
Addition
,
will
make
the
summe
of
the
next
columne
548
,
and
the
totall
548
lb
-
8
Ounces
-
2
Drams
;
which
according
to
the
greater
or
grosser
weight
may
be
thus
exprest
4
C.
7
St.
2
lb
8
℥
.
2
ʒ.
Example
6.
Addition
of
mixt
numbers
in
long
or
radicall
Measures
.
The
Measures
in
England
are
Pe.
|
Ya
.
|
Fe.
|
318
|
5
|
1
|
299
|
4
|
2
|
48
|
5
|
1
|
167
|
0
|
2
|
98
|
4
|
1
|
319
|
5
|
1
|
1253
|
3
|
2
|
more
various
than
either
of
the
former
two
,
in
this
example
I
will
onely
use
Perches
,
Yards
,
and
Feet
,
and
the
least
of
these
admitting
of
many
subdivisions
,
but
in
finding
or
measuring
of
great
distances
quite
unnecessary
;
as
a
League
containing
3
Miles
:
one
Mile
8
Furlongs
:
one
Furlong
40
Pole
or
Perches
:
one
Perch
5
Yards
and
a
half
:
and
3
Feet
a
Yard
:
in
this
example
a
statute
Perch
is
the
integer
,
and
't
is
supposed
that
the
distance
measured
between
6
stations
are
here
set
down
,
whose
summe
is
required
,
which
to
finde
,
begin
first
with
the
least
denomination
,
where
I
finde
8
Feet
,
that
is
2
Yards
or
integers
in
the
next
columne
;
so
setting
down
2
feet
in
its
proper
columne
,
I
adde
the
other
6
feet
as
2
yards
or
integers
to
the
next
columne
,
which
will
make
the
summe
there
25
,
that
is
3
Yards
and
4
Perches
:
where
I
subscribe
the
parts
,
and
adde
the
22
yards
as
4
Perches
to
the
next
columne
,
which
will
make
the
first
rank
of
figures
there
53
,
and
these
being
in
this
columne
,
all
of
one
denomination
,
I
subscribe
3
,
and
goe
5
decimalls
in
the
next
rank
,
as
by
the
former
method
,
and
so
continue
on
,
untill
they
be
all
added
together
,
whose
totall
summe
is
1253
Porches
,
3
Yards
,
2
Feet
,
for
the
distance
of
the
6
stations
;
and
being
320
Porches
make
one
Mile
,
containing
8
Furlongs
,
and
every
one
of
those
40
Pole
,
you
may
write
the
totall
of
the
distance
thus
,
3
Miles
,
7
Furlongs
,
13
Poles
,
3
Yards
,
2
Feet
:
here
you
must
observe
in
subscribing
the
parts
to
note
when
you
have
done
,
whether
the
fractions
or
parts
of
severall
denominations
will
not
make
an
integer
,
as
sometimes
't
will
happen
,
when
a
part
of
a
denomination
must
be
had
to
make
an
integer
;
as
in
this
case
where
5
yards
,
1
foot
,
and
6
inches
,
makes
a
Porch
,
or
16
feet
and
a
halfe
,
and
so
it
would
have
happen'd
here
,
if
the
summe
remaining
in
the
columne
of
yards
had
been
5
;
and
so
in
divers
other
cases
might
be
instanc'd
.
SUBTRACTION
.
Parag.
3.
The
second
species
.
THe
property
of
this
Species
,
is
to
finde
the
difference
between
any
two
numbers
given
;
which
is
found
by
drawing
,
or
taking
one
number
from
another
,
the
residue
or
remainder
is
the
difference
between
them
,
which
consequently
being
added
to
the
lesser
,
will
make
a
number
equall
to
the
greater
,
the
difference
being
the
excesse
between
them
:
as
if
1
were
to
be
taken
from
9
,
the
remainder
or
difference
will
be
8
,
which
excesse
added
to
1
,
will
make
it
9
,
equall
to
the
greater
number
given
:
so
if
6
were
substracted
from
8
,
the
remainder
will
be
2
,
which
difference
added
to
6
,
makes
8
the
greater
number
,
or
equall
to
it
:
for
nothing
that
is
compounded
can
be
the
same
:
Yet
2
Pintes
will
make
1
Quart
,
and
4
Quarts
1
Gallon
,
but
leaving
this
for
Logicians
to
discusse
,
I
will
proceed
.
When
any
subtraction
is
to
be
made
,
the
lesser
must
be
deducted
from
the
greater
,
or
equall
things
from
other
quantities
that
are
equall
,
but
a
greater
from
a
lesse
cannot
be
deducted
in
the
whole
,
yet
in
particulars
they
may
,
when
the
praecedent
numbers
towards
the
left
hand
shall
be
greater
,
from
whence
the
subtraction
is
to
be
made
,
as
shall
be
illustrated
by
examples
,
and
first
observe
to
write
in
figures
the
greater
number
,
and
under
that
the
lesser
,
and
in
such
order
as
was
prescrib'd
in
Addition
,
according
to
their
ranks
and
places
,
whether
they
be
Cyphers
or
significant
figures
as
unites
under
unites
,
tennes
under
tennes
,
&c.
Yet
notwithstanding
when
the
lesser
number
is
under
the
greater
according
to
their
places
,
a
significant
figure
may
be
under
a
cypher
,
or
a
greater
under
the
lesse
;
in
such
cases
,
take
an
unite
from
the
place
before
it
,
towards
the
left
hand
,
it
will
be
10
in
that
ranke
,
which
adde
to
it
,
and
then
subtract
the
former
number
,
and
keep
the
unite
which
was
borrow'd
,
in
your
minde
,
and
adde
that
to
the
lower
number
in
the
next
place
;
for
the
upper
number
was
to
have
been
an
unite
lesse
,
therefore
if
the
lower
precedent
number
be
made
a
unite
greater
it
is
all
one
:
as
if
9
were
to
be
subtracted
from
10
;
and
plac'd
according
to
my
direction
,
it
cannot
be
taken
from
a
cypher
alone
,
but
as
a
decimall
it
may
,
and
1
will
remaine
.
Againe
,
admit
19
to
be
subtracted
from
21
,
9
cannot
be
taken
from
1
,
therefore
suppose
an
unite
borrow'd
of
2
&
then
the
former
1
will
be
made
11
,
from
whence
take
9
,
and
there
will
remaine
2
,
now
't
is
all
one
whether
I
make
2
in
the
21
but
1
,
or
the
Decimall
1
I
borrow'd
be
added
to
1
in
the
19
,
and
so
make
that
2
,
which
is
the
best
way
,
and
then
take
2
from
2
and
nothing
remaines
but
2
,
the
first
subtraction
;
for
the
difference
betwixt
21
and
19.
If
severall
numbers
were
to
be
subtracted
,
from
divers
other
sums
,
collect
them
all
into
one
by
Addition
,
and
likewise
those
which
are
to
be
subtracted
,
and
then
finde
the
difference
between
them
;
for
this
rule
or
Species
admits
but
of
two
summes
;
these
principles
observ'd
,
the
rest
will
be
clear'd
by
examples
following
.
Example
1.
Arithmeticall
and
Geometricall
demonstrations
.
Number
in
Arithmetick
,
and
a
Line
in
Geometrie
have
an
undoubted
Symmetrie
,
or
proportion
in
all
their
parts
,
being
both
of
the
first
quantity
,
as
was
said
and
prov'd
already
:
and
here
in
this
example
,
if
the
line
C
were
to
be
taken
from
the
line
A
or
laid
upon
it
,
the
line
C
would
extend
it selfe
to
B
;
from
whence
't
is
evident
,
A
,
B
,
would
be
the
difference
betwixt
them
,
which
remainder
is
equall
to
B
,
Againe
,
suppose
the
line
A
were
a
Statute
pearch
or
Pole
,
and
divided
into
inches
,
it
would
containe
198
of
those
equall
parts
,
and
admit
the
line
C
were
12
foot
or
144
inches
,
to
be
taken
from
A
,
having
plac'd
them
right
under
one
an
other
,
begin
with
the
unites
,
and
say
;
take
4
from
8
and
there
will
remaine
4.
then
goe
to
the
rank
of
decimalls
and
say
,
take
4
out
of
9
and
there
will
remaine
5
,
which
place
under
the
line
in
the
second
rank
,
then
proceed
to
the
place
of
hundreds
,
where
you
will
finde
1
taken
from
1
,
and
nothing
will
remaine
;
so
54
inches
is
the
residue
,
or
difference
between
the
two
lines
which
was
required
.
Example
2.
Henry
the
8
,
in
the
yeare
Anno
Domini
salvatoris
|
1653
|
Anno
Domini
salvatoris
|
1536
|
The
difference
of
yeares
|
117
|
of
our
Lord
&
Saviour
1536
,
made
himself
the
Head
of
the
Protestant
Church
in
his
own
Dominions
,
and
Thomas
Lord
Cromwell
his
Vicar
Generall
under
him
:
and
in
the
yeare
of
Christ
1653
,
Oliver
Cromwell
Lord
Generall
for
the
new
State
,
was
made
Protector
over
all
the
three
Kingdomes
;
and
here
it
is
required
the
annuall
revolutions
,
or
the
yeares
elapsed
between
them
.
First
,
set
downe
the
yeare
present
,
which
is
the
greater
number
,
and
under
that
the
lesse
,
each
figure
according
to
his
degree
or
place
,
then
say
,
take
6
out
of
3
,
which
cannot
be
done
,
without
borrowing
one
of
5
,
that
is
10
in
that
place
,
which
added
to
3
will
make
13
,
so
take
6
from
13
and
there
will
remaine
7
,
which
place
beneath
the
line
,
drawn
to
part
the
remainder
,
from
the
numbers
to
be
subtracted
,
and
under
6
,
then
say
1
and
3
is
4
,
which
taken
from
5
,
and
there
remaines
1
,
which
write
under
3
,
then
take
5
out
of
6
and
there
will
remaine
1
,
which
set
downe
in
its
place
,
then
take
1
from
1
and
nothing
will
remaine
,
so
the
difference
between
these
yeares
is
117
,
and
so
many
annuall
revolutions
of
the
Sun
are
elapsed
.
Example
3.
There
was
a
Merchant
|
lb.
|
Money
borrowed
|
5000
|
Money
repaid
|
4975
|
Remaining
due
|
025
|
who
had
borrowed
at
one
time
3200
pound
,
and
yet
to
goe
forward
with
his
designes
,
was
constrained
to
take
up
more
as
1500
,
and
presently
after
that
300
l
,
upon
the
returne
of
his
adventure
,
the
Merchant
sold
his
goods
so
soon
as
he
could
,
to
pay
his
Creditor
,
whom
he
brought
at
one
time
4975
li.
&
it
was
required
to
know
what
the
difference
was
between
the
mony
borrowed
,
and
the
summe
paid
.
First
,
collect
together
all
the
severall
summes
that
were
lent
,
which
will
make
5000
lb
and
set
it
down
,
and
being
't
is
evident
,
the
money
repaid
was
lesse
,
subscribe
that
under
the
greater
,
then
draw
a
line
and
subtract
them
,
saying
,
take
5
from
10
and
there
will
remain
5
,
and
8
out
of
10
there
will
remain
2
,
which
write
in
the
second
place
:
and
proceed
,
saying
,
1
which
was
borrowed
,
and
9
will
make
10
,
take
10
from
10
and
nothing
remains
;
then
again
say
1
and
4
makes
5
,
take
5
out
5
and
nothing
will
remain
:
the
last
cypher
was
set
down
to
keep
the
place
,
being
there
were
more
to
be
subtracted
,
and
something
might
have
remained
,
which
is
not
at
the
first
seen
to
young
practitioners
,
and
no
prejudice
to
any
:
the
difference
is
25
li.
and
so
much
money
is
remaining
due
to
the
Creditor
,
besides
Interest
,
which
according
to
the
time
is
to
be
considered
.
Example
4.
There
was
a
Merchant
|
lb.
|
The
Banke
|
1900
|
The
sūme
known
|
892
|
The
difference
|
1008
|
who
upon
a
forreign
imployment
made
two
of
his
servants
Cash-keepers
in
his
absence
,
and
commanded
them
to
put
all
the
money
they
should
receive
into
a
Counter
or
common
Bank
,
whereof
he
took
the
key
himself
,
and
left
in
his
treasurie
95
li.
before
his
return
,
one
of
the
Cash-keepers
died
,
and
calling
the
other
to
an
account
of
what
money
he
had
received
,
he
made
it
appear
by
Bonds
and
other
cancell'd
Deeds
,
that
there
was
797
li.
put
into
the
common
Treasure
by
him
;
but
what
his
fellow
servant
had
done
he
knew
not
:
the
Merchant
opening
his
Counter
found
in
it
1900
li.
and
it
is
required
to
know
how
much
money
the
servant
which
was
dead
had
received
:
first
adde
the
stock
which
was
left
,
viz:
95
li.
to
the
money
received
797
li.
the
summe
is
892
,
which
as
in
the
example
,
set
underneath
the
totall
summe
found
in
the
Banke
,
viz:
1900
li.
the
difference
will
shew
1008
li.
the
true
summe
,
which
the
other
servant
had
received
.
Example
5.
Subtraction
of
mixt
numbers
in
Coine
.
There
was
a
Farmer
,
who
had
L
|
S
|
D
|
Q
|
1000
|
0
|
0
|
0
|
999
|
19
|
11
|
3
|
0000
|
00
|
00
|
1
|
borrowed
1000
li.
of
an
Usurer
;
or
at
least
wrackt
up
,
by
extortion
,
with
Interest
upon
interest
untill
extended
to
that
summe
as
that
the
Farmer
knew
not
what
to
doe
;
the
Usurer
having
more
Law
of
his
side
than
equity
,
lesse
mercy
than
a
she
Wolfe
upon
her
prey
,
and
now
an
opportunity
by
the
extremity
of
Law
,
and
vertue
of
his
Bonds
,
the
day
of
forfeiture
neer
at
hand
;
not
commiserating
the
poor
mans
case
,
or
complaints
,
or
admitting
of
delayes
,
in
prolonging
time
,
any
more
than
death
:
the
Farmer
in
this
exigent
made
recourse
to
his
friends
,
engaging
them
,
pawning
his
goods
and
credit
,
so
at
last
with
the
help
of
his
Wives
butter-money
,
he
raised
the
summe
subscribed
in
the
margent
:
the
Usurer
seeing
999
li.
refused
to
receive
it
,
gaping
still
after
the
forfeiture
,
but
in
vaine
,
for
with
the
broken
money
't
was
right
to
a
farthing
,
as
by
Subtraction
will
appear
in
the
example
;
take
3
farthings
out
of
nothing
you
cannot
,
then
borrow
an
intiger
in
the
next
place
which
is
1
D
,
or
4
Q
,
then
say
,
take
3
out
of
4
and
1
Q
will
remain
;
again
1
D
borrowed
and
11
D
is
1
shilling
,
which
take
from
nothing
you
cannot
,
but
from
an
intiger
borrowed
in
the
next
columne
you
may
,
and
nothing
will
remain
:
then
proceed
and
say
,
1
S
borrowed
added
to
19
S
makes
1
Li.
or
20
S.
and
being
there
is
a
cypher
over
it
,
borrow
an
integer
in
the
next
place
which
are
L.
1
being
20
S.
take
20
out
of
20
and
nothing
remains
,
therefore
subscribe
a
cypher
as
before
:
then
in
the
columne
of
pounds
sterling
,
say
,
1
borrowed
and
9
is
10
,
take
10
out
of
10
and
nothing
remains
;
set
down
a
cypher
,
and
say
again
1
borrowed
and
9
is
10
,
take
10
out
of
nothing
I
cannot
,
borow
1
of
the
next
place
,
as
before
,
(
these
being
all
of
one
denomination
)
then
take
10
from
10
and
nothing
remains
,
and
so
proceed
till
all
be
subtracted
and
nothing
remains
,
so
in
conclusion
there
is
onely
remaining
1
Q.
which
stands
for
Quadrans
;
in
our
Coine
it
is
taken
for
the
fourth
part
of
a
Peny
,
that
is
,
a
Farthing
,
and
he
who
payes
to
that
,
the
Proverb
declares
an
honest
man
.
Example
6.
Subtraction
of
mixt
numbers
in
Weight
.
The
letters
in
the
head
of
this
lb
|
O
|
P
|
G
|
670
|
3
|
18
|
20
|
369
|
8
|
19
|
4
|
300
|
6
|
19
|
16
|
Table
denotes
in
each
columne
the
pounds
and
parts
according
to
Troy
weight
,
lb
pounds
;
O
ounces
,
whereof
12
makes
one
pound
;
P
for
peny-weight
,
twenty
of
those
makes
an
ounce
;
G
graines
,
24
of
them
doe
make
a
peny-weight
.
There
was
a
Merchant
that
brought
over
two
quantities
of
unrefined
silver
,
one
Masse
weighed
670
lb
.
3
O.
18
P.
20
G.
The
lesser
Masse
369
lb.
8
O.
19
P.
4
G.
when
they
were
refined
,
and
in
Lingots
;
the
lesser
quantity
at
5
shillings
the
ounce
,
did
come
but
to
so
much
as
was
their
difference
in
weight
between
either
Masse
,
which
is
here
required
;
having
plac'd
them
according
to
their
pounds
,
and
parts
,
as
in
the
Table
;
subtract
one
from
the
other
,
beginning
with
the
least
first
,
as
in
the
other
examples
,
and
say
,
take
4
G.
from
20
,
and
there
will
remain
16
;
then
to
the
next
columne
,
where
19
Penny
weight
is
to
be
taken
from
18
,
but
cannot
,
therefore
take
19
from
38
,
and
there
will
remain
19
P.
1
ounce
which
was
borrowed
and
8
will
be
9
,
take
9
from
3
is
impossible
,
take
9
from
15
and
there
will
remain
6
ounces
,
1
lib.
borrowed
and
9
is
10
,
take
10
from
10
and
nothing
remains
,
1
&
6
will
be
7
,
take
7
out
of
7
and
nothing
will
remain
,
subscribe
a
cypher
as
before
;
and
lastly
,
take
3
from
6
and
3
will
remain
;
so
the
difference
is
300
lib.
6
O.
19
P.
16
G.
which
at
5
shillings
the
ounce
,
that
is
3
li.
sterling
the
lib.
comes
unto
into
money
901
L.
14
S.
11
D.
Example
7.
Subtraction
of
mixt
numbers
in
Measure
.
This
examp●e
is
of
Land
measure
,
Acres
|
R
|
P
|
9012
|
1
|
8
|
912
|
1
|
3●
|
8099
|
3
|
18
|
wherein
observe
generally
throughout
England
,
that
although
there
be
Customary
&
Statute-measure
,
whereof
the
first
is
very
numerous
,
yet
in
all
and
every
one
,
there
are
40
square
Pole
in
a
Rood
,
and
4
Rood
in
an
Acre
,
and
the
question
is
here
of
a
Man
who
had
two
sonnes
,
and
in
his
last
Will
bequeathed
to
his
eldest
sonne
9
Lordships
but
no
Manours
,
which
were
in
content
upon
the
survey
9012
Acres
,
1
Rood
and
8
Perches
;
and
to
his
youngest
sonne
he
gave
Nonsuch
and
Drownland
in
a
County
of
the
Moone
,
containing
by
supposition
912
Acres
,
1
Rood
,
and
30
Pole
,
survey'd
by
Telescopes
,
and
't
is
required
to
know
how
much
one
brother
had
in
content
more
than
the
other
;
having
placed
them
,
as
in
the
Table
,
take
30
P.
from
48
and
there
will
remain
18
,
1
I
borrowed
and
1
makes
2
,
take
2
from
5
and
there
will
remain
3
R.
then
say
again
1
&
2
is
3
,
which
taken
from
12
and
there
will
remain
9
,
then
1
&
1
makes
2
,
that
taken
from
11
,
and
9
remains
,
which
subscribe
as
before
;
then
say
,
1
&
9
makes
10
,
take
10
out
of
10
and
nothing
will
remain
,
there
subscribe
a
cypher
,
and
being
there
are
no
more
figures
in
the
lesser
number
,
subtract
the
last
1
which
was
borrowed
out
of
9
and
the
remainder
will
be
8
,
so
the
residue
or
difference
between
them
is
8099
A.
3
R.
18
P.
as
in
the
Table
you
may
see
.
An
Examine
of
Addition
and
Subtraction
,
Or
a
triall
whether
you
have
done
right
or
no
,
there
be
divers
rules
,
but
the
best
and
generall
way
,
is
to
make
one
of
these
Species
prove
the
other
;
as
when
divers
numbers
are
added
together
,
from
the
summe
subtract
the
parts
,
and
you
will
produce
the
former
numbers
;
and
where
a
subtraction
is
made
,
to
try
that
;
adde
the
difference
or
remainder
,
to
the
numbers
subtracted
which
is
least
,
and
the
summe
will
give
the
greater
number
;
or
from
any
number
subtracted
,
take
the
difference
from
the
greater
given
number
,
and
the
remainder
will
be
the
lesser
number
that
was
subtracted
:
for
alwayes
,
where
there
are
2
numbers
subtracted
,
as
the
remainder
is
the
difference
betwixt
those
numbers
given
,
so
the
lesser
given
number
is
the
difference
between
the
remainder
,
and
the
greater
given
number
;
for
every
remainder
must
be
lesse
then
the
greater
number
,
by
the
quantity
of
the
lesser
,
therefore
't
is
the
difference
between
them
;
yet
though
t
is
evident
I
will
illustrate
it
with
examples
in
both
Species
,
how
they
mutually
do
try
one
another
.
In
the
first
example
of
Addition
,
if
from
the
sum
of
792
you
take
C
,
231
,
the
remainder
will
be
561
,
the
summe
of
A
and
B
,
from
whence
subtract
either
of
their
parts
,
and
the
other
will
remaine
,
as
take
away
A
320
,
and
there
will
remaine
241
for
B
,
or
if
241
had
been
subtracted
from
561
,
there
had
remained
320
for
A
,
and
thus
you
may
proceed
in
any
other
,
either
of
one
denomination
or
of
many
.
To
try
whether
Subtraction
made
,
be
true
,
take
the
last
example
of
mixt
numbers
,
and
say
18
and
30
is
48
,
that
is
8
Pole
and
1
intiger
in
the
next
columne
,
so
proceed
,
1
and
3
makes
4
and
1
will
be
5
,
that
is
1
Roode
and
goe
11
Acres
;
then
1
and
9
is
10
and
2
will
be
12
that
is
2
and
goe
1
decimall
,
which
with
9
and
2
will
be
11
,
subscribe
1
and
goe
1
,
which
added
to
9
makes
10
,
a
cypher
and
goe
1
,
lastly
1
and
8
makes
9
,
so
the
summe
is
9012
A
,
1
R
,
8
P
,
as
before
:
or
to
try
this
by
subtraction
,
take
the
remainder
from
the
greater
number
,
their
difference
will
be
the
lesser
number
which
was
subtracted
,
if
it
were
rightly
perform'd
,
as
take
18
from
48
and
there
will
remaine
30
P
,
take
4
Roode
out
of
5
and
there
will
remaine
1
R
,
take
10
Acres
out
of
12
and
there
will
remaine
2
,
take
10
out
of
11
and
there
will
remaine
1
,
then
take
1
out
of
10
and
the
remainder
will
be
9
,
lastly
1
and
8
is
9
,
take
9
from
9
,
and
nothing
remaines
,
and
the
other
figures
subscribed
will
make
912
A.
1
R.
30
P.
the
lesser
given
number
:
and
thus
by
finding
either
of
them
by
the
remainder
,
it
proves
the
worke
undoubtedly
true
.
MULTIPLICATION
.
The
third
Species
,
and
fourth
Paragraph
.
MVltiplication
is
the
encreasing
of
any
number
by
another
,
so
often
as
there
be
unites
in
one
of
the
numbers
;
or
from
any
two
numbers
given
to
finde
a
third
,
which
shall
contain
one
of
the
numbers
as
many
times
as
there
be
unites
in
the
other
.
To
Multiplication
there
appertains
three
principle
members
,
viz:
the
Multiplicand
,
the
Multiplier
,
and
the
Product
.
The
Multiplicand
,
is
the
number
given
for
to
be
multiplied
,
and
is
properly
the
greater
number
in
respect
of
order
,
but
of
no
necessity
;
for
't
is
all
one
whether
4
be
multiplied
by
3
,
or
3
encreased
by
4
,
't
will
be
either
way
12.
The
Multiplier
is
usually
the
least
,
and
alwayes
the
lower
number
:
it
is
to
encrease
the
Multiplicand
as
many
times
as
there
be
unites
contained
in
the
Multiplier
;
as
if
6
were
to
be
multiplied
by
4
,
they
would
produce
24
,
and
so
would
6
times
4
;
that
is
4
added
6
times
together
,
for
the
number
of
unites
contained
in
the
Multiplier
.
The
Product
is
the
number
produced
by
the
mutuall
encreasing
one
number
by
another
,
as
the
Multiplier
by
the
Multiplicand
;
as
if
9
were
multiplied
by
2
,
the
Product
will
be
18
,
or
8
times
9
is
72.
All
numbers
encreasing
but
according
to
the
unites
contained
in
them
,
you
may
divide
the
Multiplier
,
the
Multiplicand
,
or
both
of
them
into
what
parts
you
please
,
then
multiply
those
parts
,
and
adde
them
all
together
,
they
will
produce
the
same
number
;
for
the
whole
is
in
the
same
proportion
to
the
whole
as
all
the
parts
are
,
as
in
the
last
example
;
divide
9
the
Multiplicand
into
what
parts
you
please
,
as
2
,
3
,
4
,
these
making
9
in
all
,
then
encrease
all
those
parts
by
8
the
Multiplier
,
and
they
will
produce
16
,
24
,
32
,
whose
summe
is
72
,
and
so
is
9
multiplied
by
8
,
which
Multiplier
divide
into
what
parts
you
will
,
as
admit
in
two
,
viz:
3
&
5
,
and
then
multiply
the
former
parts
of
the
Multiplicand
,
which
were
2
,
3
,
4
,
by
3
,
and
then
by
5
;
the
Products
will
be
6
,
9
,
12
,
&
10
,
15
,
20
,
all
which
Products
added
together
make
72
,
that
is
8
times
9
as
before
,
and
the
like
of
any
other
;
for
as
an
unite
or
1
is
to
the
Multiplier
,
so
will
the
Multiplicand
be
in
proportion
to
the
Product
,
and
the
contrary
,
as
by
examples
shall
be
illustrated
.
A
Demonstration
.
A
Table
of
Multiplication
.
A
|
|
|
|
F
|
|
|
|
B
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
2
|
4
|
6
|
8
|
10
|
12
|
14
|
16
|
18
|
3
|
6
|
9
|
12
|
15
|
18
|
21
|
24
|
27
|
4
|
8
|
12
|
16
|
20
|
24
|
28
|
32
|
36
|
5
|
10
|
15
|
20
|
25
|
30
|
35
|
40
|
45
|
6
|
12
|
18
|
24
|
30
|
36
|
42
|
48
|
54
|
7
|
14
|
21
|
28
|
35
|
42
|
49
|
56
|
63
|
8
|
16
|
24
|
32
|
40
|
48
|
56
|
64
|
72
|
9
|
18
|
27
|
36
|
45
|
54
|
63
|
72
|
81
|
D
|
|
|
|
E
|
|
|
|
C
|
This
Table
consists
of
the
mutuall
multiplication
of
all
the
nine
significant
figures
,
proceeding
from
an
unite
to
9
either
way
;
that
is
,
from
A
to
B
,
and
from
A
to
D
,
according
to
the
orderly
succession
of
these
numerall
characters
,
every
one
having
their
severall
Products
,
or
number
of
unites
contained
in
their
multiplications
,
which
ought
for
to
be
learned
without
book
,
and
imprinted
in
your
memory
;
yet
to
facilitate
the
labour
of
young
beginners
,
I
have
here
subscribed
this
Table
,
easie
to
be
understood
,
and
a
readily
to
be
remembred
,
divided
into
91
little
squares
;
every
one
containing
the
Product
,
made
by
the
simple
figure
in
the
head
of
the
Table
over
it
,
and
that
in
the
side
right
against
it
,
on
the
left
hand
;
every
one
encreasing
,
or
decreasing
in
his
columne
,
according
to
the
unites
in
the
Multiplier
,
or
Multiplicand
;
as
for
example
,
admit
5
&
6
were
to
be
multiplied
,
I
look
the
one
number
in
the
head
of
the
Table
,
and
the
other
in
the
side
,
in
their
common
square
or
angle
you
will
finde
30
,
their
true
Product
;
and
observe
how
that
columne
under
6
encreases
and
decreases
by
6
,
and
that
of
5
by
5
:
thus
,
9
times
9
will
be
found
81
,
and
their
columnes
encreased
by
9
,
and
so
the
like
of
any
others
.
And
farther
you
may
here
observe
how
every
simple
figure
multiplied
in
its
selfe
,
is
terminated
,
untill
it
returns
to
its
own
character
again
;
as
for
example
,
1
multiplies
nothing
,
but
still
remains
an
unite
,
being
no
number
of
it selfe
;
2
multiplied
by
2
ends
in
4
,
8
,
6
,
and
fourthly
is
terminated
in
2
its
root
;
3
by
3
ends
in
9
,
7
,
1
,
and
fourthly
in
3
again
;
4
by
4
ends
in
6
,
and
secondly
in
4
,
next
5
&
6
multiplied
in
themselves
,
are
terminated
in
their
own
figures
,
and
from
thence
are
called
Circular
numbers
;
7
multiplied
by
7
ends
in
9
,
3
,
1
,
and
fourthly
,
in
its
self
;
8
by
8
ends
in
4
,
2
,
6
,
and
fourthly
,
in
its
selfe
again
;
9
multiplied
by
9
will
end
in
1
,
and
in
every
second
multiplication
is
terminated
in
9
,
it
s
own
root
,
or
number
again
produced
.
Multiplication
is
a
quadrature
and
hath
this
Analogy
or
proportion
with
a
superficiall
square
call'd
in
Geometrie
the
second
quantity
,
a
figure
composed
of
lines
,
whose
sides
are
divided
into
parts
,
and
intersected
with
paralle's
,
or
equidistant
lines
,
as
is
the
last
figure
A
,
B
,
C
,
D
,
making
thus
both
Squares
,
and
right
angled
Parallelograms
,
equall
to
the
numbers
multiplied
in
themselves
together
;
as
for
example
,
the
Table
or
square
A
,
C
,
hath
every
side
divided
into
9
equall
parts
,
as
admit
Poles
,
Yards
,
Feet
,
Inches
or
what
you
please
;
suppose
these
sides
as
A
,
B
,
and
A
,
D
,
in
Feet
,
and
intersected
with
lines
,
the
whole
Square
will
containe
81
square
Feet
,
for
the
true
superficiall
content
of
it
,
so
if
9
were
multiplied
by
9
,
the
Product
will
be
the
same
,
or
if
the
long
Square
A
,
F
,
E
,
D
,
were
given
,
the
superficiall
content
of
this
Geometricall
figure
,
call'd
a
Parallelogram
,
would
be
45
square
Feet
;
and
so
is
the
product
of
9
multiplied
by
5
:
If
it
were
required
to
know
how
many
feet
there
were
in
a
Yard
square
,
3
Feet
makes
a
yard
in
length
,
therefore
if
every
side
of
such
a
square
were
divided
into
3
equall
parts
,
and
intersected
with
right
lines
,
there
will
be
found
9
square
feet
,
as
in
the
Table
will
appeare
;
and
so
3
multiplied
by
3
will
produce
9
,
and
in
these
,
the
proportion
is
continued
;
as
an
unite
is
unto
the
number
,
or
side
of
the
Square
given
,
so
will
the
side
be
to
the
whole
Square
;
or
as
1
to
the
Multiplier
,
so
will
the
Multiplicand
be
unto
the
Product
,
as
in
the
former
examples
;
as
1
to
A
,
B
,
9
,
so
will
A
,
D
,
9
be
to
81
the
whole
Square
or
Product
,
or
as
1
to
A
,
F
,
5
,
so
A
,
D.
9
in
proportion
to
45
the
long
Square
A
,
F
,
E
,
D
,
and
so
the
like
of
any
other
in
this
kind
whether
greater
or
lesse
,
and
so
much
for
the
Sympathy
betwixt
Multiplication
,
and
Geometricall
Squares
of
the
second
quantity
.
The
way
and
form
of
Multiplication
when
there
is
more
than
one
significant
figure
in
the
Multiplicand
,
Multiplier
,
or
in
both
of
them
.
When
there
be
severall
figures
to
be
multiplied
both
in
the
Multiplier
and
Multiplicand
,
set
down
first
the
greater
number
,
and
under
that
the
lesser
,
according
to
their
degrees
or
places
,
as
unites
under
unites
,
tennes
under
tennes
,
hundreds
underneath
hundreds
,
&c.
as
hath
been
directed
in
divers
other
examples
already
;
this
done
,
draw
a
line
underneath
them
both
whereby
to
separate
them
from
the
numbers
encreasing
by
their
mutuall
multiplications
,
proceeding
from
the
right
hand
in
order
towards
the
left
;
every
figure
of
the
Multiplier
,
must
be
encreased
or
multiplied
,
into
every
particular
figure
of
the
Multiplicand
,
from
whence
there
will
arise
so
many
Products
,
as
there
be
figures
in
the
Multiplier
;
the
first
figure
in
every
Product
ought
to
be
placed
exactly
under
that
figure
which
multiplies
,
and
so
in
order
with
a
convenient
distance
towards
the
left
hand
,
for
all
the
severall
Products
must
be
added
together
,
whereby
to
finde
the
result
or
totall
of
them
:
and
in
every
figure
as
you
multiply
,
set
down
the
Product
if
under
10
;
but
if
a
decimall
or
decimalls
,
subscribe
a
cypher
;
if
a
compound
number
,
write
down
the
significant
figure
,
and
keep
the
decimall
or
decimalls
in
your
minde
,
and
as
you
multiply
the
next
figure
,
adde
them
in
as
unites
;
and
so
proceed
untill
every
figure
of
the
Multiplier
be
encreased
,
by
all
the
figures
of
the
Multiplicand
in
order
as
was
said
already
,
and
shall
be
now
illustrated
with
severall
Paradigma's
following
.
To
finde
how
many
dayes
there
be
contained
in
a
common
Year
consisting
of
52
Weeks
and
one
day
.
Example
1.
Having
set
down
the
52
Multiplicand
|
52
|
The
Multiplier
|
7
|
The
Product
|
364
|
The
odde
day
|
1
|
The
totall
of
days
in
a
year
|
365
|
Weeks
,
subscribe
underneath
it
,
the
dayes
in
one
Week
,
which
are
7
,
and
being
a
simple
number
,
I
set
it
under
2
,
in
the
unite
place
;
and
since
there
are
so
many
dayes
in
one
week
,
there
must
be
52
times
so
many
in
a
yeare
,
besides
the
odde
day
:
then
say
7
times
2
makes
14
,
for
which
subscribe
4
below
the
line
,
and
right
under
7
,
and
keep
the
decimall
in
your
minde
,
they
say
7
times
5
(
as
by
the
Table
of
Multiplication
)
will
make
35
,
and
the
1
decimall
in
minde
makes
36
,
therefore
subscribe
6
right
underneath
the
5
,
and
keep
the
3
decimalls
in
minde
;
but
since
there
is
no
more
,
write
it
down
,
and
the
product
is
364
,
the
number
of
dayes
in
52
weeks
;
and
being
there
is
one
day
more
in
a
yeare
,
adde
that
in
,
and
then
the
totall
will
be
365
,
the
true
number
of
dayes
in
the
Suns
annuall
revolution
,
the
thing
required
.
The
dayes
and
parts
of
a
yeare
being
known
,
to
finde
the
number
of
houres
contained
in
it
.
Example
2.
First
set
down
the
greater
Multiplicand
|
365
|
Multiplier
|
24
|
The
Products
|
1460
|
|
730
|
The
totall
|
8760
|
number
,
and
then
the
Multiplier
,
according
to
the
places
of
the
figures
,
in
this
24
being
the
houres
contained
in
a
naturall
day
,
and
then
say
4
times
5
is
20
,
for
which
subscribe
a
cypher
,
and
keep
2
decimalls
in
minde
;
then
4
times
6
is
24
&
2
is
26
,
that
is
6
and
goe
2
,
then
say
4
times
3
is
12
,
and
2
in
minde
is
14
,
and
being
there
are
no
more
figures
in
the
Multiplicand
,
write
them
down
,
and
then
the
first
Product
will
be
1460
;
then
goe
to
the
second
figure
of
the
Multiplier
,
which
here
is
2
;
and
say
,
2
times
5
is
10
,
subscribe
a
cypher
under
2
the
Multiplier
,
and
goe
1
decimall
,
then
2
times
6
is
12
and
1
makes
13
,
that
is
3
and
goe
1
,
then
2
times
3
is
6
,
and
1
in
minde
makes
7
,
which
write
down
under
the
1
,
and
then
adde
the
Products
together
,
the
totall
will
be
8760
,
the
number
of
hours
in
365
dayes
,
and
since
the
Julian
account
,
makes
the
magnitude
of
the
yeare
for
to
consist
of
6
hours
more
,
adde
6
unto
the
Product
,
and
the
totall
will
be
8766
hours
contained
in
a
common
year
.
To
finde
how
many
minutes
are
contain'd
in
a
year
.
Example
3.
Here
you
are
to
know
the
Multiplicand
|
8766
|
Multiplier
|
60
|
|
0000
|
|
52596
|
Product
|
525960
|
parts
or
minutes
of
an
hour
,
which
are
60
,
those
must
be
the
Multiplier
,
and
8766
,
the
houres
contained
in
a
yeare
the
multiplicand
,
then
say
no
times
6
,
or
6
times
nothing
,
is
nothing
,
therefore
subscribe
so
many
cyphers
,
as
there
be
figures
in
the
Multiplicand
,
and
then
begin
with
6
,
and
say
6
times
6
is
36
,
there
set
downe
6
beneath
the
line
right
underneath
the
Multiplier
6
;
then
say
6
times
6
makes
36
,
and
3
decimalls
in
minde
will
be
39
,
there
subscribe
9
and
goe
3
,
then
6
times
7
will
be
42
,
and
3
in
minde
is
45
,
that
is
5
and
goe
4
,
then
6
times
8
is
48
,
and
4
decimalls
in
minde
will
make
52
,
which
subscribe
,
there
being
no
more
figures
to
be
Multiplied
;
the
product
of
these
is
525960
,
and
so
many
minutes
there
are
in
a
vulgar
yeare
,
but
this
and
all
such
questions
are
performed
a
readier
way
,
as
you
shall
see
by
the
examples
following
.
Breviats
,
or
compendious
Abridgements
in
Multiplication
exemplified
.
Example
4.
If
any
number
be
given
Multiplicand
|
1000
|
Multiplier
|
25
|
The
Product
|
25000
|
for
to
be
multiplied
,
which
hath
onely
cyphers
,
with
an
unite
upon
the
left
hand
;
being
1
is
no
number
,
and
consequently
multiplies
not
,
annex
all
the
cyphers
to
the
significant
figures
towards
the
right
hand
,
and
the
worke
is
ended
,
as
suppose
1000
Souldiers
,
and
every
,
one
received
for
his
pay
25
Crowns
,
the
product
will
be
25000
Crowns
for
the
whole
summe
paid
;
and
so
for
any
other
,
as
10
,
100
,
1000
,
10000
,
&c.
but
if
there
be
any
other
number
given
to
be
multiplied
,
with
cyphers
following
any
significant
figure
or
figures
,
in
all
such
cases
neglect
the
cyphers
,
and
making
the
lesser
number
then
Multiplier
,
and
proceed
as
before
,
and
to
the
product
annex
all
the
cyphers
that
were
towards
the
right
hand
either
in
the
Multiplier
,
Multiplicand
,
or
in
both
of
them
,
as
in
the
former
example
where
60
was
the
Multiplier
,
if
8766
had
been
multiplied
by
6
,
the
Product
would
have
been
52596
,
to
which
if
a
cypher
had
been
annext
,
the
summe
would
have
been
525960
,
as
before
.
When
cyphers
in
any
Multiplier
are
included
between
significant
figures
,
the
multiplication
may
be
easily
abreviated
thus
.
Example
5.
Admit
9080
were
a
number
Multiplicand
|
908
|
Multiplier
|
309
|
|
8172
|
|
2724
|
Product
|
280572
|
given
for
to
be
multiplied
by
3090
,
here
according
unto
the
last
Paradigma
,
I
doe
neglect
the
cyphers
in
both
numbers
,
and
so
set
down
908
,
and
under
that
309
;
then
draw
a
line
underneath
them
as
in
the
Table
,
and
beginning
with
9
the
first
Multiplier
,
say
.
9
times
8
is
72
,
both
which
figures
subscribe
,
because
a
cypher
follows
in
the
Multiplicand
,
then
say
9
times
9
is
81
,
which
also
set
down
,
there
being
no
more
to
come
the
cypher
is
next
,
which
makes
nothing
,
but
onely
assumes
a
place
;
therefore
goe
unto
the
next
significant
figure
which
is
3
,
and
by
reason
of
the
cypher
in
the
Multiplicand
you
may
say
,
3
times
8
is
24
,
and
3
times
9
is
27
,
which
subscribe
in
order
observing
punctually
to
place
the
first
figure
of
that
Product
under
the
figure
of
the
Multiplier
,
as
in
this
4
under
3
,
and
so
in
order
proceed
towards
the
left
hand
,
by
this
means
the
place
of
the
cypher
is
gained
,
and
being
between
the
first
and
last
Product
can
be
nothing
,
nor
have
a
place
in
adding
them
together
:
the
Product
of
these
figures
is
280572
,
and
being
the
Multiplier
and
Multiplicand
had
either
of
them
a
cypher
in
the
first
place
on
the
right
hand
,
annex
the
two
cyphers
to
this
summe
,
and
then
the
totall
will
be
28,057,200
,
the
solution
required
.
When
any
number
is
given
for
to
be
multiplied
by
one
figure
,
or
by
two
,
if
one
be
an
unite
,
it
may
be
performed
compendiously
thus
.
Example
6.
It
is
required
to
know
|
112
|
Multiplicand
|
A
896
|
The
Product
|
B
5376
|
The
totall
|
A
,
B
,
14336
|
how
many
ounces
there
be
in
800
li.
of
the
Merchants
weight
;
first
set
down
their
hundred
,
which
is
112
li.
then
draw
a
line
underneath
it
,
and
8
being
the
Multiplier
,
you
may
keep
it
in
minde
,
and
say
8
times
2
is
16
,
that
is
6
and
goe
1
:
then
8
times
1
is
8
&
1
will
be
9
,
which
subscribe
,
then
say
again
8
times
1
is
but
8
,
which
having
set
down
,
the
Product
of
8
hundred
weight
is
896
lib.
as
in
the
Table
,
which
is
to
be
reduced
into
ounces
,
16
of
them
making
1
li.
an
unite
being
here
in
the
second
place
towards
the
left
hand
multiplies
not
,
but
will
produce
the
Multiplicand
,
and
remove
it
one
degree
farther
(
according
to
my
former
prescribed
rules
of
Multiplication
)
therefore
in
all
such
cases
as
these
,
you
may
neglect
the
unite
,
and
keep
the
other
figure
in
minde
for
your
Multiplier
,
and
place
your
Product
a
degree
backward
,
as
in
this
example
,
A
is
the
Multiplicand
for
to
be
encreased
by
16
,
or
6
kept
in
minde
,
then
say
6
times
6
is
36
,
that
is
6
and
goe
3
decimalls
,
then
6
times
9
is
54
and
3
will
be
57
;
which
is
7
and
reserve
5
;
then
say
6
times
8
will
make
48
and
5
is
53
,
which
figures
subscribe
,
there
being
no
more
to
be
multiplied
,
so
this
Product
is
at
B
5376
,
which
is
placed
one
degree
towards
the
right
hand
,
whereby
the
unite
place
in
A
the
Multiplicand
,
may
stand
in
the
decimall
place
of
the
Product
,
and
so
it
would
have
done
according
to
the
common
way
of
Multiplication
,
but
then
beneath
it
,
as
't
is
now
above
it
:
these
two
numbers
must
be
added
together
,
whose
totall
is
A
,
B
,
14,336
,
the
true
number
of
ounces
in
8
C
lib.
grosse
weight
;
the
line
drawn
between
A
&
B
is
here
for
directions
onely
,
and
in
your
work
is
better
omitted
.
How
to
multiply
the
shortest
way
with
two
figures
,
when
the
first
figure
on
the
right
hand
is
an
unite
.
Example
7.
It
is
required
to
finde
Multiplicand
|
6336
|
Product
|
25344
|
The
totall
|
259776
|
how
many
inches
there
be
in
41
Miles
,
one
containing
320
Perches
,
every
Perch
in
length
16
feet
and
a
half
,
and
12
inches
the
length
of
one
Foot
;
now
in
a
Pole
there
is
33
half
feet
,
which
multiplied
by
6
inches
the
Product
will
be
198
,
the
number
of
inches
in
a
Perch
or
Pole
,
that
multiplied
by
320
,
the
Perches
in
a
Mile
,
or
by
32
,
the
Product
will
be
6336
,
to
which
adde
a
cypher
and
the
Product
is
then
63,360
,
the
number
of
inches
in
a
Mile
,
and
41
is
the
distance
given
:
now
omitting
the
cypher
,
set
down
6336
which
is
the
Multiplicand
,
and
41
is
to
be
the
Multiplier
,
an
unite
can
onely
produce
6336
,
and
being
the
1
is
to
the
right
hand
,
set
the
Product
of
the
other
figure
one
degree
towards
the
left
hand
,
keeping
that
Multiplier
in
your
minde
,
and
say
,
4
times
6
is
24
,
that
is
4
and
goe
2
,
then
4
times
3
is
12
,
and
2
makes
14
,
that
is
4
and
goe
1
,
then
4
times
3
is
12
and
1
decimall
in
minde
is
13
,
that
is
3
and
goe
1
,
then
4
times
6
is
24
and
1
is
25
;
the
totall
is
259776
,
to
which
annex
the
cypher
omitted
on
the
right
hand
,
the
totall
is
2,597,760
,
the
number
of
inches
contained
in
41
Miles
,
the
thing
sought
.
Example
8.
If
you
are
to
multiply
by
11
,
28800
|
28800
|
28800
|
28800
|
316800
|
316800
|
it
is
all
one
whether
you
place
the
number
given
as
the
Product
,
a
degree
forward
or
backward
,
for
it
comprehends
both
the
last
examples
in
one
,
as
in
this
Paradigma
will
appeare
,
there
are
11
halfe
Yards
or
Cubes
,
in
the
length
of
a
Perch
,
320
makes
a
Mile
,
3
Miles
a
League
,
and
the
distance
from
London
to
Norwich
is
30
Leagues
,
and
how
many
Cubes
or
halfe
Yards
are
there
conta●n'd
between
those
Cities
,
in
one
League
there
are
960
Perches
,
in
30
Leagues
28,800
Perches
,
and
this
Multiplied
by
11
is
the
question
required
,
in
all
such
cases
you
have
no
more
to
doe
,
but
subscribe
the
same
number
againe
,
which
was
to
be
Multiplied
,
observing
to
remove
all
the
figures
one
place
or
degree
either
backwards
or
forwards
,
the
summe
of
those
two
numbers
will
give
the
product
,
as
you
may
see
in
the
example
;
all
these
wayes
are
as
easie
as
shorr
,
if
well
observ'd
:
but
otherwise
much
more
subject
to
errour
then
the
common
way
.
Two
severall
explications
of
Multiplication
continued
.
Example
9.
These
which
some
calls
continuall
Multiplication
,
is
nothing
but
the
Multiplying
of
severall
numbers
into
one
product
;
that
is
,
the
third
number
multiplied
into
the
product
of
the
first
and
second
,
and
that
into
a
fourth
,
&c.
As
for
example
.
4
,
7
,
10
,
13
,
admit
the
numbers
given
,
't
is
all
one
where
you
begin
but
let
it
be
at
4
,
and
7
multiplied
makes
28
,
that
by
10
will
be
280
,
and
this
product
multiplied
by
13
makes
3640
,
the
first
question
solved
;
secondly
when
the
Multiplicand
consists
of
severall
denominations
,
as
5
L.
6
S.
8
D.
to
be
increased
by
6
produceth
32
L.
viz.
6
by
8
is
48
,
a
cypher
and
goe
4
S.
then
6
times
6
is
36
,
and
4
in
mind
is
40
S.
a
cypher
and
goe
2
L.
then
6
times
5
is
30
,
and
2
is
32
L.
DIVISION
.
Parag:
5.
The
fourth
species
.
DIvision
is
to
separate
any
quantity
given
into
any
parts
assign'd
,
or
to
finde
how
often
one
number
is
contained
in
any
other
;
or
from
any
two
numbers
given
to
finde
a
third
,
that
shall
consist
of
so
many
unites
as
the
one
number
is
comprehended
in
the
other
.
To
Division
there
belongs
three
principle
members
or
parts
,
viz.
the
Dividend
,
the
Divisor
,
and
the
Quotient
.
The
Dividend
is
the
number
given
for
to
be
divided
,
and
is
the
greater
number
.
The
Divisor
is
the
number
by
which
you
must
divide
,
and
is
the
lesser
number
.
The
Quotient
is
that
number
,
which
is
produced
by
division
,
and
containes
so
many
unites
,
as
the
Divisor
is
comprehended
in
the
Dividend
,
from
whence
this
Analogy
or
proportion
proceeds
,
viz.
as
the
Dividend
is
to
the
Divisor
,
so
will
the
Quotient
be
unto
an
unite
,
and
the
contrary
;
as
the
Divisor
is
to
an
unite
,
so
will
the
Dividend
be
to
the
Quotient
.
Division
proved
by
Arithmeticall
and
Geometricall
Demonstrations
.
To
illustrate
this
Species
,
there
was
a
man
had
6
Daughters
,
to
whom
he
bequeath'd
30
fields
or
inclosures
,
all
of
equall
value
and
content
;
and
so
,
as
af-after
his
de●th
,
every
one
should
have
an
equall
portion
or
share
in
the
land
,
and
if
any
died
before
they
married
,
the
rest
were
to
have
it
divided
amongst
them
equally
;
suppose
all
the
fields
to
be
inclos'd
with
the
square
A
,
B
,
C
,
D
,
the
Father
departed
,
the
land
was
thus
divided
:
30
is
the
number
of
fields
,
which
is
the
Dividend
,
6
the
number
of
Daughters
must
be
the
Divisor
,
as
A
,
B
,
then
will
A
,
D
,
5
or
B
,
C
,
be
the
Quotient
,
for
6
is
contained
in
30
,
5
times
,
that
is
so
often
as
there
be
unites
in
the
Quotient
,
as
by
the
number
of
little
squares
is
evident
,
within
a
yeare
,
5
of
these
coheires
married
,
and
the
youngest
was
prevented
against
her
will
you
know
by
untimely
Death
who
by
this
meanes
caus'd
more
divisions
in
the
estate
,
for
now
5
was
the
Divisor
,
and
6
the
Quotient
,
for
so
many
times
5
is
contained
in
30
,
which
is
in
proportion
to
A
,
D
,
5
as
A
,
B
,
6
is
unto
an
unite
,
this
is
directly
contrary
to
Multiplication
,
for
A
,
B
,
multiplied
into
A
,
D
,
or
which
is
all
one
C
,
B
,
in
C
,
D
,
will
make
30
,
for
the
whole
square
A
,
B
,
C
,
D
,
which
divided
by
one
of
the
sides
will
produce
the
other
,
and
so
any
number
whatsoever
that
is
multiplied
by
an
other
,
will
be
divided
by
the
same
,
and
Multiplication
and
Division
,
doe
try
and
prove
one
an
other
,
as
shall
be
showne
hereafter
.
This
is
the
foundation
,
and
now
I
will
shew
the
operation
of
it
,
illustrated
with
divers
Paradigmas
.
Example
1.
There
was
a
man
who
by
his
industry
had
gained
963
li.
and
falling
sick
(
past
hope
of
recovery
)
made
his
Will
;
he
had
three
Sonnes
,
to
whom
he
bequeathed
his
estate
impartially
,
not
swayed
by
the
custome
of
the
Countrey
,
nor
over-ruled
by
his
affection
,
not
knowing
which
of
them
would
deserve
it
best
,
he
distributed
his
estate
equally
to
them
all
,
which
after
his
departure
was
accordingly
divided
;
and
first
of
all
in
division
set
down
the
Dividend
,
which
in
this
example
is
963
li:
the
Divisor
is
3
,
which
place
under
the
first
figure
on
the
left
hand
viz.
9
,
then
see
how
many
times
3
is
contain'd
in
9
,
which
will
be
found
3
times
;
and
having
made
an
arch
,
or
such
a
line
,
as
in
the
example
(
upon
the
right
hand
of
the
Dividend
)
set
downe
the
Quotient
,
which
must
be
multiplied
into
the
Divisor
,
and
the
product
subtracted
from
the
Dividend
as
3
times
3
is
9
,
take
9
out
of
9
,
and
nothing
remaines
;
then
remove
the
Divisor
one
place
or
degree
towards
the
right
hand
,
as
under
6
;
then
finde
how
many
times
3
will
be
contain'd
in
6
,
which
is
twice
,
therefore
put
2
in
the
Quotient
,
which
multiplied
by
3
makes
6
,
which
take
from
6
the
Dividend
,
and
nothing
remaines
,
remove
3
againe
to
the
next
place
,
as
under
3
,
which
is
there
contain'd
but
once
,
put
1
in
the
Quotient
,
which
times
3
is
but
3
,
and
that
taken
from
3
the
Dividend
,
nothing
remaines
;
every
ones
share
being
321
L
,
and
so
many
times
3
the
Divisor
is
contain'd
in
963
L
,
the
Dividend
,
if
any
thing
remaines
in
making
of
subtraction
place
it
over
the
figure
or
figures
,
from
whence
subtracted
,
and
cancell
those
figures
you
have
done
with
all
,
With
a
light
dash
of
your
pen
,
and
if
any
thing
remaines
at
last
,
place
it
beneath
your
worke
,
in
this
example
nothing
remaines
,
in
whi●h
you
may
see
the
forme
,
and
precedents
in
those
following
.
Example
2.
There
was
a
man
beyond
sea
who
had
received
of
a
Merchants
Factor
by
Bills
of
Exchange
1726
Crownes
,
and
't
is
required
to
know
how
many
pounds
Sterling
it
is
;
4
of
those
pieces
making
20
shillings
or
1
lb.
therefore
4
must
be
the
Divisor
,
having
set
down
1726
the
Dividend
,
and
made
a
place
for
the
Quotient
,
4
being
not
contained
in
an
unite
,
therefore
set
4
under
7
,
and
see
how
many
times
it
will
be
had
in
the
compound
number
over
it
,
viz:
in
17
,
and
you
will
find
4
times
,
then
put
4
in
the
Quotient
,
which
multiplied
into
the
Divisor
,
the
Product
is
16
,
which
taken
from
17
,
there
will
remain
main
1
,
which
place
over
7
,
and
cancell
17
of
the
Dividend
;
and
4
the
Divisor
,
place
again
under
2
;
then
seek
how
many
times
it
will
be
contained
in
12
(
which
is
over
it
uncancelled
)
and
you
will
finde
3
,
which
put
into
the
Quotient
,
and
that
multiplied
by
4
,
the
Product
is
12
,
which
subtracted
from
12
nothing
remains
,
for
order
place
a
cypher
over
2
,
and
cancell
the
12
:
lastly
,
remove
4
the
Divisor
under
6
,
in
which
,
4
is
but
once
contained
,
therefore
put
1
in
the
Quotient
,
and
subtract
4
from
6
,
and
2
will
remain
,
which
write
down
under
all
the
work
;
where
you
may
see
1726
Crowns
,
being
divided
by
4
(
the
parts
of
a
pound
Sterling
)
the
Quotient
is
431
lib.
and
2
Crowns
or
10
shillings
more
,
as
in
the
operation
appears
.
Example
3.
Here
is
given
2000
Drams
,
and
it
is
required
to
know
how
many
Ounces
there
be
contained
in
that
weight
;
being
8
Drams
doe
make
an
Ounce
,
8
is
the
Divisor
,
which
cannot
be
had
in
2
,
the
first
figure
of
the
Dividend
,
therefore
set
it
under
the
second
place
towards
the
right
hand
,
and
then
see
how
many
times
it
is
contained
in
both
those
figures
,
viz:
in
20
,
3
will
be
too
great
,
because
3
times
8
is
24
,
therefore
take
2
,
and
set
it
in
the
Quotient
,
and
multiply
it
into
the
Divisor
,
saying
,
2
times
8
is
16
,
which
Product
subtract
from
20
and
4
remains
,
which
place
over
the
cypher
;
then
remove
8
the
Divisor
to
the
next
place
towards
the
right
hand
,
and
look
how
often
8
will
be
contained
in
40
,
and
that
is
found
5
times
just
,
which
put
in
the
Quotient
,
and
say
5
times
8
is
40
,
take
40
out
of
40
,
and
nothing
remaines
,
and
cancell
as
you
goe
all
the
figures
you
have
done
withall
;
then
remove
8
to
the
next
place
,
which
is
under
the
last
cypher
,
and
being
there
is
nothing
more
,
put
a
cypher
in
the
Quotient
,
and
the
worke
is
ended
,
the
Quotient
250
Ounces
,
nothing
remaining
,
and
so
many
will
be
found
in
2000
Drams
,
the
question
solved
,
hitherto
hath
been
examples
with
but
one
figure
in
the
Divisor
,
the
next
with
many
.
How
to
divide
,
when
the
Divisor
consists
of
more
figures
than
one
.
Example
4.
There
is
a
distance
given
between
two
places
containing
in
feet
15000
,
and
it
is
required
to
know
how
many
stadiums
it
is
,
one
consisting
of
625
feet
,
8
of
them
making
an
Italian
Mile
,
or
a
thousand
paces
,
3
of
those
Miles
a
League
.
Here
in
this
question
15000
feet
is
the
Dividend
,
or
number
to
be
divided
,
625
the
Divisor
,
the
number
of
feet
in
one
stadium
:
observe
(
as
before
)
to
place
the
formost
figure
of
the
Divisor
under
the
first
figure
upon
the
sinister
hand
of
the
Dividend
,
if
it
will
contain
it
,
and
the
other
figures
following
;
if
not
,
remove
it
one
place
towards
the
right
hand
,
and
then
set
down
the
rest
of
the
Divisor
in
order
:
this
done
,
you
are
to
chuse
such
a
figure
for
the
Quotient
,
as
that
being
multiplied
into
all
the
figures
of
the
Divisor
,
the
Product
shall
be
equall
to
the
figures
over
them
,
or
the
nearest
lesse
;
which
you
may
finde
by
Multiplication
,
but
by
the
first
figure
of
the
Divisor
sooner
,
and
as
certaine
,
if
you
observe
the
figures
following
,
and
by
practice
will
be
made
easie
,
in
this
example
625
is
the
Divisor
,
the
first
figure
6
,
cannot
be
had
in
1
,
the
formost
of
the
Dividend
,
therefore
place
6
under
5
and
the
rest
in
order
,
under
the
other
figures
:
then
looke
how
many
times
6
will
be
found
in
15
,
3
will
be
too
great
,
for
3
times
6
is
18
;
then
take
2
,
which
put
in
the
Quotient
,
and
then
multiply
it
,
into
the
first
figure
on
the
right
hand
of
the
Divisor
viz.
2
times
5
is
10
,
over
which
stands
a
cypher
,
subtract
10
from
10
and
nothing
remaines
,
set
a
cypher
and
goe
1
,
then
say
2
times
2
is
4
and
1
borrowed
is
5
,
take
5
from
10
and
5
will
remaine
,
which
set
over
the
cypher
,
and
cancell
0
,
and
goe
1
,
then
say
2
times
6
is
12
and
1
borrowed
is
13
,
take
13
from
15
and
2
will
remaine
,
cancell
15
;
and
the
Divisor
as
you
make
subtraction
:
this
done
,
remove
the
Divisor
one
degree
farther
,
as
in
this
example
,
set
6
below
2
of
the
former
divisor
,
2
under
5
,
and
5
,
under
the
last
cypher
here
of
the
Dividend
:
then
see
how
often
will
6
be
in
25
(
as
yet
uncancell'd
)
and
't
will
be
found
4
times
,
put
4
in
the
Quotient
,
and
then
say
4
times
5
is
20
,
over
which
stands
a
cypher
,
take
20
from
20
and
nothing
remaines
,
set
a
cypher
over
it
,
and
keep
2
decimalls
in
minde
,
then
say
4
times
2
makes
8
,
and
2
will
be
10
,
which
subtracted
from
10
nothing
remaines
,
place
a
cypher
over
it
,
and
cancell
the
figures
when
subtraction
is
made
,
then
say
4
times
6
is
24
and
1
in
minde
is
25
,
which
taken
from
25
nothing
remains
,
the
reason
,
of
placing
cyphers
when
nothing
remaines
is
,
that
there
may
be
a
significant
figure
remaining
after
them
to
the
left
hand
,
which
is
in
value
according
to
its
place
or
degree
;
and
observe
that
the
remainder
is
lesse
then
the
Divisor
,
or
the
worke
is
false
in
multiplying
or
subtracting
,
in
this
example
24
stadiums
is
the
Quotient
,
that
is
one
League
,
or
3
Miles
,
for
so
many
times
is
8
contained
in
24
,
the
thing
required
.
To
divide
a
number
of
divers
denominations
,
into
any
parts
assigned
.
Example
5.
To
effect
this
,
|
Li.
|
S.
|
D
|
Q.
|
A
|
71024
|
16
|
3
|
0
|
B
|
3945
|
16
|
5
|
2
|
there
are
2
wayes
,
the
one
is
for
to
reduce
them
into
one
and
the
least
denomination
,
and
then
for
to
divide
that
summe
by
the
number
given
:
and
if
there
be
any
remainder
,
you
may
reject
it
,
as
not
of
common
use
:
when
this
is
done
,
reduce
the
Quotient
into
severall
denominations
again
:
as
admit
A
were
the
number
given
in
pounds
,
shillings
,
and
pence
,
for
to
be
divided
into
18
equall
parts
:
this
reduced
into
the
least
denomination
of
Coyne
(
as
Farthings
)
will
be
in
one
summe
68,183,820
Q.
this
divided
by
18
,
will
be
3,787,990
Q.
which
divided
by
4
,
the
number
of
Farthings
in
a
Peny
,
the
Quotient
will
be
946
997
D.
2
Q.
the
Pence
divided
by
12
will
produce
78916
,
S
,
5
D
,
the
shillings
againe
divided
by
20
(
so
many
making
a
Pound
sterling
)
the
Quotient
will
be
3945
L
,
16
S
the
remainder
,
so
the
18
part
of
A
71,024
L.
16
S.
3
D.
will
be
B
3945
L.
16
S.
5
D.
2
Q.
This
is
a
common
way
to
those
who
are
not
conversant
in
fractions
,
but
this
following
is
better
,
shorter
and
not
so
subject
to
errour
,
and
thus
.
A
more
compendious
way
.
First
set
down
the
greatest
denomination
if
the
Divisor
will
be
contained
in
it
;
if
not
,
multiply
it
into
a
lesse
denomination
by
the
parts
,
and
to
the
Product
adde
the
parts
of
that
denomination
(
if
there
be
any
)
and
then
divide
it
by
the
given
Divisor
;
and
if
there
proves
a
remainder
,
multiply
that
by
the
parts
of
the
next
denomination
,
and
so
proceed
to
the
last
,
as
in
this
example
71024
li.
is
the
Dividend
,
18
the
Divisor
:
the
operation
will
be
as
in
the
former
Paradigma's
,
by
which
you
will
finde
here
a
Quotient
of
3945
li.
and
the
remainder
14
li.
The
last
remainder
was
14
li.
which
multiplied
by
20
shillings
(
the
parts
of
a
pound
sterling
)
the
Product
will
be
280
shillings
,
to
which
adde
16
S.
the
summe
296
for
the
Dividend
,
which
divided
by
18
,
the
Quotient
will
be
16
shillings
,
and
the
remainder
8
shillings
.
In
the
last
division
8
s.
was
the
remainder
,
and
being
there
are
12
pence
in
1
shilling
,
that
multiplied
by
12
the
Product
is
96
pence
,
to
which
adde
in
the
3
D
in
the
question
,
the
summe
will
be
99
for
the
Dividend
,
under
which
subscribe
the
Divisor
18
and
divide
:
the
Quotient
is
5
D.
the
remainder
9
D.
The
remainder
last
being
9
Den.
which
multiplied
by
4
makes
36
,
the
Product
in
Farthings
,
which
divided
by
18
,
the
Quotient
is
found
2
Farthings
,
and
nothing
remaining
,
so
the
18
part
of
71024
Lib.
16
S.
3
D.
is
found
to
be
3945
Lib.
16
S.
5
D.
2
Q.
as
before
.
A
succinct
and
perfect
way
of
Division
,
without
removing
the
Divisor
,
or
cancelling
a
figure
.
Example
6.
This
way
of
all
oother
that
I
have
seene
is
the
most
compendious
,
facile
,
and
conformable
to
practice
,
this
species
of
Division
being
contrary
to
Multiplication
,
as
in
the
effect
,
so
in
operation
,
as
by
their
trialls
will
more
evidently
appeare
,
and
first
of
all
,
you
are
to
observe
to
set
downe
the
Dividend
,
and
upon
the
left
hand
of
it
(
in
the
same
line
)
the
Divisor
;
then
draw
a
line
right
downe
with
your
pen
,
to
part
the
numbers
,
and
draw
an
other
line
unto
that
,
for
to
separate
the
Divisor
from
the
Quotient
,
which
is
to
stand
under
it
,
as
in
this
example
,
where
1638
yeares
or
dayes
are
given
for
to
be
divided
by
7
,
make
a
point
with
your
pen
,
under
that
figure
of
the
Dividend
,
which
first
containes
the
Divisor
,
reckoned
from
the
formost
figure
of
the
left
hand
,
supposing
the
point
for
to
be
a
period
to
that
number
;
as
here
I
make
a
point
under
6
,
for
7
cannot
be
had
in
one
,
but
in
16
it
may
,
and
twice
,
set
2
in
the
Quotient
under
the
line
,
beneath
the
Divisor
7
,
which
multiplied
together
is
14
,
that
subtracted
from
16
the
remainder
is
2
which
place
under
6
,
in
this
forme
and
method
you
must
proceed
,
never
subscribing
the
products
,
but
committing
them
to
memory
,
for
they
serve
but
till
subtraction
can
be
made
,
therefore
unnecessary
,
and
would
incumber
the
worke
,
with
supernumerarie
figures
,
for
which
cause
they
were
omitted
in
all
my
former
examples
,
and
yet
this
a
briefer
way
,
as
will
evidently
appear
,
make
an
other
point
even
with
the
remainder
2
,
and
under
the
next
place
of
the
Dividend
as
under
3
,
and
aske
how
many
times
7
will
be
found
in
23
,
the
answer
will
be
3
,
then
say
3
times
7
is
21
,
which
taken
from
23
,
and
2
will
remaine
,
which
subscribe
under
the
last
point
,
and
make
an
other
against
it
under
8
;
then
see
how
often
7
will
be
had
in
28
,
which
will
be
found
4
times
,
which
multiplied
by
the
Divisor
7
,
the
product
will
be
28
,
which
taken
from
28
nothing
remaines
,
the
Quotient
234.
To
finde
the
Cycle
of
the
Moon
,
or
Golden
number
.
Example
7.
This
Prime
or
Cycle
of
the
Moon
is
a
number
proceeding
from
an
unite
encreasing
every
year
1
,
unto
19
,
and
then
begins
again
,
this
revolution
is
to
finde
the
difference
betwixt
the
Solar
and
Lunary
yeare
;
for
in
this
period
of
time
all
the
Lunations
and
Aspects
of
the
two
Luminaries
doe
returne
againe
to
the
same
places
as
they
had
been
before
;
but
this
computation
exceeds
the
truth
in
time
1
houre
and
a
halfe
in
19
yeares
or
very
neere
,
this
number
may
be
alwayes
thus
found
:
Our
Blessed
Saviour
was
borne
in
the
first
yeare
of
this
Number
,
which
you
will
finde
,
if
you
adde
one
to
the
yeare
of
ou●
Lord
Jesus
,
and
divide
the
summe
by
19
,
the
remainder
is
the
Golden
number
,
and
when
nothing
remaines
't
is
19
,
the
Quotient
is
the
revolution
or
period
of
this
Cycle
,
since
the
Sacred
Virgin
was
a
Mother
,
and
for
the
more
illustration
of
it
,
take
this
Paradigma
;
in
the
yeare
of
Grace
,
1654.
the
Prime
is
required
,
adde
1
to
it
,
and
then
the
number
is
1655
,
to
be
divided
by
19
,
which
divisor
place
on
the
left
hand
of
it
,
parted
with
a
line
both
from
thence
,
and
from
the
Quotient
to
be
found
,
as
in
the
example
;
then
under
the
first
5
make
a
point
,
for
although
1
is
contained
in
1
,
9
cannot
be
had
in
6
,
this
done
see
how
often
1
will
be
found
in
16
,
if
I
take
9
(
which
is
the
greatest
number
can
be
had
at
one
time
)
't
will
be
too
great
,
for
9
times
1
out
of
16
,
there
will
remaine
but
7
,
when
9
times
9
will
be
81
out
of
75
,
therefore
set
8
in
the
Quotient
,
and
say
8
times
9
is
72
,
taken
from
75
,
and
3
will
remaine
,
which
subscribe
under
the
point
,
then
say
8
times
1
is
8
,
and
7
decimalls
in
minde
is
15
,
which
taken
from
16
,
and
1
remaines
,
that
subscrib'd
,
make
another
point
against
3
,
under
the
next
place
,
&
see
how
often
19
will
be
contained
in
135
,
or
1
in
13
(
the
rest
of
the
Divisor
considered
)
and
you
will
finde
7
the
neerest
,
which
orderly
place
in
the
Quotient
,
and
with
that
multiply
alwayes
by
the
first
figure
in
the
Divisor
upon
the
right
hand
,
as
7
times
9
produceth
63
,
which
taken
from
65
;
or
which
is
all
one
,
take
3
out
of
5
and
2
remaines
,
which
place
under
the
point
,
then
say
7
times
1
is
7
,
and
6
decimalls
in
minde
makes
13
,
which
taken
from
13
nothing
remaines
;
so
there
is
onely
2
remaining
for
the
Golden
number
this
present
yeare
;
and
87
revolutions
of
this
Cycle
have
been
compleated
,
and
1
yeare
more
since
the
Nativity
of
Christ
.
To
finde
the
Cycle
of
the
Sun
for
the
Dominicall
Letter
any
yeare
.
Example
8.
This
is
a
number
proceeding
by
unites
from
1
,
2
,
3
,
&c.
And
so
to
28
;
it
hath
no
relation
to
the
motion
of
the
Sunne
,
but
to
the
Sunday
Letter
,
which
are
the
first
7
of
the
Alphaber
,
serving
by
turnes
in
a
retrograde
order
,
and
every
fourth
yeare
hath
2
Dominicall
Letters
,
so
4
times
7
is
the
period
of
this
number
,
every
letter
having
serv'd
in
course
,
both
in
the
common
,
and
Leap
yeares
,
and
when
this
number
hath
had
a
period
of
28
,
the
next
yeare
it
begins
againe
at
1
:
the
Worlds
Redeemer
was
born
in
the
ninth
yeare
of
this
number
;
therefore
adde
9
to
the
yeare
of
our
Lord
,
in
which
it
is
required
,
divide
the
summe
by
28
,
and
if
nothing
remaines
't
is
the
number
sought
,
and
if
there
does
,
that
reremainder
is
the
number
required
for
the
yeare
currant
in
the
Julian
Kalender
;
and
the
Quotient
will
shew
the
revolutions
past
,
as
in
this
example
,
where
the
Cycle
of
the
Sun
is
required
for
the
yeare
of
Christ
1654
;
unto
which
adde
9
,
and
the
summe
will
be
1663
for
the
Dividend
,
and
28
the
Divisor
:
make
a
point
under
the
second
6
from
1
,
and
then
see
how
many
times
2
may
be
contained
in
16
;
6
will
be
too
great
,
for
6
multiplied
by
28
the
product
will
be
168
,
whereas
the
3
first
figures
are
but
166
,
therefore
take
5
for
the
first
Quotient
,
which
in
order
multiplied
by
the
former
rules
into
the
Divisor
,
and
the
product
subtracted
from
the
Dividend
,
the
remainder
is
26
,
before
which
make
an
other
point
under
3
the
last
figure
of
the
Dividend
,
and
finde
a
new
Quotient
,
which
in
this
will
be
9
,
that
multiplied
,
and
the
product
subtracted
,
the
true
remainder
will
be
11
,
the
number
desired
for
the
yeare
1654
by
which
Cycle
,
the
Dominicall
Letter
will
be
found
A
,
the
Quotient
59
,
the
number
of
those
Circles
revolutions
since
our
Saviours
Birth
:
but
here
I
will
say
no
more
of
this
,
lest
I
have
said
too
much
already
,
intending
here
Arithmetick
,
and
not
the
computation
of
Kalenders
;
but
these
two
questions
or
propositions
I
rather
chose
,
to
shew
how
ready
for
use
the
remainder
stands
,
being
in
these
the
thing
chiefly
required
,
whereas
in
other
questions
(
excepting
such
as
are
of
severall
denominations
)
the
remainder
will
be
unknowne
,
without
the
knowledge
of
Fractions
.
A
number
of
divers
denominations
given
,
for
to
be
divided
into
any
parts
required
.
Example
9.
Dayes
|
Hours
|
′
|
″
|
‴
|
'
'
'
'
|
|
730
|
11
|
37
|
0
|
0
|
0
|
A
|
The
magnitude
of
two
Yeares
.
|
A
Solar
Moneth
contains
|
30
|
10
|
29
|
2
|
30
|
|
B
|
Here
is
given
the
space
of
2
yeares
,
and
it
is
required
for
to
know
the
greatnesse
of
a
Solar
moneth
,
vulgarly
accounted
the
space
of
30
dayes
,
a
naturall
day
containing
24
houres
every
houre
60
minutes
,
commonly
noted
with
such
a
dash
over
it
as
thus
,
60
of
them
making
one
Second
noted
with
2
dashes
of
a
pen
,
every
Second
is
divided
into
60
parts
call'd
Thirds
sign'd
with
3
dashes
;
Fourths
with
4
&c.
There
are
in
2
yeares
space
24
Moneths
,
therefore
the
number
given
,
divided
into
24
equall
parts
solves
the
question
here
propounded
;
this
may
be
effected
by
the
fift
Example
,
but
better
by
the
way
following
,
in
division
of
each
particular
part
or
denomination
thus
.
The
number
of
dayes
in
2
yeares
are
730
,
which
must
be
divided
by
24
,
the
number
of
moneths
contained
in
that
space
of
time
,
having
placed
your
numbers
in
order
prescribed
,
make
a
point
under
3
in
the
Dividend
,
which
done
you
will
finde
3
for
the
Quotient
,
that
multiplied
into
the
Divisor
,
and
the
Product
subtracted
from
the
Dividend
,
the
remainder
will
be
found
,
then
make
an
other
point
a
place
farther
,
as
under
the
cypher
,
and
being
24
is
not
to
be
had
in
10
,
put
a
cypher
in
the
Quotient
,
which
is
30
dayes
,
and
the
remainder
10
,
set
down
30
in
the
former
Table
under
dayes
.
The
last
remainder
was
10
dayes
,
which
convert
into
hours
the
next
denomination
lesse
,
by
multiplying
of
it
into
24
(
the
houres
contained
in
a
naturall
day
)
the
Product
will
be
240
,
to
which
adde
11
houres
(
being
parts
of
the
2
yeares
given
)
the
summe
will
be
251
hours
:
which
divide
by
24
,
the
Quotient
will
be
10
houres
,
and
the
remainder
11
,
place
10
in
the
first
Table
against
B
under
the
title
of
Hours
.
The
last
remainder
was
11
houres
,
which
multiplied
by
60
,
the
minutes
in
one
houre
,
the
Product
is
660
,
to
which
adde
37
minutes
against
A
,
the
summe
will
be
697
for
the
Dividend
,
which
divided
by
24
the
Quotient
will
be
29
minutes
,
and
the
remainder
1
,
set
29′
against
B
in
the
columne
of
minutes
.
In
the
last
operation
there
was
but
1
minute
remaining
,
which
is
60
seconds
,
that
divided
by
24
,
the
Quotient
will
be
2
seconds
,
and
the
remainder
12
:
place
the
2
,
in
the
first
Table
against
B
,
under
the
title
of
Seconds
.
The
12
Seconds
which
did
last
remaine
,
multiplied
into
Thirds
(
the
next
denomination
lesse
)
the
Product
will
be
720
,
which
divided
by
24
,
the
Quotient
will
be
30
Thirds
,
and
nothing
remaining
,
if
any
thing
bad
,
you
might
have
continued
them
in
this
manner
to
Fourths
&c.
The
last
Quotient
put
into
the
Table
,
you
will
finde
against
B
,
that
a
Solar
Moneth
containes
30
Dayes
,
10
Hours
,
29
Minutes
,
2
Seconds
,
and
30
Thirds
.
Breviates
,
or
compendious
wayes
and
observations
in
Division
exemplified
.
Example
10.
If
you
are
to
divide
by
|
5754
|
7101
|
|
1
|
|
|
4
|
|
2877
|
789
|
|
|
13701
|
10,000
|
|
2
|
|
|
3
|
|
4567
|
2500
|
|
any
significant
figure
onely
,
it
is
unnecessary
for
to
set
down
the
Divisor
,
but
to
keep
it
in
minde
and
the
product
as
before
,
while
subtraction's
made
,
the
Multiplier
will
be
the
Quotient
in
all
such
cases
,
as
by
these
4
Paradigma's
will
appear
;
the
first
is
the
yeare
since
the
Creation
of
the
World
,
for
to
be
divided
into
two
equall
parts
;
make
a
point
under
5
,
wherein
2
is
twice
contained
and
1
remains
;
then
point
forward
:
2
will
be
in
17
,
8
times
:
then
2
in
15
,
7
times
:
and
lastly
,
2
in
14
,
7
times
,
the
Quotient
2877.
In
the
second
Table
there
is
given
13701
Nobles
,
one
being
in
value
6
s.
8
d.
3
of
them
makes
20
s.
and
it
is
required
to
know
how
many
pounds
sterling
it
will
make
in
all
:
divide
by
3
,
and
say
3
will
be
found
in
13
,
4
times
;
in
17
,
5
times
;
in
20
,
6
times
;
and
in
21
,
7
times
;
the
Quotient
is
4567
li.
In
the
third
Example
10,000
Crownes
are
received
beyond
sea
,
and
it
is
required
to
know
how
many
pounds
sterling
must
be
paid
for
them
in
England
,
4
of
them
making
20
shillings
,
make
a
point
under
the
cypher
where
4
will
be
had
in
10
,
2
times
;
in
20
,
5
times
;
and
being
nothing
remains
but
2
cyphers
more
towards
the
right
hand
,
annex
them
to
it
,
and
it
is
done
,
the
Quotient
being
2500
li.
Fourthly
,
there
is
7101
li.
for
to
be
equally
distributed
unto
9
men
;
9
cannot
be
had
in
7
,
therefore
make
a
point
under
the
next
place
,
and
then
9
will
be
found
in
71
,
7
times
;
and
next
in
80
,
8
times
;
and
lastly
,
in
81
,
9
times
;
the
Quotient
789
li.
And
so
much
does
every
one
of
their
shares
come
unto
.
When
there
is
an
unite
with
cyphers
annext
unto
a
Divisor
,
cut
off
so
many
places
upon
the
right
hand
of
the
Dividend
,
as
there
were
cyphers
in
the
Divisor
:
as
for
example
,
120
li.
is
to
be
distributed
equally
to
10
men
,
for
the
cypher
in
the
Divisor
cut
off
one
place
in
the
Dividend
,
and
12
li.
will
be
every
ones
part
or
the
Quotient
;
or
if
100
Souldiers
were
to
receive
for
their
pay
625
li.
every
ones
share
will
be
6
li.
and
25
li.
over
,
which
by
the
former
rules
converted
into
shillings
,
by
being
multiplied
by
20
,
the
product
will
be
500
s.
from
whence
cut
off
the
2
cyphers
,
and
there
will
remaine
5
s.
So
6
li.
5
s.
every
Soldier
must
receive
.
And
so
for
any
other
of
this
kinde
:
the
reason
is
evident
,
for
1
divides
nothing
,
and
the
Quotient
must
have
so
many
places
lesse
than
the
Dividend
,
by
the
number
of
cyphers
in
the
right
hand
of
the
Divisor
,
as
by
the
common
way
of
Division
will
plainly
appear
.
If
the
Divisor
consists
of
any
significant
figure
or
figures
in
the
formost
place
,
and
a
cypher
or
cyphers
to
the
right
hand
,
leave
out
the
cyphers
in
the
Divisor
,
and
cut
off
so
many
places
upon
the
right
hand
of
the
Dividend
,
and
with
the
residue
divide
,
and
when
the
division
is
done
,
annex
the
cyphers
to
the
D
visor
,
and
to
the
remainder
,
the
figures
that
were
severed
from
the
Dividend
,
both
of
them
constituting
a
Fraction
,
or
true
part
of
the
Integer
.
Example
11.
Admit
the
distance
from
London
to
the
City
of
Yorke
were
48,080
Poles
or
Pearches
,
and
it
is
required
to
know
how
many
miles
they
are
a
sunder
:
40
Poles
makes
a
Furlong
,
and
8
of
them
a
Mile
,
or
320
Poles
,
which
is
here
the
Divisor
,
and
cutting
off
the
cypher
on
the
right
hand
,
I
must
doe
so
in
the
Dividend
,
which
will
be
then
4808
for
to
be
divided
by
32
,
which
accordingly
done
the
Quotient
will
be
150
Miles
,
and
8
remaining
,
to
which
annex
the
cypher
cut
off
it
will
be
80
Poles
,
or
a
quarter
of
a
Mile
,
for
the
distance
desired
.
An
Examine
of
Multiplication
and
Division
.
These
two
species
doe
trie
one
the
other
,
as
Addition
and
Subtraction
did
:
for
in
any
number
that
is
multiplied
,
if
the
Product
be
divided
by
the
Multiplier
,
the
Quotient
will
be
the
Multiplicand
,
as
before
:
and
so
likewise
any
number
that
is
divided
,
if
you
multiply
the
Quotient
and
Divisor
together
,
and
to
the
Product
adde
in
the
Remainder
(
if
there
be
any
)
the
summe
or
Product
will
be
the
Dividend
again
,
if
your
work
be
true
.
Example
12.
The
Julian
,
or
old
account
An.
Dom.
.
|
Bissextiles
|
1654
|
413
|
|
1652
|
413
|
2
|
Rem
.
2
|
1654
|
1
|
2
|
did
make
the
magnitude
of
the
year
for
to
consist
of
365
dayes
and
6
houres
;
24
making
a
natural
day
,
for
which
cause
every
fourth
year
contained
366
dayes
,
commonly
called
Leap-yeare
or
Bissextile
;
in
one
of
those
was
our
sacred
Redeemer
borne
;
now
to
finde
this
Bissextile
for
any
yeare
since
,
or
to
come
(
according
to
the
old
Kalender
)
divide
the
yeare
given
by
4
,
the
Quotient
shews
the
revolution
of
those
Leap-yeares
,
since
His
happy
Birth
,
the
remainder
are
the
yeares
elapsed
since
the
last
;
and
if
nothing
remains
it
is
Leap-yeare
;
in
this
example
is
given
the
yeare
1654
,
which
divided
according
to
the
tenth
Example
,
the
Quotient
is
413
,
the
number
of
Leap-yeares
past
,
since
the
blessed
Virgin
Mary
was
a
Mother
:
and
the
Remainder
is
2
,
and
so
many
yeares
are
elapsed
since
the
last
Bissextile
,
as
by
the
first
example
in
the
margent
;
the
second
shewes
,
whether
the
division
be
right
,
or
no
;
the
Quotient
is
413
the
number
of
Bissextiles
,
which
now
I
make
the
Multiplicand
,
and
4
(
which
was
the
Divisor
)
the
Multiplier
,
whose
product
is
1652
,
unto
which
adde
the
Remainder
2
in
the
Division
,
the
summe
will
be
1654
,
as
before
,
in
the
second
Table
of
the
margent
is
evident
.
And
thus
is
Division
tried
by
Multiplication
.
Example
13.
By
the
third
example
of
Multiplication
,
Minutes
|
525960
|
|
|
Quotient
|
8766
|
Houres
|
8766
|
it
was
desired
to
know
the
number
of
minutes
in
a
vulgar
yeare
containing
365
dayes
and
6
houres
,
24
making
a
naturall
day
,
that
is
8766
houres
in
a
year
,
which
was
there
the
Multiplicand
,
and
60
,
the
minutes
in
an
houre
,
was
the
Multiplier
,
which
here
I
make
the
Divisor
and
the
Product
525960
minutes
,
the
Dividend
which
by
the
tenth
example
of
Division
may
be
divided
by
6
cutting
off
a
cypher
upon
the
right
hand
of
the
Dividend
,
and
then
the
Quotient
will
be
found
8766
houres
,
the
Multiplicand
as
before
,
which
proves
the
Multiplication
true
;
and
so
the
like
of
any
other
:
if
any
thing
had
remained
,
the
last
place
cut
off
in
the
Dividend
must
have
been
restored
unto
it
,
and
the
cypher
likewise
to
the
Divisor
.
A
Memorandum
.
Observe
in
this
last
way
of
Division
,
that
how
many
points
there
be
in
the
Dividend
,
so
many
figures
,
or
cyphers
there
must
be
in
the
Quotient
,
and
that
every
remainder
must
be
lesse
then
the
Divisor
,
otherwise
the
Quotient
is
too
little
,
or
the
operations
wrong
,
and
when
any
number
is
given
for
to
be
divided
,
if
you
can
finde
a
number
,
that
will
divide
both
the
Dividend
and
Divisor
,
without
leaving
any
remainder
,
they
will
remaine
in
the
same
proportion
,
as
when
cyphers
are
cut
off
from
either
,
and
if
their
Quotients
doe
divide
one
an
other
,
they
will
produce
an
other
Quotient
equall
to
the
first
,
and
their
remainder
(
if
there
be
any
)
shall
have
still
the
same
proportion
,
as
for
example
,
if
48
were
to
be
divided
by
12
the
Quotient
would
be
4
,
and
't
will
be
so
if
you
take
the
halfe
of
these
,
as
24
to
be
divided
by
6
,
or
12
by
3
;
the
Quotient
will
be
4
,
as
you
shall
see
more
at
large
hereafter
,
in
all
wayes
of
Division
,
if
the
Dividend
ends
with
an
odde
number
in
the
place
of
unites
,
viz.
1
,
3
,
5
,
7
,
9
,
and
the
Divisor
even
,
there
must
be
a
remainder
when
the
division
is
done
;
for
any
odde
number
Multiplied
by
one
that
is
even
,
the
product
will
be
even
,
although
the
Quotient
be
odde
,
all
numbers
may
be
divided
into
2
equall
parts
,
if
the
figure
be
even
in
the
unite
place
,
if
odde
it
cannot
without
a
fraction
;
if
any
number
hath
5
or
0
in
the
unite
place
,
5
will
divide
it
,
without
any
remainder
;
but
when
any
thing
does
remaine
,
after
Division
is
ended
,
although
it
were
a
part
of
the
Dividend
,
yet
as
it
hath
relation
to
the
Divisor
,
it
must
be
of
the
same
denomination
with
the
Quotient
as
it
is
a
fraction
,
or
part
of
an
integer
;
as
if
10
s.
were
to
be
divided
betwixt
4
men
,
the
Quotient
will
be
2
and
the
remainder
2
s.
but
as
it
hath
relation
to
four
men
it
is
but
6
d.
or
two
fourths
of
a
shilling
,
the
Quotients
denomination
,
and
should
be
annext
unto
it
as
a
fraction
according
to
the
next
Section
and
Paragraphs
instructions
.