THE
ELEMENTS
OF
ARITHMETICKE
MOST
METHODICAL
LIE
DELIVERED
.
CHAP.
I.
The
definition
and
subiect
of
Arithmeticke
.
ARithmetick
is
the
Art
of
numbring
well
.
This
definition
is
short
&
plaine
,
being
drawen
from
the
end
of
Arithmetike
,
after
which
sort
other
Arts
also
are
defined
.
To
number
well
,
is
to
expresse
and
practise
the
whole
force
,
properties
,
and
vse
of
numbers
.
It
appeareth
then
that
the
Subiect
of
Arithmetike
(
that
is
to
say
,
the
thing
whereabout
that
Art
is
wholly
occupied
and
whereunto
all
the
rules
thereof
are
to
be
referred
)
is
number
.
Number
is
a
Multitude
consisting
of
vnities
.
It
is
the
definition
of
the
seuenth
booke
of
Euclide
.
Aristotle
in
his
Categories
seemeth
to
define
it
thus
.
A
number
is
a
discreet
quantitie
not
hauing
position
but
order
.
The
which
definition
agreeth
with
that
of
Euclid
,
sauing
that
it
distinguisheth
an
vnitie
from
a
point
by
a
certaine
contrary
qualitie
which
is
in
them
;
attributing
to
a
point
position
or
place
,
to
an
vnitie
only
order
&
consequence
.
Whereby
it
appeareth
that
a
number
is
the
ioyning
togither
of
things
following
one
an
other
in
order
not
touching
one
the
other
much
lesse
cleauing
togither
,
and
that
in
such
sort
that
you
can
not
determine
any
bond
or
common
knot
in
them
,
whereby
they
should
be
knit
togither
and
continued
.
The
beginning
of
number
in
an
Vnitie
.
An
vnitie
is
that
,
whereby
euery
thing
that
is
,
is
said
to
be
one
.
An
vnitie
properly
is
no
number
,
because
it
is
no
multitude
,
for
multitudes
onely
are
numbred
:
neither
is
it
a
part
of
a
number
because
that
euery
part
of
a
number
ought
also
to
be
a
number
.
It
answeareth
in
proportion
to
a
moment
,
and
to
a
point
:
whereof
the
one
is
the
beginning
of
time
,
the
other
of
magnitude
,
and
yet
no
part
of
them
:
as
Aristotle
proueth
in
the
fifth
and
in
the
beginning
of
the
sixth
booke
of
Phisicks
:
yet
in
Arithmeticall
numeration
it
is
wont
to
be
taken
for
a
number
,
and
that
the
least
that
may
be
.
The
diuision
of
number
shall
be
drawen
out
of
numeration
.
The
Scholemen
deuide
numbers
into
Digites
,
articles
&
mixt
numbers
:
to
this
end
as
I
think
that
they
might
make
a
distinction
in
the
naturall
consequence
of
numbers
:
wherin
first
of
all
the
nine
simple
figures
are
to
be
considered
,
the
which
the
number
10.
succeedeth
as
it
were
a
knot
knitting
them
to
the
mixt
numbers
following
.
But
for
so
much
as
this
is
no
lawfull
diuision
of
numbers
,
we
will
draw
an
other
out
of
the
last
part
of
Numeration
,
which
we
call
Diuision
,
the
vse
whereof
is
exceeding
great
.
In
the
meane
time
if
the
former
two
numbers
metaphorically
so
termed
(
which
haue
giuen
occasion
to
the
rest
vsed
of
the
Latines
who
say
Arithmeticum
pauciores
habere
digitos
,
plures
autem
articulos
homine
)
if
that
diuision
I
say
do
displease
you
,
you
may
peraduenture
more
fitly
deuide
numbers
into
simple
,
decades
and
compounde
numbers
.
CHAP.
II.
Of
Notation
,
and
of
the
first
part
thereof
.
-
Simple
,
consisting
of
-
Notation
which
hath
two
parts
-
One
more
generall
,
wherein
the
figures
are
considered
,
-
By
themseluyes
,
and
so
there
are
-
Nine
signifying
figure
.
-
One
without
any
signification
.
-
Or
-
One
with
another
,
where
ther
are
to
be
considered
,
-
Degrees
,
which
are
-
Simple
,
containing
single
numbers
.
-
Multiplied
containing
-
Tens
.
-
hundreds
.
-
and
-
Periods
,
whereof
some
are
-
Single
-
Compound
both
being
-
perfect
-
or
-
imperfect
.
-
The
other
more
special
,
wherin
the
numbers
to
be
considered
are
either
of
-
One
terme
,
as
whole
numbers
-
More
termes
,
and
they
be
either
-
Of
one
kind
,
as
partes
,
otherwise
called
fractions
.
-
Of
diuerse
kindes
,
as
mixt
numbers
.
-
Numerration
.
A.
-
Compared
.
E.
ARithmeticke
is
double
:
Simple
and
compared
,
Simple
is
that
which
teacheth
the
practise
of
simple
numbers
without
comparison
.
This
partition
springeth
of
the
diuersitie
of
the
subiect
.
For
numbers
are
considered
either
by
themselues
alone
absolutly
,
or
else
one
with
an
other
in
respect
of
their
proportionalitie
,
or
proportion
which
they
haue
one
to
an
other
.
Hereon
commeth
this
foresayd
diuision
containing
all
things
belonging
to
this
Art
.
Simple
Aruthmeticke
consisteth
of
two
parts
,
Notation
and
Numeration
.
Notation
is
the
way
haw
to
set
downe
numbers
in
writing
,
and
to
expresse
them
being
written
.
The
Greekes
called
it
Semeiosis
,
and
it
is
more
fitly
called
Notation
then
NUmeration
,
because
it
teacheth
vs
how
to
note
,
that
is
,
to
write
euery
number
,
and
to
expresse
it
when
it
is
written
.
There
be
two
parts
of
it
.
The
first
expresseth
generally
the
value
of
the
figures
wherewith
the
numbers
are
written
.
The
figures
wherewith
numbers
are
written
are
ten
.
As
1.
2.
3.
4.
5.
6.
7.
8.
9.
0.
Neither
the
Greekes
nor
the
Latins
vsed
these
figures
.
For
the
Rom●●nes
vsed
onely
thesE
seuen
letters
,
M.
D.
C.
L.
X.
V.
I
but
the
Grecians
vsed
all
the
letters
of
the
Alphabet
,
being
marked
with
certaine
prickes
.
Many
therefore
thinke
that
these
figures
were
inuented
by
the
Arabians
,
the
which
people
in
old
time
practised
this
Art
and
the
whole
science
of
the
Mathematickes
very
diligently
,
where
of
beside
the
word
Algorithme
,
whereby
they
expressed
the
Art
of
numbring
,
many
other
words
left
vnto
vs
in
this
kinde
of
knowledge
may
be
a
sufficient
testimonie
.
For
the
Latins
in
the
former
rude
and
barbarous
age
forsaking
the
wel-springs
of
the
Grecians
,
followed
after
the
small
riuers
of
the
Arabians
.
Valla
and
Cardane
thinke
them
to
be
inuented
by
the
Indians
.
Some
there
be
that
attribute
them
to
the
Phoenicians
.
They
haue
bin
retained
by
the
common
consent
almost
of
all
people
,
because
they
be
most
conuenient
to
number
withall
,
and
most
fit
to
receaue
all
kinde
of
practise
vsed
in
this
Art
.
The
figures
are
considered
eyther
by
themselues
alone
,
or
one
with
another
.
By
themselues
alone
,
when
they
are
without
composition
,
or
as
it
were
without
composition
,
for
then
the
first
nine
haue
a
certaine
value
.
The
tenth
which
hath
the
forme
of
a
circle
signifieth
nothing
standing
by
it selfe
alone
,
but
being
placed
with
the
rest
,
it
maketh
to
the
increase
of
their
value
.
As
,
1.
is
asmuch
as
one
,
2.
two
,
3.
three
,
4.
four
,
5.
fiue
,
6.
six
,
7.
seuen
,
8.
eight
,
9.
nine
,
0.
of
it
own
nature
is
without
value
,
yet
hauing
his
place
wherein
none
of
the
other
signifying
figures
is
placed
,
it
encreaseth
the
value
of
them
which
follow
:
for
it
is
neuer
set
in
the
last
place
.
It
is
called
commonly
a
Cyphar
.
Figures
are
considered
one
with
an
other
being
ioyned
together
,
and
then
the
processe
and
order
of
their
places
encreaseth
their
value
.
For
so
much
as
euery
number
may
be
encreased
infinitely
,
therefore
by
reason
of
their
diuers
encrease
amounting
to
tens
,
hundreds
,
thousands
ten
thousands
,
&c.
it
is
vnpossible
to
write
euery
one
of
them
with
a
diuers
figure
.
For
then
should
the
varietie
of
the
figures
be
also
infinite
.
But
what
is
wanting
in
the
figures
,
that
their
places
through
often
repeating
of
them
,
maketh
vp
.
In
continuance
of
which
places
the
figures
take
vnto
them
a
diuers
signification
,
&
that
so
much
the
greater
by
how
much
the
more
in
ioyning
the
numbers
togither
they
come
nearer
to
the
left
hand
.
The
places
of
numbers
are
distinguished
by
degrees
and
periods
.
A
degree
is
a
place
whereby
the
value
of
each
number
is
gathered
.
A
degree
is
of
two
sorts
:
the
first
is
of
simple
numbers
,
the
other
of
multiplyed
numbers
,
to
wit
,
the
second
degree
is
of
tens
,
the
thirde
of
hundreds
,
and
so
foorth
after
the
same
manner
in
the
rest
.
Numbers
according
to
the
custome
of
all
people
proceede
from
the
left
hand
toward
the
right
,
so
that
the
least
number
occupieth
the
last
place
next
to
the
right
hand
.
Wherefore
the
last
figure
in
value
is
the
first
,
&
euery
figure
in
the
first
place
keepeth
his
first
and
proper
signification
.
In
the
other
places
euery
one
is
armed
with
a
double
power
,
to
wit
,
with
his
owne
and
with
an
other
borrowed
for
a
time
.
For
this
consideration
is
generally
to
be
obserued
in
the
places
:
that
,
that
which
followeth
is
twise
as
much
as
that
which
went
before
.
As
for
example
:
the
figure
4.
by
it selfe
alone
is
foure
:
and
so
much
also
it
signifieth
in
the
first
degree
.
But
being
remoued
into
the
second
,
it
signifieth
fortie
,
as
you
see
here
,
40.
Againe
,
being
set
in
the
third
degree
it
is
ten
times
so
much
as
it
was
in
the
second
:
that
is
,
foure
hundred
,
as
here
,
400.
And
so
forth
after
the
same
manner
both
in
the
other
figures
and
degrees
.
A
periode
is
that
which
comprehendeth
the
degrees
,
and
is
eyther
perfect
,
consisting
of
three
degrees
,
or
imperfect
contayning
lesse
then
three
.
To
conclude
,
that
Period
which
consisteth
but
of
one
trinitie
or
ternarie
of
degrees
is
single
,
but
that
which
consisteth
of
more
then
one
,
is
compound
.
As
1
2
3
is
a
perfect
single
period
:
1
2
3
4
5
6
is
a
perfect
compounde
periode
.
Also
1
2
is
an
vnperfect
single
periode
:
1
2
3
4
,
1
2
3
4
5
is
an
vnperfect
compound
period
.
1
2
3
4
5
6
7
8.
is
an
vnperfect
triple
periode
.
In
numbers
the
degrees
are
to
the
periodes
,
as
the
commata
&
cola
are
the
periodes
in
an
oration
.
CHAP.
III.
Of
the
other
part
of
Notation
.
THus
much
concerning
Notation
in
generall
:
now
followeth
the
other
part
which
expresseth
the
numbers
written
more
particularly
.
And
here
we
haue
to
deale
with
numbers
,
either
of
one
terme
,
or
more
.
Whole
numbers
are
written
with
one
terme
.
As
20
,
24
,
360
,
1234
,
1578
,
they
be
also
called
numbers
of
one
bande
or
rancke
.
How
whole
numbers
of
a
single
periode
should
be
set
downe
,
may
be
gathered
by
tha
which
hath
beene
saide
before
.
But
that
the
compound
numbers
may
be
expressed
,
they
must
with
prickes
or
small
lines
be
deuided
into
single
periods
,
and
be
so
vttered
that
at
the
first
pricke
we
name
a
thousand
,
at
the
second
a
thousand
thousand
,
at
the
third
a
thousand
thousand
thousand
,
and
so
forth
going
forward
after
the
same
order
,
till
we
haue
numbred
the
whole
.
Numbers
be
they
neuer
so
great
may
easily
be
expressed
,
if
distinguishing
them
with
periodes
you
vtter
them
by
partes
,
in
such
sort
that
both
the
value
of
the
figures
and
the
power
of
the
degrees
,
may
gouerne
them
when
they
be
vttered
.
So
that
at
the
end
of
the
first
periode
you
shall
name
a
thousand
,
at
the
end
of
the
second
a
thousand
thousand
(
which
some
call
a
million
)
and
to
be
short
,
at
the
end
of
the
third
periode
,
you
shall
name
a
thousand
thousand
thousand
:
obseruing
the
same
encrease
in
all
the
periodes
following
,
be
they
neuer
so
many
.
Hereupon
it
commeth
to
passe
that
the
least
signifying
figure
of
the
periode
following
,
is
greater
then
any
of
the
periode
going
before
.
Moreouer
the
ready
vttering
of
numbers
maketh
a
man
to
write
them
speedily
,
&
contrariwise
the
speedy
writing
of
them
maketh
a
man
to
vtter
them
readily
.
For
example
,
hauing
distinguished
this
number
34˙567˙890
with
prickes
you
shall
expresse
it
thus
beginning
at
the
left
hand
.
At
the
first
periode
you
shall
say
thirtie
foure
thousand
thousand
or
else
thirty
foure
millions
.
At
the
second
,
fiue
hundred
sixtie
seuen
thousand
.
At
the
third
eight
hundred
and
nintie
.
The
number
following
you
shall
expresse
thus
2˙016˙542˙009˙873
.
Two
thousand
thousand
thousand
thousand
:
then
sixteen
thousand
thousand
thousand
,
fiue
hundred
fourtie
two
thousand
thousand
,
9.
thousand
eight
hundred
seuentie
three
.
The
Romaines
were
wont
to
number
by
hundred
thousands
.
The
which
fashion
if
you
list
to
follow
you
shal
set
down
a
pricke
next
to
the
second
degree
of
the
second
periode
&
expresse
the
other
numbers
following
,
as
you
did
the
first
.
As
in
this
number
77˙89˙320
,
you
shall
say
thus
seuenty
seuen
hundred
foure
score
and
nine
thousand
three
hundred
and
twentie
.
After
this
manner
Plinie
in
his
second
booke
108.
Chap.
affirmeth
according
to
the
saying
of
Artemidorus
that
the
longitude
of
the
earth
inhabited
West
ward
from
the
pillers
of
Hercules
in
India
,
is
eighty
fiue
hundred
seuentie
and
eight
thousand
miles
.
The
which
number
must
be
set
downe
thus
.
8578000
.
Thus
much
for
the
notation
of
numbers
of
one
terme
:
now
follow
the
numbers
of
manie
termes
.
And
they
be
eyther
of
one
kinde
as
partes
(
which
we
commonly
call
fractions
,
)
or
of
diuerse
kinds
as
mixt
numbers
.
Partes
are
set
downe
with
two
termes
deuided
one
from
an
other
with
a
line
:
whereof
the
vppermost
numbreth
the
parts
,
the
neathermost
nameth
them
.
Although
an
vnitie
by
it self
of
it
own
nature
can
not
be
deuided
,
yet
if
we
regarde
the
subiect
thereof
,
or
the
magnitude
whereto
it
is
applied
,
it
may
be
deuided
.
And
yet
the
vnities
of
the
number
proceeding
of
this
diuision
do
not
arise
of
the
very
vnitie
it selfe
,
but
of
the
deuiding
of
the
magnitude
,
that
is
:
they
come
not
of
the
vnitie
as
it
is
by
it selfe
alone
,
but
as
it
is
ioyned
to
some
other
thing
.
As
a
shilling
can
not
be
deuided
as
it
is
one
,
but
as
it
is
a
peece
of
money
,
which
customablie
is
deuided
into
testons
,
groates
,
three
pences
,
two
pences
,
pence
and
half
pence
,
&c.
Wherfore
although
partes
be
called
by
the
name
of
numbers
,
and
are
written
with
the
figures
belonging
to
whole
numbers
:
yet
in
very
deede
they
are
no
numbers
,
if
Euclides
definition
of
a
number
be
true
.
But
in
the
notation
of
these
partes
there
are
two
things
to
be
considered
,
the
quantitie
of
the
parts
(
for
a
whole
number
may
haue
many
parts
,
both
great
and
finall
)
and
the
number
of
them
.
Hereupon
it
commeth
that
when
they
be
written
they
require
two
termes
or
numbers
:
whereof
the
vppermost
declareth
the
number
of
the
parts
the
which
are
to
be
found
out
,
whereupon
it
is
called
the
Number
,
or
Numerator
.
But
the
neathermost
naming
the
parts
of
the
whole
,
expresseth
their
quantitie
:
wherefore
it
is
called
the
name
,
or
the
Denominator
.
As
1/2
signifieth
one
part
of
the
whole
deuided
into
two
halfes
,
that
is
;
one
halfe
.
1/3
one
third
part
3/4
three
fourth
partes
,
or
three
quarters
.
5/6
fiue
sixth
partes
,
that
is
,
fiue
partes
of
the
whole
deuided
into
sixe
parts
.
Hereby
it
appeareth
that
the
greater
the
Denominator
is
,
the
lesse
is
the
quantitie
of
the
partes
.
Partes
are
eyther
principall
,
or
of
the
second
sort
.
The
one
haue
their
originall
of
the
first
diuision
of
the
whole
,
the
other
are
parcels
of
the
principall
parts
.
They
may
be
called
Simple
&
Deriuatiue
parts
:
whereof
the
one
ariseth
of
the
first
diuision
of
whole
numbers
,
the
other
of
breaking
the
parts
into
smaller
parcels
.
Their
notation
and
value
differeth
much
.
The
principall
parts
are
written
after
the
maner
before
named
.
But
the
parcels
,
or
the
parts
of
parts
called
commonly
fractions
of
fractions
,
are
set
downe
by
the
first
toward
the
left
hand
,
without
any
line
betweene
them
for
difference
sake
,
as
5/12
1/3
fiue
twelue
parts
of
one
third
part
,
3/4
7/8
three
fourth
parts
,
of
seuen
eight
parts
:
or
else
otherwise
by
the
preposition
5/6
out
of
3/10.
Moreouer
,
partes
are
eyther
proper
or
improper
.
They
are
lesse
then
the
whole
:
these
are
eyther
equall
vnto
,
or
greater
then
the
whole
.
True
and
proper
partes
in
deede
are
lesse
then
the
whole
,
and
therefore
that
which
they
signifie
can
not
be
expressed
by
an
vnitie
,
whereupon
they
haue
the
numerator
alwaies
lesse
then
the
Denominator
.
But
those
parts
whose
Numerator
is
equall
or
bigger
then
the
Denominator
,
are
improper
partes
,
for
that
they
may
be
expressed
by
an
whole
or
mixt
number
:
as
3/3
are
as
much
as
one
whole
:
4/2
are
two
whole
,
6/5
are
one
whole
,
and
1/5
more
then
the
whole
.
Wherefore
although
whole
numbers
be
sometime
set
downe
as
partes
,
yet
when
the
Numeration
is
once
finished
,
they
are
neuer
written
after
this
manner
,
but
must
be
reduced
to
whole
numbers
:
for
you
cannot
fitly
say
,
2/2
two
halfes
,
or
4/3
foure
third
partes
,
or
8/4
eight
quarters
:
but
one
whole
,
one
whole
with
a
third
part
,
&c.
There
remaine
as
yet
mixt
numbers
,
which
are
whole
numbers
with
partes
.
But
their
notation
may
be
easily
gathered
out
of
the
former
kindes
.
In
Geometrie
they
are
not
accounted
for
numbers
:
but
when
the
measure
of
a
thing
cannot
be
expressed
but
by
a
mixt
number
,
it
is
called
irrationall
,
as
though
it
were
not
indeede
to
be
expressed
by
a
number
.
Herevpon
commeth
the
name
of
Surde
numbers
.
CHAP.
IIII.
Of
Numeration
,
the
other
part
of
simple
Arithmeticke
,
and
of
the
first
kind
thereof
.
Numeration
is
double
.
-
Simple
,
or
of
numbers
of
one
kind
which
is
either
of
-
Whole
numbers
,
and
is
either
-
Prime
,
as
-
Addition
.
-
Subduction
.
-
Second
as
-
Multiplication
.
-
Diuision
.
-
Parts
.
-
Mixt
,
or
of
numbers
of
diuerse
kindes
.
HItherto
we
haue
entreated
of
Notation
,
nowe
followeth
Numeration
:
which
of
two
numbers
giuen
findeth
out
the
third
.
As
the
framing
of
an
argument
is
in
Logicke
,
euen
so
is
numeration
in
this
art
.
For
as
the
Logicians
in
reasoning
doe
inferre
the
conclusion
by
the
premisses
,
euen
so
the
Arithmeticians
by
numbering
doe
inferre
a
diuerse
number
from
the
numbers
giuen
.
The
numbers
giuen
are
either
of
one
and
the
selfe
same
kind
,
as
whole
numbers
,
or
partes
,
or
of
diuerse
kinds
as
mixt
numbers
.
Numeration
of
whole
numbers
is
when
the
numbers
giuen
are
whole
numbers
.
And
this
Numeration
is
double
.
Prime
:
which
numbreth
one
number
with
an
other
onely
once
,
and
that
eyther
by
adding
or
subtracting
them
:
whereupon
it
is
called
Addition
or
Subtraction
.
As
if
these
numbers
360.
and
15.
were
giuen
either
to
be
added
or
subtracted
,
you
must
either
adde
360.
once
to
15.
or
else
subtract
them
once
and
not
many
times
as
in
the
second
numeration
.
Wherefore
this
may
well
be
called
single
numeration
,
and
the
other
manifold
numeration
.
Addition
is
a
Prime
Numeration
,
which
ioyning
one
number
to
an
other
,
findeth
out
the
totall
of
the
numbers
giuen
.
Touching
Addition
of
simple
figures
the
rule
of
reason
teacheth
vs
how
to
doe
it
,
but
the
addition
of
compound
figures
may
be
learned
by
this
rule
.
If
,
proceeding
orderly
from
the
first
of
the
numbers
giuen
to
those
which
folow
,
you
set
the
particular
summes
(
being
lesse
then
ten
)
of
the
figures
of
euery
degree
added
together
alone
by
themselues
,
each
one
vnder
his
owne
degree
,
putting
the
ouerplus
(
that
is
,
for
euerie
ten
,
an
vnitie
)
to
the
degree
following
:
you
shall
finde
out
the
totall
of
the
numbers
giuen
.
Addition
of
single
numbers
may
be
wrought
by
the
direction
of
nature
only
,
wherefore
it
may
be
practised
in
our
mindes
,
and
brought
hither
for
the
speedie
dispatching
of
our
worke
.
As
5
and
6
are
11
,
7
and
8
are
15
,
9
and
9
are
18.
But
the
Addition
of
compound
numbers
goeth
on
by
peecemeale
from
part
to
part
,
so
that
it
requireth
the
helpe
of
art
.
The
first
precept
to
be
obserued
in
this
place
,
is
to
number
all
the
figures
of
euerie
place
alone
by
themselues
,
as
though
they
were
but
single
figures
:
and
this
must
be
done
in
all
the
partes
of
Numeration
following
.
The
seconde
is
to
number
the
figures
of
one
and
the
same
degree
onely
together
,
that
you
confound
not
the
one
with
the
other
.
Last
of
all
,
you
must
set
downe
the
totall
made
of
the
parts
,
in
such
sort
,
that
if
they
increase
to
ten
or
more
,
onely
the
figure
next
to
the
right
hand
must
be
written
vnder
that
degree
,
the
other
which
remaineth
must
be
added
to
the
degree
following
.
As
for
example
,
adde
983.
to
402.
First
,
setting
downe
the
numbers
in
such
sort
one
vnder
an
other
,
that
the
figures
of
euery
degree
may
answere
may
answere
one
to
an
other
,
I
must
beginne
at
the
first
and
say
,
2
and
3
are
5
,
the
which
I
set
downe
vnder
that
degree
:
then
0
and
8
are
8
,
the
which
I
write
vnder
the
second
place
.
Then
4
and
9
are
13
,
which
number
for
that
it
is
in
the
last
place
,
I
set
it
downe
whole
.
So
that
the
totall
of
the
numbers
giuen
is
,
1385.
The
example
standeth
thus
.
Item
adde
89647
vnto
78450.
The
numbers
being
duely
set
downe
one
vnder
an
other
,
according
to
the
order
of
their
degrees
,
and
a
line
drawne
vnderneath
them
,
I
begin
to
number
at
the
right
hand
saying
,
0
&
7
are
7.
Then
5
and
4
are
9.
Item
4
and
6
are
10.
But
for
so
much
as
the
totall
is
written
with
two
figures
,
I
set
downe
the
cypher
only
vnder
that
degree
,
and
keepe
the
vnitie
in
my
minde
.
Then
8
and
9
are
17
,
and
the
vnitie
which
I
kept
maketh
18
,
where
againe
I
set
downe
8
because
it
is
next
to
the
right
hande
and
keepe
the
vnitie
next
to
the
left
hand
in
my
minde
.
Lastly
7
and
8
are
15
,
and
one
which
I
kept
are
16
,
the
which
I
set
down
wholly
.
So
the
totall
of
the
numbers
giuen
is
,
168097.
The
example
is
to
be
set
downe
in
this
order
.
In
this
part
of
numeration
and
in
the
rest
,
there
are
said
to
be
but
only
two
numbers
giuen
,
although
sometime
there
be
more
to
be
added
together
,
for
that
the
two
first
numbers
onely
are
added
together
,
and
then
if
there
be
more
,
the
totall
is
added
to
that
which
followeth
,
as
in
the
example
following
.
678921
|
54086
|
560
|
9522
|
743089
|
Here
,
I
say
,
2
and
6
are
8
,
8
and
1
are
9
,
taking
8
,
which
is
the
totall
made
of
2
and
6
,
for
one
simple
number
,
and
adding
it
to
1
standing
aboue
it
,
&c.
Subduction
is
a
prime
Numeration
,
which
taking
one
number
from
another
,
findeth
out
the
Remainder
of
the
numbers
giuen
.
The
vse
of
it
is
to
find
out
the
difference
betwene
two
numbers
:
whereupon
it
is
requisite
that
the
numbers
giuen
,
should
be
vnequall
,
and
that
from
which
the
subtraction
is
made
,
must
be
the
greater
.
Subduction
of
simple
numbers
may
be
drawen
of
the
table
of
our
mind
.
But
compound
numbers
are
to
be
subducted
by
the
the
Theoreme
following
.
If
proceeding
orderly
from
the
first
of
the
numbers
giuen
,
you
set
the
remainder
of
euery
figure
of
one
&
the
same
degree
,
subtracted
one
from
the
other
(
either
by
it selfe
alone
,
or
by
borrowing
ten
of
the
degree
folowing
)
seuerally
ech
one
vnder
his
own
degree
:
the
remainder
of
the
numbers
giuen
may
be
found
.
In
euery
part
of
numeration
there
must
a
certaine
practise
go
before
it
:
the
which
must
not
then
be
to
seeke
when
we
should
worke
,
but
it
must
be
drawen
out
of
the
threasure
of
our
minds
,
and
ministred
vnto
vs
.
So
that
in
subduction
there
is
required
of
vs
a
ready
foresight
,
that
we
may
know
what
number
will
remaine
,
when
any
one
of
the
figures
is
taken
from
another
by
it selfe
alone
,
or
els
added
vnto
ten
.
As
5
from
8
,
there
remaine
3
,
4
from
9
there
remaine
5.
Or
by
adding
ten
to
the
single
figure
:
as
9
from
17
,
there
remaine
8
,
6
from
15
,
there
remaine
9
,
&c.
In
compound
and
great
numbers
,
it
is
most
conuenient
to
place
the
greater
vppermost
,
and
the
lesse
beneath
,
and
then
to
number
by
the
parts
going
on
towarde
the
left
hande
.
For
that
way
is
more
easie
then
the
other
which
proceedeth
from
the
left
hande
to
the
right
,
as
in
the
examples
following
.
In
the
first
example
euery
thing
is
plaine
.
In
the
second
I
beginne
to
number
,
2
from
4
remaine
2
,
3
from
3
remaine
0.
Then
for
so
much
as
8
can
not
be
taken
from
7
,
the
figure
of
the
same
degree
,
I
borrow
an
vnitie
of
the
degree
following
,
and
ioyne
it
to
7
thus
17
,
and
drawe
8
out
of
17
there
remaines
9
to
be
noted
vnder
neath
.
This
may
be
done
,
if
(
as
some
are
wont
)
I
take
the
distance
of
the
neather
number
from
10.
&
adde
it
to
the
vppermost
number
,
or
set
downe
the
vppermost
number
only
,
if
the
neathermost
be
10.
As
for
example
,
because
8
differeth
2
from
10
,
I
take
2
and
adde
them
to
7
which
make
9
to
be
written
vnder
the
third
degree
.
But
for
so
much
as
I
borrowed
1
of
8
in
the
vppermost
number
following
,
there
remaine
but
7
,
I
draw
therefore
9
from
17
(
taking
an
vnitie
from
5
following
)
there
remaine
8.
In
the
last
degree
,
because
there
is
no
figure
beneath
which
aunswereth
to
that
aboue
,
I
set
not
downe
the
whole
5
but
4
,
because
I
borrowed
one
of
it
before
.
In
the
third
example
,
that
the
signifying
figures
may
be
subducted
out
of
the
cyphers
,
we
must
borrow
of
the
numbers
following
an
vnitie
,
as
we
did
before
.
First
take
4
from
5
there
remaines
1
,
then
because
you
cannot
take
8
from
6
,
you
must
subtract
it
from
16
,
there
remaines
8.
For
although
the
figures
following
,
from
whence
I
borrow
the
vnitie
,
be
a
cyphar
,
and
therefore
without
signification
,
yet
for
so
much
as
it
hath
a
signifying
figure
before
it
,
I
may
borrow
an
vnitie
of
it
.
Then
I
go
on
and
take
9
from
9
,
there
remaineth
nothing
.
Moreouer
I
take
1
,
not
from
10
,
because
there
was
an
vnitie
borrowed
of
it
before
,
but
from
9
,
there
remaine
8.
And
for
so
much
as
there
is
no
figure
to
be
drawen
out
of
that
which
followeth
,
I
set
downe
9.
Likewise
I
write
downe
the
last
figure
lesse
by
one
then
it
is
,
because
I
presuppose
that
it
wanteth
that
vnitie
which
I
tooke
from
the
first
cyphar
.
CHAP.
V.
Of
the
second
kind
of
Numeration
.
HEtherto
of
prime
Numeration
.
Now
followeth
second
Numeration
,
which
numbreth
one
number
with
an
other
,
so
often
as
one
of
the
two
numbers
giuen
requireth
,
and
that
eyther
by
multiplying
or
diuiding
:
whereupon
it
is
called
Multiplication
or
Diuision
.
Second
Numeration
is
either
often
Addition
,
as
Multiplication
:
or
often
Subtraction
,
as
Diuision
.
For
one
of
the
numbers
giuen
is
so
often
eyther
encreased
or
diminished
,
as
the
other
,
eyther
the
multiplyer
or
diuisor
requireth
,
according
to
the
number
of
the
vnities
contayned
in
them
.
Multiplication
is
a
seconde
Numeration
,
which
ioyning
together
the
multiplicande
,
so
often
as
there
bee
vnities
in
the
Multiplier
,
bringeth
foorth
the
facit
.
This
is
the
15.
definition
of
the
7.
Booke
of
Euclide
.
It
maketh
no
matter
which
number
you
make
the
multiplicande
,
or
which
number
you
make
the
multiplier
,
as
it
appeareth
by
the
16.
prop.
of
the
seuenth
Booke
of
Euclide
,
which
sayth
:
If
two
numbers
multiplied
together
,
the
one
into
the
other
,
produce
any
numbers
,
the
numbers
produced
are
equall
the
one
to
the
other
:
that
is
,
they
make
one
and
the
same
number
.
Notwithstanding
the
Scholemen
set
the
greatest
number
vppermost
for
the
multiplicande
,
and
the
least
neathermost
for
the
multiplier
,
for
that
it
seemeth
most
conuenient
for
young
beginners
.
Multiplication
of
single
numbers
is
to
be
conceiued
in
our
mindes
by
this
Theoreme
.
If
eyther
of
the
two
numbers
giuen
be
diuided
into
certaine
partes
,
the
product
comming
of
the
whole
numbers
multiplied
together
,
is
equall
to
the
product
made
by
one
of
the
whole
numbers
and
the
parts
of
the
other
number
so
diuided
.
The
1.
proposition
of
the
second
booke
,
or
else
,
if
both
the
numbers
giuen
be
diuided
into
certaine
parts
,
that
which
is
made
of
the
whole
number
,
is
equall
to
the
product
made
of
their
parts
.
That
we
may
multiply
easily
and
readily
,
we
must
haue
in
our
mindes
,
a
table
whereby
we
may
know
what
the
pruduct
of
euery
simple
figure
is
,
being
multiplied
one
by
an
other
:
the
which
thing
may
be
easily
done
by
the
rule
of
whole
numbers
and
their
partes
.
Euery
child
by
the
direction
of
nature
can
tell
how
many
twise
foure
,
or
foure
times
fiue
,
or
thrise
sixe
doe
make
.
But
if
you
happen
to
sticke
in
greater
numbers
,
a
litle
exercise
will
make
this
table
very
readie
.
As
if
you
would
knowe
how
much
seuen
times
eight
is
:
diuide
eyther
of
the
two
numbers
giuen
,
into
as
many
partes
as
you
list
,
as
7.
into
2.
3.
2.
Then
multiply
8.
by
euery
one
of
these
parts
,
and
adde
the
particular
productes
together
,
and
you
shall
haue
56.
How
many
are
eight
times
9.
diuide
9.
into
3.
3.
3.
and
multiply
8.
by
those
partes
,
adding
the
products
together
,
so
you
shall
make
72.
The
same
may
be
done
if
you
diuide
both
the
numbers
.
The
examples
must
be
set
downe
thus
.
The
Scholemen
frame
this
Table
by
this
Theoreme
.
Two
numbers
being
giuen
which
ioyntly
together
are
more
then
tenne
,
if
you
multiply
the
difference
of
each
of
them
from
tenne
one
by
the
other
,
&
then
subduct
cross-wayes
one
of
the
differences
out
of
one
of
the
numbers
giuen
,
the
product
and
the
remainder
parted
into
diuerse
degrees
,
shall
be
the
product
of
the
numbers
giuen
.
As
you
see
in
these
examples
.
Multiplication
of
compound
figures
is
drawen
out
of
the
Theoreme
following
.
If
proceeding
from
the
first
of
the
numbers
giuen
toward
the
left
hand
,
you
multiply
the
figures
of
the
multiplier
,
into
euery
one
of
the
multiplicand
,
and
ioyne
together
the
particular
products
being
lesse
then
ten
,
setting
them
orderly
vnder
their
multiplier
,
and
put
the
ouerplus
to
the
degree
following
:
the
totall
made
of
the
partes
,
is
the
product
of
the
numbers
giuen
.
For
example
sake
,
multiply
,
365
by
3.
Here
the
numeration
must
proceede
from
part
to
part
.
Hauing
therefore
set
downe
the
numbers
,
as
you
did
in
the
former
kindes
of
numeration
,
multiply
the
neathermost
figure
seuerally
into
euery
one
of
the
vppermost
thus
:
three
times
5
are
15
,
I
set
downe
5
,
and
keepe
1
:
thrise
6
are
18
,
and
one
which
I
kept
,
are
19
,
I
set
downe
9
and
keepe
one
to
be
added
to
the
degree
following
:
thrise
3
are
9
,
and
one
are
10
,
which
I
set
downe
wholy
.
The
example
standeth
thus
.
Now
take
an
example
of
multiplication
to
be
wrought
by
the
parts
of
both
the
numbers
giuen
.
Multiply
1568
by
54
:
Item
multiply
3508476
by
2509.
In
both
these
numbers
you
shall
haue
so
many
rowes
of
numbers
,
as
there
be
figures
in
the
multiplier
.
Wherefore
we
must
take
diligent
heede
,
that
we
confound
not
the
particular
products
,
and
beware
that
in
distinguishing
the
products
,
we
set
the
first
figure
of
euery
one
of
them
vnder
his
particular
multiplier
,
as
you
see
here
.
An
abridgement
of
multiplication
.
Neither
are
the
shortest
wayes
to
be
neglected
:
this
therefore
is
an
especiall
abridgement
of
multiplication
.
If
one
,
or
both
the
numbers
giuen
,
haue
cyphers
in
the
beginning
,
then
if
multiplying
only
the
signifying
figures
together
,
you
put
the
cyphers
to
the
totall
,
you
shall
haue
the
product
of
the
numbers
giuen
.
The
correllary
of
this
abridgement
may
be
this
.
If
the
last
figure
of
the
multiplier
be
an
vnitie
,
and
the
other
cyphers
,
then
setting
the
cyphers
before
the
multiplicand
,
you
shall
haue
the
product
of
the
numbers
giuen
.
The
exsamples
.
Diuision
is
a
numeration
,
which
drawing
one
number
from
another
,
as
often
as
may
be
,
findeth
out
the
quotient
of
the
greater
.
In
diuision
there
are
three
numbers
to
be
considered
,
the
Diuidende
,
the
Diuisor
,
&
the
Quotient
,
the
which
must
be
so
placed
,
that
the
Diuidend
may
stande
aboue
,
the
Diuisor
beneath
,
and
the
Quotient
at
the
side
,
or
betwene
them
both
.
The
vse
of
diuision
is
to
declare
how
many
times
the
lesse
is
contained
in
the
greater
.
The
Artisicers
terme
diuision
by
a
Geometricall
phrase
,
calling
it
Comparison
,
which
is
the
applying
of
a
measure
giuen
to
any
right
line
,
as
here
,
A
—
‑
—
‑
—
‑
—
‑
—
B
C
—
D
the
line
CD
,
is
compared
with
the
line
AB
to
see
how
many
times
it
may
be
contained
in
it
.
Or
it
is
the
ioyning
of
one
side
to
an
other
side
,
giuen
to
make
a
rightangled
parallelogram
.
As
therfore
in
multiplication
,
the
multiplying
of
two
sides
togither
maketh
a
right
angled
figure
,
if
the
lines
meete
perpendicularly
,
euen
so
in
diuision
,
the
diuiding
of
the
area
or
platforme
of
the
right
angled
figure
giuen
by
the
length
as
it
were
by
the
diuisor
findeth
out
the
breadth
which
is
represented
by
the
quotient
,
as
you
see
here
.
Multiply
the
side
A
C
,
by
the
side
C
D
,
the
product
will
be
the
area
or
platforme
of
the
right
angled
figure
A
B
C
D.
Likewise
diuide
the
platforme
of
the
right
angled
figure
E
F
G
H
,
by
the
side
G
H
which
is
the
length
,
you
shall
find
out
the
other
side
E
G
which
is
the
breadth
which
answereth
to
the
quotient
in
diuision
.
Hereupon
it
is
called
the
breadth
of
the
comparison
or
of
the
compared
figure
.
A
quotient
is
a
part
of
the
diuidende
,
hauing
the
same
denomination
with
the
diuisor
.
This
definition
is
taken
out
of
the
39
prop.
of
the
seuenth
booke
of
Euclide
which
sayth
:
If
a
number
measure
any
number
,
the
number
measured
shall
haue
a
part
of
the
same
Denomination
with
the
number
which
measureth
it
.
And
contrariwise
,
as
it
is
in
the
40.
prop.
If
it
haue
a
part
,
the
number
whereof
the
part
taketh
his
denomination
,
shall
measure
it
.
For
Euclide
taketh
measuring
here
for
diuiding
:
as
for
example
,
because
48
may
be
diuided
by
6
into
8
the
quotient
,
I
say
that
8
being
the
quotient
,
is
the
6
part
of
48
,
and
taketh
his
denomination
of
the
Diuisor
.
And
contrariwise
,
because
8
is
the
6
part
of
48
,
therefore
6
diuide
48.
Likewise
if
6
diuide
54
,
the
quotient
will
be
9
,
and
9
therefore
is
the
6
part
of
54
,
and
because
9
is
the
sixth
part
of
54
,
therefore
6
diuide
54
,
&c.
A
part
of
a
number
is
a
lesse
number
in
respect
of
a
greater
,
when
the
lesse
measureth
the
greater
.
This
is
the
first
definition
of
the
5
and
the
third
def.
of
the
seuenth
booke
.
Euclide
will
not
haue
euery
number
that
is
lesse
then
a
greater
number
,
to
be
a
part
of
the
greater
,
but
onely
that
which
being
taken
oftentimes
by
it selfe
alone
,
measureth
the
greater
,
that
is
,
eyther
maketh
it
,
or
taketh
it
cleane
away
:
As
two
is
the
third
part
of
6
,
fiue
is
the
halfe
part
often
.
For
that
2
being
three
times
taken
,
eyther
maketh
or
taketh
away
6
,
and
5
twice
taken
make
10
or
destroyeth
ten
.
This
kind
of
part
is
called
commonly
,
pars
metiens
,
or
mensurans
.
A
measuring
part
and
of
the
barbarous
it
is
called
pars
aliquota
,
an
aliquot
part
,
that
is
,
a
quotient
:
but
the
number
which
by
it selfe
alone
being
oftentimes
taken
,
maketh
not
a
number
without
the
helpe
of
some
other
part
or
number
,
is
the
parts
of
the
whole
,
vsing
the
worde
parts
in
the
plurall
number
for
distinction
sake
.
As
2
are
the
parts
of
fiue
,
because
two
make
not
fiue
alone
,
without
the
helpe
of
three
,
or
some
other
numbers
.
This
kind
of
part
they
commonly
call
,
pars
constituens
,
or
componens
:
of
the
barbarous
it
is
called
,
pars
aliquanta
.
Diuision
of
a
simple
quotient
may
be
taken
out
of
the
table
of
Multiplication
.
For
by
the
same
numbers
that
a
number
is
made
,
by
the
same
it
may
be
diuided
.
The
first
principle
to
be
obserued
in
diuision
is
to
consider
wittily
,
and
to
know
readily
by
what
number
,
and
what
number
euery
one
of
the
single
figures
do
diuide
another
,
as
sixe
times
7
are
42
,
therefore
7
diuide
42
by
6
,
and
6
diuide
it
by
7.
Item
seuen
times
9
are
63
,
therefore
9
diuide
63
,
and
the
quotient
is
7
,
or
7
diuide
it
,
and
the
quotient
is
9.
The
diuision
of
a
number
that
hath
many
figures
,
is
wrought
by
the
Theoreme
following
.
If
,
beginning
at
the
last
figure
of
the
numbers
giuen
,
you
multiply
the
particular
quotient
of
the
diuisor
contayned
in
the
diuidend
(
setting
it
downe
aside
by
it selfe
)
into
the
Diuisor
,
and
then
subtract
the
product
from
the
diuidend
,
doing
this
as
often
as
may
be
,
by
setting
forwarde
the
Diuisor
towarde
the
right
hand
,
till
you
come
to
the
first
figure
of
the
diuidend
,
the
quotient
of
the
parts
will
be
the
quotient
of
the
numbers
giuen
.
Although
diuision
commonly
be
wrought
by
finding
out
the
quotient
by
multiplication
&
subduction
:
yet
the
principall
worke
wherein
the
whole
force
or
vertue
of
diuision
consisteth
,
is
onely
the
finding
out
of
the
quotient
,
which
being
once
found
,
the
diuision
in
minde
is
supposed
to
be
done
.
But
when
we
haue
found
out
the
quotient
,
we
are
to
thinke
and
consider
with
our selues
into
how
many
partes
the
diuisor
cutteth
the
number
which
is
set
ouer
it
,
and
how
often
it
may
be
drawen
out
of
it
.
That
this
therefore
may
be
set
before
our
eyes
,
and
that
in
remouing
forward
the
Diuisor
,
the
remainder
(
wherein
afterwarde
the
quotient
must
be
sought
for
)
may
more
manifestly
appeare
,
the
truth
of
that
which
we
conceiued
before
,
is
proued
by
multiplication
and
subtraction
.
To
expresse
this
by
examples
,
diuide
4936
by
4
,
we
must
seeke
how
often
the
Diuisor
4
being
set
vnder
the
last
figure
of
the
diuidend
,
is
contayned
in
the
same
.
I
say
then
that
4
is
contayned
in
4
once
:
wherefore
I
set
downe
1
in
the
quotient
,
and
take
4
out
of
4
,
there
remaines
nothing
.
Then
set
I
the
Diuisor
one
degree
forwarde
towarde
the
right
hand
,
where
4
are
contayned
in
9
twice
:
wherfore
I
note
2
in
the
quotient
,
and
multiply
the
Diuisor
by
it
,
the
product
is
8
,
which
being
subducted
from
9
,
there
remaine
1.
The
figures
out
of
which
the
subtraction
is
made
,
must
straight
way
be
blotted
out
.
Againe
set
forwarde
the
diuisor
:
there
4
may
be
taken
from
13
three
times
,
wherefore
I
set
3
in
the
quotient
,
and
multiply
4
by
it
,
the
product
is
12
,
which
being
drawen
out
of
13
,
there
remaines
1.
To
conclude
,
I
find
4
in
16
foure
times
,
therefore
I
set
downe
4
for
the
quotient
,
and
multiply
4
by
it
,
which
being
taken
out
of
16
,
there
is
nothing
left
.
The
example
.
Againe
deuide
1008
by
36.
In
this
example
there
be
many
things
to
be
taken
heede
of
.
First
,
because
the
last
figure
of
the
diuidend
is
lesse
then
the
last
of
the
Diuisor
,
therfore
I
set
not
2
vnder
1
,
but
one
place
farther
towarde
the
right
hande
vnder
the
o.
And
first
I
consider
with
my sefe
,
how
often
three
may
be
had
in
ten
,
I
find
it
to
be
thrise
,
and
there
remaineth
one
.
But
for
so
much
as
I
cannot
subduct
the
Product
made
by
multiplying
the
quotient
into
the
whole
diuisor
,
out
of
the
number
standing
aboue
it
,
to
wit
,
108
out
of
100
,
therefore
I
take
a
quotient
lesse
then
that
by
one
,
that
is
2
,
whereby
the
diuisor
being
multiplied
,
there
arise
72
,
which
being
taken
out
of
100
,
there
remaine
28
to
be
written
ouer
the
head
.
Then
the
diuisor
being
set
forward
,
I
see
three
to
be
contained
in
28
nine
times
,
but
because
I
must
haue
regard
also
of
the
figure
following
,
I
set
but
8
in
the
quotient
,
which
being
multiplied
into
the
diuisor
,
make
288
to
be
drawen
out
of
the
number
set
ouer
it
.
The
example
is
thus
.
It
appeareth
therefore
by
this
,
that
the
question
must
be
made
not
of
the
whole
Diuisor
,
vnlesse
it
be
a
single
number
,
but
only
of
the
last
figure
of
the
Diuisor
,
and
euery
particular
must
be
but
a
single
figure
,
as
if
you
shold
seeke
how
often
2
were
in
21
,
you
can
not
take
it
tenne
times
,
but
9
times
,
but
9
times
at
the
most
.
Moreouer
the
nature
of
the
thing
requireth
that
we
find
out
alwayes
such
a
quotient
,
as
being
multiplied
by
the
Diuisor
,
maketh
no
greater
a
number
then
the
diuidend
is
.
See
the
examples
following
.
An
abridgement
of
diuision
.
If
the
Diuisor
end
in
cyphers
,
the
worke
may
be
wrought
by
the
signifying
figures
alone
setting
the
cyphers
in
the
meane
time
,
vnder
the
vtmost
figures
of
the
diuidend
,
next
to
the
right
hand
.
But
if
the
last
figure
of
the
Diuisor
be
an
vnitie
,
and
the
rest
cyphers
,
then
setting
the
Diuisor
as
before
,
and
taking
the
figures
that
haue
no
cyuphers
vnderneath
them
for
the
quotient
,
the
diuision
is
dispatched
.
the
figures
that
remaine
after
the
diuision
is
done
,
must
be
set
downe
as
the
partes
are
with
the
diuisor
vnderneath
them
for
their
denominator
.
As
for
example
,
diuide
165968
by
360.
Item
6734
by
100
:
the
example
shall
stand
thus
.
CHAP.
VI.
Of
the
double
diuision
of
numbers
.
Numbers
by
diuision
are
distinguished
2.
manner
of
wayes
,
-
First
they
are
either
-
Odde
.
-
Or
-
Euen
:
which
may
be
diuided
either
,
-
One
way
only
,
and
are
either
,
-
Euenly
euen
.
-
Or
-
Oddely
euen
.
-
Many
wayes
,
and
are
both
euenly
,
and
oddely
euen
.
-
Secondly
,
as
they
be
considered
,
either
-
By
themselues
alone
,
and
then
they
are
-
Prime
.
-
or
-
Compound
.
-
One
with
an
other
,
and
then
also
they
are
-
Prime
.
-
or
-
Compound
.
NVmbers
are
diuided
two
manner
of
wayes
,
first
into
euen
and
odde
numbers
.
An
euen
number
is
that
which
may
be
diuided
by
2.
An
odde
number
is
that
which
can
not
be
diuided
by
2.
This
is
the
6
&
7
definition
of
the
seuenth
booke
of
Euclide
,
whence
this
difference
of
numbers
drawen
out
of
their
diuision
,
is
taken
,
as
that
also
which
followeth
,
whose
vse
in
Arithmeticke
is
very
great
.
Euen
numbers
are
as
these
following
,
2.
4.
6.
8.
10.
12.
14.
and
so
forth
after
the
same
order
,
alwaies
omitting
one
.
Odde
numbers
are
,
as
3.
5.
7.
9.
11.
13.
15.
&c.
Of
euen
numbers
some
may
be
diuided
but
one
way
onely
,
other
some
may
be
diuided
many
waies
.
Those
which
may
be
diuided
but
one
way
onely
,
are
euenly
euen
,
or
oddely
euen
.
A
number
euenly
euen
,
is
that
which
an
euen
number
diuideth
by
an
euen
number
.
This
is
the
eight
definition
of
the
seuenth
,
such
are
4.
8.
16.
32.
64.
&c.
that
is
to
say
,
all
the
numbers
from
2
vpward
doubled
by
2.
As
appeareth
by
the
32
prop.
of
the
9
booke
.
A
number
odly
euen
,
is
that
which
an
odde
number
measureth
by
an
euen
number
.
As
it
is
in
the
9
definition
of
the
seuenth
booke
.
Euclide
calleth
it
an
euenly
odde
number
,
as
6.
10.
14.
18.
22.
For
3
an
odde
number
diuideth
6
by
2
an
euen
number
:
5
an
odde
nūder
diuideth
10.
by
2
an
euen
number
.
Such
are
all
the
numbers
whose
moytie
or
halfe
is
an
odde
number
,
as
appeareth
by
the
33
prop.
of
the
seuenth
booke
.
The
numbers
which
may
be
diuided
manie
wayes
,
are
euenly
euen
and
odde
:
which
may
be
diuided
both
by
an
euen
number
,
and
by
an
odde
,
into
an
euen
number
.
This
third
kind
of
numbers
,
is
let
passe
of
Euclide
among
the
definitions
of
the
seuenth
booke
,
but
yet
not
neglected
in
the
propositions
of
the
ninth
booke
.
The
examples
therof
are
these
,
12.
20.
24.
28.
56.
144.
As
2
an
euen
number
diuideth
12
by
6
an
euen
number
,
and
3
an
odde
number
diuideth
it
by
4
an
euen
number
.
Item
2
an
euen
number
diuideth
20
into
10
an
euen
number
:
and
5
an
odde
number
diuideth
it
into
4
an
euen
number
.
Such
are
all
those
numbers
which
are
neither
doubled
by
two
from
2
vpward
,
nor
haue
their
halfe
an
odde
number
as
it
is
in
the
34.
prop.
of
the
9
booke
.
Againe
,
numbers
be
distinguished
otherwaies
being
considered
both
by
themselues
,
and
one
with
another
.
Being
considered
by
themselues
,
they
are
eyther
prime
,
which
may
be
diuided
by
an
vnitie
onely
:
or
compound
,
which
may
be
diuided
by
some
other
number
.
This
is
the
12
and
13
definition
of
the
seuenth
booke
,
where
Euclide
diuideth
numbers
into
prime
&
compound
numbers
.
The
matter
of
this
diuision
is
taken
out
of
the
34
prop
of
the
seuenth
booke
,
which
sayth
:
That
euery
number
is
eyther
a
prime
number
,
or
else
diuided
by
a
prime
number
:
that
is
,
a
compound
number
,
as
appeareth
by
the
33
prop.
of
the
same
booke
.
A
prime
number
is
that
which
no
other
number
diuideth
besides
an
vnitie
,
sauing
that
it
measureth
it selfe
.
It
may
be
called
an
vncompound
number
for
that
it
is
made
of
no
number
:
as
2.
3.
5.
7.
11.
13.
&c.
A
compound
number
is
that
which
some
other
number
maketh
,
being
taken
certaine
times
,
as
4.
6.
10.
12.
Of
this
sort
are
first
of
all
,
all
euen
numbers
,
then
all
those
odde
numbers
which
in
the
eleuenth
definition
of
the
seuenth
booke
are
called
oddely
odde
:
which
may
be
diuided
by
an
odde
number
into
an
odde
number
,
as
9.
15.
21.
25.
27.
&c.
Numbers
compared
one
with
an
other
,
some
are
Prime
in
respect
one
of
an
other
,
which
can
not
commonly
be
diuided
by
any
other
number
,
but
by
an
vnitie
.
Other
some
are
compound
in
respect
one
of
an
other
,
which
may
be
commonly
diuided
by
one
or
moe
numbers
.
This
difference
of
numbers
compared
togeather
,
is
drawen
out
of
the
former
,
and
is
set
downe
in
the
twelfth
and
eleuenth
definition
of
the
seuenth
booke
.
Numbers
prime
in
respect
one
of
another
,
haue
no
common
diuisor
beside
an
vnitie
,
which
measureth
all
numbers
:
As
3
and
8
,
6
and
7
,
5
and
12.
Therefore
when
the
numbers
giuen
are
prime
one
to
an
other
,
there
can
no
lesse
numbers
be
giuen
in
the
same
proportions
.
So
that
prime
numbers
are
the
least
,
and
the
least
are
prime
numbers
,
as
appeareth
in
the
twentie
and
twentie
foure
proposition
of
the
seuenth
booke
.
And
therefore
,
when
the
numbers
giuen
are
prime
,
they
are
also
the
termes
of
the
proportion
giuen
,
as
we
shall
see
afterward
.
Compound
numbers
in
respect
one
of
an
other
,
may
commonly
be
diuided
by
one
,
or
many
numbers
,
as
3
and
9
,
9
and
15
:
for
3
is
the
common
Diuisor
of
them
both
.
Item
12
,
18
,
24
,
may
be
commonly
diuided
not
onely
by
6
,
but
also
by
3
and
2.
If
you
would
knowe
whether
the
numbers
giuen
,
be
prime
or
compounde
one
to
an
other
,
you
may
doe
it
by
the
Theoreme
following
.
Two
vnequall
numbers
being
giuen
,
if
in
subducting
the
lesse
from
the
greater
as
often
as
may
be
,
the
remainder
diuideth
not
that
which
went
before
it
,
vntill
it
come
to
an
vnitie
,
the
numbers
giuen
are
prime
one
to
an
other
.
1.
prop.
7.
As
in
4
and
11
,
18
and
7
,
35
and
12
,
in
subducting
the
one
from
the
other
continually
,
you
shall
come
to
an
vnitie
:
wherefore
I
say
,
that
they
be
prime
in
respect
one
of
an
other
.
And
hereby
we
may
easily
conclude
,
if
after
the
continuall
subducting
of
the
one
from
the
other
,
we
come
not
downe
to
an
vnitie
,
but
meete
with
some
one
number
diuiding
that
which
went
before
:
the
numbers
giuen
are
compound
in
respect
one
of
an
other
.
As
in
14
and
6
,
30
and
18
,
49
and
14.
An
abridgement
of
the
former
worke
.
For
so
much
as
diuision
is
nothing
but
an
often
subduction
,
if
you
diuide
the
greater
number
giuen
by
the
lesse
,
you
may
doe
the
former
worke
more
readily
.
Yet
in
diuiding
the
numbers
after
this
maner
,
you
must
haue
no
regard
to
the
quotient
,
but
compare
the
Diuisor
with
the
remainder
.
As
I
proue
that
234
and
17
are
prime
numbers
one
to
the
other
,
by
diuision
thus
.
In
compound
numbers
take
these
for
example
144
and
27.
CHAP.
VII
.
Of
the
greatest
common
Diuisor
,
and
the
least
common
diuidend
.
NVmbers
are
diuided
after
the
manner
before
named
.
Moreouer
out
of
the
difference
of
numbers
compared
together
,
there
ariseth
a
double
inuention
:
one
is
the
finding
out
of
the
greatest
common
diuisor
,
the
other
of
the
least
common
diuidend
.
For
so
much
as
compound
numbers
may
be
diuided
often
times
by
many
diuisors
,
the
drift
of
the
first
inuention
is
out
of
many
to
choose
the
greatest
number
,
which
may
be
the
common
diuisor
,
or
as
Euclide
termeth
it
the
common
measure
of
them
all
.
The
numbers
therefore
giuen
Are
either
-
Two
onely
,
and
no
more
,
as
in
the
first
Rule
.
I.
-
Three
or
more
,
&
in
them
we
meete
with
the
greatest
common
diuisor
,
either
.
-
At
the
first
.
II.
-
Or
at
the
second
time
.
III.
The
finding
out
of
the
greatest
common
diuisor
is
to
be
learned
by
the
Theoreme
following
.
I.
If
after
the
continuall
subduction
of
two
numbers
giuen
,
the
one
from
the
other
,
as
often
as
may
be
(
which
may
more
briefly
be
done
by
diuision
)
there
remaine
some
one
number
which
will
exactly
diuide
them
both
:
The
number
remaining
shall
be
their
greatest
common
diuisor
.
As
in
76
and
20
,
their
greatest
common
diuisor
is
4
,
as
appeareth
by
this
.
Item
of
these
compound
numbers
63
and
4
,
their
greatest
common
diuisor
is
7.
II.
When
there
are
three
numbers
giuen
,
if
the
greatest
common
diuisor
of
the
two
formost
,
being
compared
with
the
third
,
do
diuide
it
also
,
that
diuisor
shall
be
the
greatest
common
diuisor
of
them
all
:
and
so
forth
in
the
rest
,
be
they
neuer
so
many
.
3
prop.
7.
As
14.
21.
35.
63.
|
Compound
numbers
.
|
7
|
The
greatest
common
diuisor
.
|
2.
3.
5.
9.
|
The
quotients
of
the
numbers
,
being
prime
numbers
one
to
an
other
.
|
Here
7
being
the
greatest
common
diuisor
of
the
two
for
most
14
and
21
,
compared
with
the
third
number
35
,
diuideth
it
also
and
in
like
manner
the
fourth
63
:
wherefore
it
is
the
greatest
common
diuisor
of
them
all
.
III.
But
when
the
greatest
common
diuisor
of
the
formost
,
diuideth
not
those
which
follow
,
then
comparing
the
first
diuisor
,
found
out
with
the
third
number
,
the
greatest
common
diuisor
of
the
numbers
compared
together
shall
be
the
common
diuisor
of
the
numbers
giuen
:
and
so
foorth
in
the
rest
be
they
neuer
so
many
3
prop.
7.
As
betwene
18.
12.
9.
The
greatest
common
diuisor
betwene
18
&
12
,
is
6
,
the
which
number
for
so
much
as
it
cannot
diuide
9
,
compare
9
and
6
together
by
themselues
,
and
then
because
the
greatest
common
diuisor
of
6
and
9
is
3
,
it
shall
be
also
the
greatest
common
Diuisor
of
all
the
numbers
giuen
.
The
example
is
thus
set
downe
.
18
|
12
|
9
|
The
numbers
giuen
.
|
6
|
|
The
numbers
compared
.
|
|
3
|
The
greatest
Diuisor
.
|
See
also
this
example
following
of
foure
compound
numbers
in
respect
one
of
an
other
,
wherein
there
is
a
double
comparison
made
,
and
the
greatest
common
Diuisor
diuideth
it selfe
also
.
8
|
16
|
28
|
26
|
The
numbers
giuen
.
|
8
|
|
|
The
comparisons
.
|
|
4
|
|
|
|
2
|
The
greatest
common
diuisor
.
|
The
greatest
common
diuisor
is
found
out
as
is
aforesaid
.
Now
followeth
the
finding
out
of
the
least
number
that
may
commonly
be
diuided
by
the
numbers
giuen
:
that
is
,
such
a
number
,
then
which
there
cannot
be
a
lesse
,
which
two
or
more
numbers
giuen
may
exactly
diuide
.
Euclide
calleth
it
the
least
that
may
be
diuided
by
certain
numbers
.
For
as
many
numbers
giuen
may
make
many
compound
numbers
,
euen
so
also
they
may
diuide
them
,
out
of
which
numbers
to
chose
the
least
that
may
be
diuided
,
is
greatly
auaileable
to
the
readinesse
and
easinesse
of
numbring
.
The
numbers
whose
least
common
diuidend
we
seeke
for
,
are
either
-
Only
two
alone
,
&
those
-
Prime
one
to
another
.
I.
-
or
-
Compound
.
II.
-
Many
,
wherin
we
finde
our
the
least
number
that
may
be
diuided
.
-
At
the
first
.
III.
-
Or
-
At
the
second
time
.
IIII.
The
finding
out
of
the
least
number
that
may
be
commonly
diuided
by
certaine
numbers
giuen
,
is
learned
by
the
Theoremes
following
.
I.
If
the
two
numbers
giuen
,
be
prime
one
to
an
other
,
their
product
is
the
least
number
that
may
be
commonly
diuided
by
them
.
As
15
the
product
of
3
and
5
,
being
prime
numbers
one
to
an
other
,
is
their
least
common
diuidend
.
Item
63
is
the
least
number
commonly
diuided
by
9
and
7
II.
But
if
the
two
numbers
giuen
,
be
compound
one
to
an
other
,
if
the
quotient
of
the
one
,
found
out
by
the
greatest
common
diuisor
,
multiply
the
other
,
the
product
shall
be
the
least
number
that
may
be
diuided
by
them
two
.
36
prop.
7.
As
for
example
,
I
diuide
9
and
12
,
being
compound
one
to
another
by
3
,
their
greatest
common
diuisor
,
the
quotient
will
be
3
and
4.
Then
if
I
multiply
either
9
by
4
,
or
12
by
3
crossewayes
,
the
product
36
shall
be
the
least
number
that
may
be
commonly
diuided
by
9
and
12
,
the
numbers
giuen
:
III.
But
when
there
be
more
then
two
numbers
giuen
,
if
the
thirde
diuideth
that
which
the
two
former
diuided
,
the
first
numbers
found
out
shall
be
the
least
that
may
commonly
be
diuided
by
them
all
.
As
these
three
numbers
6.
10.
15.
compound
one
to
an
other
being
giuen
,
because
30
being
the
least
number
to
be
diuided
by
6
and
10
,
may
also
be
diuided
by
15
,
which
is
the
third
number
,
therefore
it
is
the
least
that
may
be
diuided
by
them
all
.
Likewise
in
4.
7.
28
,
because
28
being
the
least
number
that
may
commonly
be
diuided
by
4
and
7
,
it
may
also
be
diuided
by
the
third
number
28
,
as
by
it selfe
,
it
shall
be
the
least
number
that
may
be
diuided
by
the
numbers
giuen
.
As
in
these
examples
.
IIII.
But
if
the
third
number
doth
not
diuide
that
number
which
the
former
numbers
did
diuide
,
then
compare
the
least
diuidend
found
out
,
with
the
third
number
:
the
number
which
may
commonly
be
diuided
by
the
numbers
thus
compared
together
,
shal
be
the
least
that
may
be
diuided
by
them
all
:
and
so
forth
in
the
rest
.
38
prop.
7.
As
in
4
,
8
,
12
,
the
least
diuidend
is
24.
Item
in
6
,
16
,
28
,
the
least
common
diuidend
is
,
336
,
as
you
see
.
6
|
|
16
|
28
|
|
2
|
|
|
3
|
|
8
|
|
48
|
|
|
4
|
|
12
|
|
7
|
|
336
|
So
,
the
least
number
that
may
be
diuided
by
1
,
2
,
3
,
4
,
5
,
6
,
10
,
12
,
15
,
30
,
is
60
,
wherevpon
it
commeth
to
passe
,
that
this
number
hath
so
greatly
pleased
the
Astronomers
in
numbring
their
parts
.
1
|
2
|
3
|
4
|
5
|
6
|
10
|
12
|
15
|
30
|
The
number
giuen
.
|
2
|
|
|
|
|
|
|
|
|
The
numbers
compared
.
|
|
6
|
|
|
|
|
|
|
|
|
|
2
|
|
|
|
|
|
|
|
|
3
|
2
|
|
|
|
|
|
|
|
|
12
|
|
|
|
|
|
|
|
|
|
60
|
|
|
|
|
|
|
|
|
|
6
|
|
|
|
|
|
|
|
|
10
|
1
|
|
|
|
|
|
|
|
|
60
|
|
|
|
|
|
|
|
|
|
10
|
|
|
|
|
|
|
|
|
6
|
1
|
|
|
|
|
|
|
|
|
60
|
|
|
|
|
|
|
|
|
|
12
|
|
|
|
|
|
|
|
|
5
|
1
|
|
|
|
|
|
|
|
|
60
|
|
|
|
|
|
|
|
|
|
15
|
|
|
|
|
|
|
|
|
4
|
1
|
|
|
|
|
|
|
|
|
60
|
|
|
|
|
|
|
|
|
|
30
|
|
|
|
|
|
|
|
|
2
|
1
|
|
|
|
|
|
|
|
|
60
|
The
least
common
diuidend
.
|
CHAP.
VIII
.
Of
the
Accidentall
numeration
of
partes
which
commonly
they
call
Reduction
.
The
numeration
of
parts
is
-
Accidentall
:
and
is
called
Reduction
,
which
is
of
partes
-
vnto
parts
either
of
-
Least
termes
.
-
One
denomination
.
-
Vnto
whole
numbers
,
which
is
either
of
-
Proper
parts
.
-
Improper
parts
.
-
Or
-
Essentiall
,
which
is
-
Prime
,
as
-
Addition
.
-
Subduction
.
-
Second
,
as
-
Multiplication
.
-
Diuision
.
HItherto
we
haue
spoken
of
the
numeration
of
whole
numbers
:
now
followeth
the
numeration
of
parts
,
wherein
the
numbers
giuen
are
onely
parts
.
And
it
is
also
double
,
Accident
all
and
Essentiall
.
Accidentall
numeration
is
that
which
by
reduction
chaungeth
the
forme
and
fashion
of
the
parts
,
chaunging
their
value
nothing
at
all
.
This
numeration
in
one
word
is
commonly
called
reduction
,
where
the
parts
are
so
ordered
,
that
they
only
take
vnto
them
another
shape
,
such
an
one
as
is
fittest
for
the
worke
ensuing
.
Reduction
is
double
:
the
one
is
of
parts
vnto
parts
,
the
other
of
parts
vnto
whole
numbers
.
Reduction
of
parts
vnto
parts
,
is
that
which
findeth
out
other
parts
proportionable
to
the
parts
giuen
,
by
reducing
them
either
to
lesse
termes
or
to
termes
of
one
denomination
.
Reduction
of
parts
to
lesse
termes
,
is
that
which
diuideth
the
termes
of
the
partes
giuen
(
being
compound
numbers
one
to
another
)
by
their
greatest
common
diuisor
:
and
taketh
their
quotients
in
stead
of
their
numbers
giuen
.
Seeing
that
parts
haue
their
value
onely
of
the
proportion
that
is
betweene
the
vpper
number
and
the
neathermost
,
and
euery
proportion
is
best
knowen
when
it
is
set
downe
in
least
termes
:
therefore
in
the
partes
also
it
shall
not
be
amisse
to
reduce
them
which
are
written
in
greater
termes
(
that
is
,
with
numbers
compound
one
to
an
other
)
or
lesser
termes
,
the
which
thing
is
done
by
the
common
diuisor
.
For
the
quotients
haue
the
same
proportion
that
the
numbers
diuided
haue
,
as
appeareth
by
the
35
prop.
7.
So
that
their
value
is
not
chaunged
.
The
proofe
also
of
this
may
be
taken
out
of
the
17
prop.
7.
which
saieth
,
If
a
number
multiply
two
other
numbers
,
the
products
shall
haue
the
same
proportion
that
the
multipliers
haue
.
Contrariwise
therefore
if
a
number
diuide
other
two
numbers
,
the
quotients
and
the
numbers
diuided
shall
haue
one
proportion
the
one
to
the
other
.
As
if
4
multiply
6
and
8
,
their
products
24
and
32
shall
haue
the
same
proportion
which
the
numbers
giuen
haue
.
Againe
,
if
you
diuide
these
products
24
and
32
by
4
,
the
same
numbers
to
wit
,
6
and
8
will
returne
againe
in
the
quotient
.
If
the
parts
giuen
therefore
be
6/21
they
must
not
be
set
downe
in
these
termes
,
but
must
be
reduced
straight
wayes
to
2/7.
Likewise
4/16
diuided
by
4
,
come
to
1/4.
So
108/252
may
be
reduced
by
36
their
greatest
common
diuisor
to
3/7.
By
this
kinde
of
Reduction
not
onely
the
termes
of
the
same
partes
,
but
also
the
termes
of
diuerse
parts
may
be
reduced
crossewayes
one
by
an
other
:
that
is
,
the
numerator
of
the
one
,
and
the
denominator
of
the
other
.
As
2/3
and
5/8
their
termes
being
reduced
crosse-wayes
will
be
2/3
5/4.
But
this
rule
concerning
the
Reduction
of
diuerse
partes
,
is
not
so
generall
as
the
other
of
one
and
the
same
parts
.
For
it
is
more
speciall
and
peculiar
,
as
it
were
to
multiplication
,
as
we
shall
see
hereafter
,
wherefore
we
must
not
thinke
that
all
such
partes
giuen
may
be
reduced
after
this
manner
.
Reduction
of
parts
to
one
denomination
is
that
,
which
(
when
the
parts
giuen
are
of
diuerse
denomitations
)
taketh
first
the
least
number
which
may
be
diuided
by
the
denominators
,
for
the
common
Denominator
,
and
then
diuiding
that
number
by
the
denominators
of
the
parts
giuen
,
setteth
down
the
products
made
by
the
quotients
,
&
by
the
numerators
of
the
parts
giuen
for
the
numerators
of
the
partswhich
we
seeke
for
.
It
is
a
thing
of
great
vse
to
reduce
the
partes
giuen
to
one
denomination
,
and
to
make
them
proportionable
one
to
an
other
in
the
least
termes
that
may
be
:
that
is
;
of
partes
of
diuerse
kindes
,
to
make
partes
of
one
and
the
same
kind
.
As
for
example
if
3/4
and
5/6
were
to
be
brought
to
one
denomination
,
I
must
first
seeke
out
the
least
number
that
may
be
diuided
by
4
and
6
,
which
is
12
,
then
I
diuide
12
by
their
denominators
4
and
6
,
and
multiply
3
and
5
the
numerators
of
the
parts
giuen
by
the
quotients
3
and
2
,
the
products
9
and
10
shall
be
the
numerators
of
the
parts
sought
out
:
and
the
least
number
which
might
be
diuided
by
the
denominators
:
to
wit
,
12
,
shall
be
the
common
denominator
to
them
both
.
As
in
this
example
.
See
also
the
examples
following
.
In
the
first
example
therefore
,
the
parts
of
one
denomination
are
9/12
10/12
,
and
they
be
likewise
the
least
proportionable
to
the
parts
giuen
.
For
as
in
3/4
4
is
to
12
,
so
is
3
to
9
:
and
in
5/6
as
6
is
to
12
,
so
is
5
to
10
,
and
so
forth
in
the
other
examples
.
By
this
rule
of
reduction
we
may
knowe
whether
the
parts
being
of
diuers
denominations
be
equal
or
vnequall
.
For
being
brought
to
one
denomination
,
if
they
haue
the
same
numerators
they
be
equall
:
if
not
,
they
be
vnequall
:
they
that
haue
the
greatest
numerator
are
the
greatest
,
and
they
that
haue
the
least
numerator
are
the
lesse
.
As
if
it
should
fall
into
question
what
proportion
there
were
betwene
5/6
and
10/12.
Item
betwene
5/7
and
13/21
,
whether
they
be
equall
or
vnequall
,
this
rule
will
take
away
the
doubt
:
and
will
declare
that
the
former
parts
are
equal
,
the
latter
vnequal
,
and
5/7
to
be
the
greater
,
13/21
the
lesser
,
as
you
see
here
.
To
this
reduction
of
partes
to
parts
,
succeedeth
the
reduction
of
parts
to
whole
numbers
,
and
it
is
either
of
proper
or
improper
parts
.
The
first
reduction
of
proper
parts
taketh
the
numerators
of
many
partes
,
hauing
one
denomination
for
the
parts
giuen
.
As
3/5
and
4/5
are
reduced
to
whole
numbers
when
the
numerators
of
the
partes
giuen
3
and
4
,
being
as
it
were
proportionable
to
the
parts
giuen
,
are
taken
for
the
parts
themselues
:
so
likewise
we
reduce
these
partes
1/9
4/9
7/9
to
whole
numbers
,
when
we
take
1
,
4
,
7
,
the
numerators
for
the
parts
themselues
.
Reduction
of
improper
parts
taketh
the
quotient
of
the
numerator
diuided
by
the
denominator
,
for
the
parts
giuen
.
This
reduction
is
vsed
when
the
partes
are
but
one
,
and
their
value
either
equall
or
greater
then
the
whole
,
as
3/3
10/5
27/9.
If
you
diuide
the
numerators
of
euery
part
seuerally
by
his
denominator
,
the
quotients
will
be
1
,
2
,
3.
Hitherto
also
may
those
parts
be
referred
whose
denominators
doe
not
exactly
diuide
the
numerators
,
as
36/7
are
reduced
to
5
1/7
and
29/6
to
4
5/6.
CHAP.
IX
.
Of
the
Essentiall
numeration
of
parts
.
WE
haue
spoken
of
the
Accident
all
numeration
of
partes
.
Now
followeth
the
Essentiall
numeration
,
which
numbreth
the
parts
either
by
encreasing
,
or
diminishing
them
.
There
be
two
kindes
of
Essentiall
numeration
.
The
first
giueth
the
same
denominator
to
the
numerator
found
out
,
which
the
partes
haue
that
be
giuen
:
and
it
is
either
Addition
or
Subduction
.
Addition
addeth
together
the
numerators
hauing
one
denomination
,
and
setteth
the
common
denominator
of
the
parts
giuen
vnder
the
totall
.
Partes
of
diuerse
denominations
the
one
being
greater
or
lesse
then
the
other
,
can
not
be
added
together
without
confounding
of
the
denominators
,
wherefore
it
is
needfull
to
reduce
them
to
one
denomination
.
As
if
you
adde
3/5
to
4/5
the
totall
will
be
7/5
,
or
by
reduction
1
2/5.
Item
adde
3/4
to
5/8
,
here
if
you
should
confound
4
with
8
,
the
totall
should
haue
his
denominator
of
neither
of
them
both
.
Wherfore
the
parts
must
first
be
brought
to
one
denomination
,
and
then
be
added
together
.
As
in
this
example
.
The
Reduction
.
The
Addition
.
The
totall
.
Likewise
in
adding
1/2
vnto
5/6
and
3/11
there
arise
106/66
,
or
by
reduction
,
1
20/33.
The
example
is
thus
.
Reduction
.
Addition
.
The
parts
found
out
.
Subduction
is
a
numeration
which
subducting
the
numerators
of
the
parts
giuen
,
hauing
one
denomination
,
one
from
the
other
,
taketh
the
common
denominator
of
the
partes
giuen
,
for
the
denominator
of
the
remainder
.
As
in
subducting
3/8
from
7/8
there
remaines
4/8
or
1/2.
Item
subduct
1/3
from
10/13.
Here
for
so
much
as
the
parts
be
of
diuerse
denominations
,
you
must
reduce
them
to
13/39
and
30/39
,
then
subducting
the
numerators
one
from
another
,
that
is
,
13
from
30
,
there
remaines
17
,
vnder
which
you
must
set
the
common
denominator
,
so
that
the
partes
found
out
are
17/39.
As
in
this
example
.
Thus
much
of
the
first
kind
of
the
Essentiall
numeration
of
parts
:
the
second
kinde
giueth
to
the
numerator
found
out
an
other
denominator
then
the
parts
haue
which
are
giuen
.
And
it
is
either
Multiplication
or
Diuision
.
Multiplication
is
that
which
multiplying
the
termes
of
one
kinde
together
(
that
is
the
numerators
by
the
numerators
,
and
the
denominators
by
the
denominators
)
taketh
the
products
for
the
termes
of
the
parts
found
out
.
This
multiplication
is
properly
the
finding
out
of
other
parts
then
are
giuen
,
which
are
to
one
of
the
partes
giuen
,
as
the
other
is
to
an
vnitie
:
and
it
is
wrought
by
multiplying
both
the
numerators
and
the
denominators
of
the
parts
giuen
one
by
an
other
,
wherby
we
may
necessarily
conclude
,
that
the
partes
found
out
are
alwayes
lesse
then
the
parts
multiplyed
.
As
if
there
were
1/2
to
be
multiplyed
by
1/3
,
multiplying
the
numerators
and
the
denominators
one
by
the
other
,
the
product
is
1/6
,
which
is
to
1/3
,
as
1/2
is
to
1.
For
as
1/2
is
the
halfe
of
one
,
so
1/6
is
the
halfe
of
1/3
,
wherefore
1/6
is
lesse
then
1/3
or
1/2.
This
perhappes
may
be
perceiued
more
easily
in
the
numbring
of
some
certaine
thing
.
As
for
example
,
in
a
degree
there
be
60
minutes
,
therefore
1/2
of
a
degree
are
30
minutes
,
1/3
20
minutes
,
1/6
10
minutes
:
as
therefore
60
are
to
30
,
so
are
20
to
10.
Againe
in
a
shilling
there
are
twelue
pence
,
therefore
1/2
of
a
shilling
are
sixe
pence
,
1/3
foure
pence
,
1/6
2
pence
:
as
therefore
a
shilling
is
to
sixe
pence
,
so
are
foure
pence
to
two
pence
.
Item
if
5/7
be
multiplied
by
3/4
the
product
will
be
15/28.
The
abridgement
of
multiplication
.
If
the
opposite
termes
of
the
partes
giuen
be
numbers
compound
one
to
an
other
,
then
in
their
stead
take
their
quotients
found
out
by
their
greatest
common
diuisor
:
and
then
worke
the
multiplication
as
before
.
As
,
multiply
3/4
by
5/18.
Here
for
so
much
as
3
and
18
being
opposite
termes
,
are
compound
one
to
another
,
you
shall
set
downe
in
their
stead
their
quotients
1
and
6
found
out
by
their
common
Diuisor
,
so
that
the
parts
thus
reduced
shall
be
1/4
and
5/6.
Now
multiplying
1
by
5
,
and
4
by
6
,
the
product
will
be
5/24.
Likewise
if
3/7
should
be
multiplyed
by
14/15
,
first
you
must
reduce
them
to
1/1
and
2/5
,
which
being
multiplied
one
by
the
other
,
make
2/5.
As
in
these
examples
.
So
if
3/10
5/6
and
1/4
be
multiplied
together
,
they
giue
1/16.
For
first
3/10
and
5/6
being
multiplied
together
,
doe
make
1/4
,
which
being
afterwarde
multiplied
with
1/4
make
1/16.
Thus
:
Correlarie
gathered
of
the
former
abridgement
.
Of
this
reduction
of
numbers
this
consequence
is
gathered
.
If
the
crosse
or
opposite
termes
of
many
parts
be
equall
one
to
the
other
,
then
neglecting
them
which
are
equall
,
the
numerator
remayning
set
ouer
the
denominator
remaining
,
is
the
product
of
them
all
.
As
if
4/5
5/6
6/7
7/8
,
were
to
be
multiplied
together
,
I
cast
away
the
middle
numbers
which
are
equall
,
and
see
4
the
numerator
ouer
8
the
denominator
,
so
that
the
product
is
4/8
or
by
reduction
1/2.
The
reason
hereof
is
this
,
because
the
crosse
termes
being
as
it
were
compound
one
to
the
other
,
are
by
reduction
to
one
denominator
brought
to
vnities
thus
,
4/1
1/1
1/1
1/8.
But
for
so
much
as
an
vnitie
in
multiplication
chaungeth
not
a
number
,
therefore
the
figure
4
multiplied
by
the
three
vnities
in
the
same
ranke
,
maketh
but
4
,
no
more
doth
8.
Moreouer
it
appeareth
by
this
proportion
which
is
in
multiplication
,
that
seconde
parts
may
well
be
reduced
to
principall
parts
by
this
meanes
,
as
1/3
out
of
3/4
are
3/12
or
1/4
of
the
whole
,
which
thing
may
be
easily
perceiued
in
the
numbring
of
any
thing
,
as
3/4
of
a
degree
are
45
minutes
,
whose
third
part
are
15
minutes
,
which
are
equall
to
1/4
of
a
degree
.
So
2/3
of
●/4
parts
of
a
crowne
are
halfe
a
crowne
.
For
3/4
are
three
shillings
nine
pence
,
whose
2/3
parts
are
two
shillings
sixe
pence
.
So
5/6
1/2
of
3/8
by
multiplication
are
brought
to
15/96
or
5/32.
And
thus
much
of
multiplication
of
parts
.
Diuision
is
that
which
in
stead
of
the
partes
giuen
diuideth
whole
numbers
proportionable
vnto
them
.
The
diuision
of
parts
is
properly
the
finding
out
of
a
quotient
which
is
in
proportion
to
the
diuidend
,
as
an
vnitie
is
to
the
diuisor
.
Whereby
it
is
manifest
,
that
the
quotient
must
needes
be
greater
then
the
parts
which
are
diuided
,
otherwise
then
it
is
in
diuision
of
whole
numbers
:
for
there
the
quotient
is
alwayes
lesse
then
the
diuidend
.
This
inuention
commeth
of
the
diuision
of
such
numbers
as
are
proportionable
to
the
parts
giuen
,
for
so
much
as
the
partes
themselues
can
not
be
diuided
.
First
therefore
diuide
these
partes
of
one
denomination
,
to
wit
,
9/10
by
3/10.
Here
therefore
you
shall
reduce
them
(
by
the
former
reduction
to
whole
numbers
)
vnto
9
and
3
,
then
shall
you
diuide
9
by
3
,
the
quotient
will
be
3
,
which
declareth
that
3/10
the
diuisor
is
contayned
in
9/10
,
the
diuidend
three
times
,
and
as
3/10
are
to
one
,
so
are
9/10
to
three
:
the
which
quotient
therefore
is
bigger
then
the
partes
diuided
.
Diuided
5/6
by
3/6
,
that
is
,
see
how
many
times
3/6
are
contayned
in
5/6.
First
you
shall
reduce
the
partes
giuen
to
whole
numbers
proportionable
vnto
them
,
that
is
,
to
5
and
3
,
then
shall
you
diuide
5
by
3
,
the
quotient
will
be
1
2/3.
Contrariwise
,
3/6
diuided
by
5/6
giue
in
the
quotient
3/5.
For
both
the
denominators
being
neglected
,
I
set
the
one
numerator
ouer
the
other
as
parts
,
because
the
one
cannot
be
diuided
by
the
other
.
Furthermore
if
you
diuide
parts
of
diuerse
denominations
,
as
3/4
by
1/3
,
you
shall
first
make
the
partes
of
one
denomination
,
reducing
them
to
9/12
and
4/12
:
then
by
the
former
reduction
you
shall
bring
them
to
whole
numbers
,
and
diuide
9
by
4
,
the
quotient
will
be
2
1/4.
Diuide
8/9
by
5/6
,
reduce
the
partes
to
one
denomination
,
that
is
,
to
16/18
and
13/18
,
then
reduce
these
partes
hauing
one
denomination
to
whole
numbers
16
and
15
,
which
being
diuided
one
by
another
,
make
in
their
quotient
1
1/15.
CHAP.
X.
Of
mixt
numeration
.
Mixt
numeration
is
either
-
Accidentall
,
and
is
reduction
of
-
Whole
numbers
,
-
Or
-
Mixt
numbers
.
-
Essentiall
,
wherof
there
are
two
kinds
-
Prime
,
as
-
Addition
.
-
Subduction
.
-
Second
,
as
-
Multiplication
.
-
Diuision
.
HEtherto
of
simple
numeration
.
It
remayneth
now
to
speake
of
mixt
numeration
wherein
the
numbers
giuen
are
whole
numbers
and
parts
together
.
And
this
numeration
also
as
that
of
the
partes
is
double
.
First
Accident
all
,
which
beside
the
former
reduction
consisteth
of
the
reduction
both
of
whole
and
mixt
numbers
vnto
parts
.
Reduction
of
whole
numbers
vnto
parts
taketh
the
product
made
of
the
whole
numbers
giuen
,
&
the
denominator
of
the
parts
giuen
,
for
the
numerator
of
the
parts
which
we
seeke
for
.
This
reduction
may
be
drawen
out
of
the
●●●uction
of
improper
parts
,
as
it
were
a
consequence
following
it
.
For
as
you
reduce
4/4
by
diuision
vnto
1
whole
,
so
contrariwise
you
shall
bring
one
whole
to
4
quarters
,
if
you
multiply
one
whole
by
4
which
is
the
denominator
of
the
parts
,
for
they
will
return
to
4/46
Item
,
as
30/5
by
diuision
are
reduced
to
6
an
whole
number
:
so
if
you
multiply
6
by
5
the
denominator
of
the
partes
,
and
take
the
product
30
for
the
numerator
,
and
5
for
the
denominator
,
there
will
be
againe
30/5
which
are
equall
to
the
whole
numbers
giuen
.
Thus
whole
numbers
as
often
as
the
vse
of
numeration
requireth
,
are
written
as
parts
,
by
setting
an
vnitie
vnderneath
them
.
As
for
example
these
whole
numbers
,
2
,
5
,
9
,
hauing
an
vnitie
set
vnderneath
them
for
their
denominator
shall
be
as
it
were
partes
after
this
manner
,
259/111
Reduction
of
mixt
numbers
taketh
the
product
made
of
the
whole
numbers
giuen
,
and
the
denominator
of
the
partes
giuen
being
added
to
the
numerator
,
for
the
numerator
of
the
partes
found
out
.
As
you
may
reduce
4
3/7
into
partes
,
if
you
multiply
4
by
7
,
the
denominator
of
the
parts
giuen
,
and
adde
3
the
numerator
,
to
28
the
product
,
taking
the
totall
31
for
the
numerator
,
and
setting
the
denominator
giuen
vnderneath
it
,
thus
,
31/7.
Likewise
reducing
6
3/4
,
there
come
forth
27/4
,
and
so
foorth
in
the
rest
.
Thus
much
concerning
the
accidentall
numeration
of
mixt
numbers
:
now
remaine
the
speciall
kindes
of
numeration
,
whereof
the
first
numbreth
the
termes
of
mixt
numbers
seuerally
by
themselues
alone
:
that
is
,
the
whole
numbers
by
themselues
,
and
the
parts
by
themselues
,
and
that
by
adding
or
subducting
them
,
whereupon
it
is
called
addition
and
subduction
.
These
kinds
of
numeration
differ
nothing
from
the
former
,
but
are
in
proportion
all
together
like
vnto
them
.
The
examples
of
Addition
.
I.
Adde
3
and
2
4/5
the
totall
will
be
5
4/5.
II.
Item
adde
3/4
to
6
6/7
,
first
the
parts
added
together
are
45/28
,
or
by
reduction
1
17/28
,
the
totall
therefore
is
7
17/28.
III.
Adde
8
7/9
to
4
1/2
,
first
the
parts
added
together
are
23/18
,
or
by
reduction
1
5/18
,
then
adde
,
1
to
4
make
5
,
which
being
added
to
8
,
are
13.
So
that
the
totall
is
13
5/18.
The
examples
of
Subduction
.
I.
Take
from
3
the
whole
number
5/9
,
here
I
reduce
an
vnitie
of
the
whole
number
to
9/9
,
and
there
remaines
2.
Afterward
I
subduct
5/9
out
of
9/9
,
there
remaine
4/9.
The
remainder
therefore
of
the
numbers
giuen
to
be
numbred
is
,
2
4/9.
II.
Item
,
take
1/4
from
8
1/5.
Here
because
1/4
cannot
be
taken
out
of
1/5
,
to
wit
,
the
greater
from
the
lesse
:
I
take
1
from
8
,
there
remaine
7
,
then
I
bring
1
and
1/5
by
the
reduction
of
mixt
numbers
vnto
6/5
,
from
whence
(
after
I
haue
brought
the
parts
to
one
denomination
)
I
take
1/4
,
there
remaine
19/10.
So
that
the
remainder
of
the
numbers
giuen
is
,
7
19/20.
III.
Take
4
from
10
1/2
,
the
remainder
will
be
6
1/2.
IIII.
Take
5
7/12
from
7
5/8.
Here
first
you
shall
make
the
partes
of
one
denomination
,
that
is
,
15/24
14/24.
Then
taking
5
from
7
there
remaine
2
,
then
also
subducting
the
partes
hauing
one
denomination
one
from
the
other
,
there
remaines
1/24.
The
remainder
therefore
of
the
parts
giuen
is
,
2
1/24.
The
latter
kinde
of
mixt
numeration
numbreth
the
termes
of
the
mixt
numbers
giuen
ioyntly
together
:
that
is
,
the
whole
numbers
with
the
partes
,
and
that
by
multiplying
or
diuiding
them
:
whereupon
it
is
called
Multiplication
or
Diuision
.
If
3/4
were
to
be
multiplyed
by
5
,
setting
downe
the
whole
numbers
after
the
manner
of
parts
,
I
multiply
5/1
by
3/4
,
the
product
is
15/4
or
by
reduction
3
3/4.
The
value
of
the
parts
is
sought
out
after
this
manner
.
As
if
you
would
knowe
what
5/8
of
a
circle
were
,
I
set
downe
the
whole
partes
of
a
circle
were
,
I
set
downe
the
whole
partes
of
a
circle
,
that
is
,
360
degrees
after
the
common
order
of
partes
,
thus
,
360/1
:
then
multiplying
the
numbers
giuen
together
,
the
product
is
1800/8
,
or
by
reduction
vnto
whole
numbers
,
225.
So
many
degrees
are
5/8
of
a
circle
.
So
likewise
if
you
would
haue
1/7
of
10
●/8
,
first
you
shal
reduce
the
last
of
the
two
numbers
giuen
by
the
reduction
of
mixt
numbers
into
83/8
,
which
being
multiplyed
by
1/7
,
make
83/56
,
or
by
reduction
1
27/56.
Likewise
if
both
the
numbers
giuen
be
mixt
,
as
4
1/3
and
5
3/11
,
the
product
made
by
multiplication
will
be
754/33
,
or
by
the
reduction
of
improper
parts
,
22
28/33.
The
examples
of
Diuision
.
If
4
were
to
be
diuided
by
7/9
,
you
must
find
out
whole
numbers
proportionable
to
the
numbers
giuen
:
for
the
whole
4
being
reduced
into
9
,
the
parts
will
be
36/9
and
7/9
,
wherby
I
see
that
the
whole
numbers
36
and
7
,
are
proportionable
vnto
the
numbers
giuen
:
and
they
being
diuided
one
by
the
other
,
make
in
the
quotient
5
1/7.
Item
if
1/2
be
diuided
by
the
whole
number
3
,
3
being
reduced
vnto
6/2
,
I
finde
the
whole
numbers
1
and
6
to
be
proportionable
to
the
numbers
giuen
:
which
being
diuided
the
one
by
the
other
,
make
in
the
quotient
1/6.
Item
,
diuide
8
2/3
by
4
3/5
:
here
reduce
the
termes
of
both
numbers
by
the
reduction
of
mixt
numbers
to
26/3
and
23/5
,
then
make
them
of
one
denomination
thus
,
130/15
69/15
,
afterwards
reduce
them
to
whole
numbers
proportionable
vnto
them
,
that
is
,
to
130
and
69
,
which
being
diuided
one
by
the
other
,
the
quotient
shall
be
1
61/69.
The
end
of
the
first
part
.
THE
SECOND
PART
OF
ARITHMETICKE
.
CHAP.
XI
.
Of
the
kindes
of
proportionalitie
or
reason
.
E.
The
second
part
of
Arithmeticke
consisteth
in
the
comparison
.
-
Either
of
the
termes
,
and
is
called
proportionality
or
reason
,
whereof
-
The
one
is
in
equall
termes
:
and
is
called
the
proportionalitie
or
reason
of
equalitie
.
-
The
other
is
in
vnequall
termes
:
&
is
called
the
proportionalitie
or
reason
of
inequalitie
,
and
is
to
be
considered
-
Either
according
to
the
difference
of
the
termes
.
-
Or
according
to
their
quotient
:
and
herein
we
are
to
note
,
-
The
kinds
:
for
of
the
proportionalitie
of
inequalitie
some
is
,
-
Of
the
greater
inequality
,
&
that
is
-
Either
Prime
,
as
-
Multiplex
.
-
or
-
Simple
,
which
is
-
Muitiplex
Superparticular
.
-
Or
-
Multiplex
Superpartiens
.
-
Of
the
lesser
inequality
,
which
is
to
be
diuided
into
as
many
kinds
as
the
former
is
diuided
into
.
-
Or
The
handling
of
the
kinds
wherin
we
are
to
cōsidertheir
-
Notation
.
-
Numeration
,
which
is
,
-
Addition
.
-
Or
-
Subduction
.
-
Or
of
the
proportionalities
or
reasons
.
F.
HIther
to
of
Simple
,
now
followeth
Compare
Arithmeticke
,
which
consisteth
in
repressing
the
nature
and
the
qualitie
of
termes
compared
together
.
In
writing
of
proportions
the
numbers
whereby
they
are
expressed
,
are
by
a
Geometricall
phrase
called
terms
:
by
which
word
are
to
be
vnderstood
such
quantities
as
are
compared
together
.
Hereupon
generally
such
things
as
are
compared
together
are
called
Termes
.
The
comparing
of
termes
together
,
is
called
Proportion
,
which
is
a
certain
respect
which
the
terms
haue
one
to
another
according
to
their
quantitie
.
This
definition
in
this
place
is
more
general
then
the
3
def.
of
the
5
book
of
Euclide
,
which
faith
,
that
Proportion
is
a
certaine
respect
of
two
magnitudes
of
one
kinde
onely
,
according
to
their
quantitie
.
For
in
that
place
it
was
requisit
that
there
shold
be
mention
made
of
magnitudes
of
one
kind
only
,
because
that
magnitudes
are
not
al
of
one
kind
.
But
here
those
words
may
be
omitted
,
because
al
numbers
are
of
one
kind
,
&
euery
one
may
be
compared
together
:
and
moreouer
the
analogie
or
respect
which
they
haue
one
to
another
,
may
be
expressed
by
an
other
number
,
which
thing
cannot
be
done
in
all
magnitudes
,
though
they
be
all
of
one
kind
itrationall
,
and
surde
magnitudes
can
not
be
expressed
by
number
.
The
comparison
of
termes
,
is
either
equall
or
vnequall
,
whereupon
it
is
called
Proportion
of
equalitie
or
vnequalitie
.
Proportion
of
equalitie
is
one
only
,
and
not
diuided
into
many
kinds
.
As
the
proportion
of
5
to
5
,
of
6
to
6
,
of
12
to
12.
For
both
in
these
,
and
in
all
other
equall
termes
,
the
respect
is
all
one
,
neither
hath
it
any
more
kindes
,
because
that
one
and
the
same
number
cannot
be
either
more
or
lesse
equall
to
another
.
Proportion
of
inequality
is
knowne
either
by
the
difference
of
the
termes
,
or
by
their
quotient
.
The
proportion
of
difference
,
is
that
whereby
one
terme
differeth
from
another
.
The
proportion
of
the
quotient
is
when
one
terme
contayneth
another
.
The
proportion
of
difference
is
known
by
subduction
,
the
proportion
of
the
quotient
is
knowne
by
diuision
.
As
in
the
naturall
order
of
the
numbers
the
difference
betwene
1
and
2
,
2
and
3
,
3
and
4
,
4
and
5
,
5
and
6
,
&c.
is
all
one
,
that
is
an
vnitie
.
Againe
,
the
difference
betweene
8
and
5
is
3
,
betweene
6
and
4
it
is
2.
But
the
proportion
of
the
quotient
is
by
an
especiall
phrase
simply
called
proportion
.
As
,
the
proportion
of
2
to
1
is
double
,
of
3
to
2
sesquialter
,
of
24
to
3
octuple
.
For
the
former
termes
(
which
commonly
are
called
the
antecedents
)
being
diuided
by
the
termes
following
(
which
are
called
the
consequents
)
the
quotients
are
1
,
1
1/2
,
8
:
whereby
we
may
gather
that
there
may
be
now
and
then
the
same
difference
betwene
the
termes
though
as
touching
the
quotient
their
proportion
differ
,
as
in
4
8
and
12
,
the
difference
betwene
4
and
8
,
and
betwene
8
and
12
,
is
all
one
,
yet
the
proportion
gathered
by
their
quotients
is
diuerse
,
for
in
the
one
it
is
2
,
double
,
in
the
other
it
is
but
sesquialter
1
1/2.
Contrariwise
also
in
termes
the
proportion
of
the
quotient
may
be
all
one
,
though
their
difference
be
diuerse
:
as
in
3
and
9
,
4
and
12
,
the
quotient
is
3
triple
in
them
both
:
yet
the
difference
is
not
a
like
,
for
in
the
former
it
is
6
,
in
the
second
it
is
8.
Prōportion
of
inequalitie
is
double
,
for
either
the
greater
terme
is
antecedent
to
the
lesse
,
or
consequent
.
Hereupon
the
one
is
called
the
proportion
of
the
greater
inequalitie
,
the
other
the
proportion
of
the
lesser
inequalitie
.
When
two
termes
are
compared
together
,
the
one
is
called
the
Antecedent
,
the
other
the
consequent
.
Therefore
,
as
often
as
the
greater
terme
is
the
antecedent
and
goeth
before
,
it
is
called
the
greater
inequalitie
.
As
when
6
is
compared
to
3
,
and
3
to
2.
But
if
the
lesser
terme
be
the
Antecedent
,
and
we
consider
what
difference
there
is
betweene
that
and
the
greater
,
it
is
called
the
lesser
inequalitie
.
Both
these
fornamed
proportions
are
either
Prime
or
Seconde
.
Prime
,
is
a
proportion
of
one
kinde
onely
,
and
is
either
simple
or
multiplex
.
Simple
proportion
,
is
when
the
greater
terme
contayneth
the
lesse
once
and
somewhat
more
,
as
proportion
Superparticular
and
Superpartiens
.
Superparticular
,
is
a
simple
proportion
wherein
the
great●●●
terme
contayneth
the
lesse
,
and
one
part
more
of
the
lesse
.
As
when
3
is
compared
to
2
,
6
to
5
,
8
to
7.
For
in
these
proportions
the
antecedent
contayneth
the
consequent
once
,
and
one
part
beside
of
the
lesser
terme
.
As
it
may
be
perceiued
by
their
quotients
when
they
be
diuided
thus
,
1
1/2
,
1
1/5
,
1
1/7.
This
kinde
contayneth
vnder
it
infinite
other
kindes
,
whereof
the
first
is
sesquialter
,
the
second
sesquitertia
,
the
fourth
sesquiquarta
.
Item
there
is
the
same
proportion
betwene
12
and
8
,
that
is
betwene
15
and
10
,
as
appeareth
by
their
termes
reduced
to
the
least
denomination
thus
,
1
1/2.
Superpartiens
is
a
simple
proportion
,
wherein
the
greater
terme
contayneth
the
lesse
,
and
●ertaine
parts
of
it
.
As
5
to
2
,
7
to
4
,
13
to
8.
For
the
antecedents
being
diuided
by
the
consequents
,
their
quotients
declare
their
proportion
on
this
manner
,
1
2/3
,
1
3/4
,
1
5/8.
Likewise
the
proportion
betwene
15
and
9
is
superpartiens
,
as
appeareth
by
the
quotient
and
the
parts
reduced
to
the
least
denomination
.
1
2/3.
Multiplex
is
a
prime
proportion
when
the
greater
terme
contayneth
the
lesse
certaine
times
exactly
.
Multiplex
is
defined
after
the
same
manner
in
the
second
defin
.
of
the
fifth
booke
of
Euclide
,
which
saith
,
that
Multiplex
is
a
greater
quantitie
in
respect
of
a
lesse
,
when
the
lesse
measureth
the
greater
.
As
the
proportions
of
4
to
2
,
9
to
3
,
16
to
4
are
multiplices
,
that
is
to
say
,
double
,
triple
,
quadruple
:
for
their
quotients
are
2
,
3
,
4.
Second
proportion
consisteth
of
two
kindes
of
proportion
,
and
is
either
multiplex
superparticularis
,
or
multiplex
superpartiens
.
It
seemeth
that
these
kinds
of
proportions
are
made
of
the
former
,
by
ioyning
two
of
their
names
together
.
For
the
one
is
the
third
kinde
of
proportion
ioyned
to
the
first
,
and
the
other
is
the
third
ioyned
to
the
second
.
Multiplex
superparticular
is
is
a
compound
proportion
,
wherein
one
terme
contayneth
another
certaine
times
,
and
one
part
besides
.
As
5
to
2
,
10
to
3
,
13
to
4.
For
the
quotients
declaring
their
proportions
,
are
2
1/29
3
1/3
,
3
1/4.
Multiplex
superpartiens
,
is
when
the
one
terme
containeth
the
other
,
&
besides
more
parts
then
one
.
Here
we
are
to
vnderstande
,
that
they
be
the
true
partes
which
are
not
equall
to
the
lesser
terme
,
as
in
11
to
4
,
12
to
5
,
22
to
6.
For
the
quotients
are
2
3/4
,
2
2/3
,
3
2/3.
CHAP.
XII
.
Of
Notation
and
Numeration
of
proportions
.
THus
much
for
the
kinds
of
proportion
,
now
followetg
the
handling
and
practise
of
them
:
which
consisteth
in
Numeration
and
Notation
.
Notation
is
the
writing
of
the
antecedents
aboue
,
and
the
consequents
beneath
.
The
Antecedents
4
8
6
Or
5
1
The
Consequents
3
3
2
Or
6
2
The
proportions
being
written
,
are
expressed
by
the
quotient
(
the
parts
if
there
be
any
,
being
reduced
to
the
least
termes
)
which
declareth
as
well
their
proportion
as
their
denomination
.
In
multiplex
proportion
it
is
called
double
,
triple
,
quadruple
,
octuple
proportion
,
where
the
quotients
are
2
,
3
,
4
,
8.
In
proportions
of
lesser
inequalitie
this
preposition
Sub
is
generally
added
in
them
all
.
As
for
example
,
it
is
called
in
Submultiplex
,
Subduple
,
Subtriple
,
Subquadruple
,
&c.
In
Superparticular
proportion
(
because
the
halfe
of
the
greater
number
is
the
proportion
)
it
is
called
after
the
manner
of
the
Latines
Sesquialtera
,
Sesquitertia
,
&c.
putting
the
worde
Sequi
in
the
beginning
,
and
the
denominator
of
the
partes
in
the
ending
as
when
the
quotients
are
1
1/2
,
1
1/3
,
1
1/4.
In
proportions
of
the
lesser
inequalitie
we
say
,
Subsesquialtera
,
Subsesquitertia
,
&c.
Likewise
Submultiplex
,
Superparticular
,
Submultiplex
Superpartiens
.
For
in
compound
proportions
the
names
are
also
compound
,
as
in
3
1/1
triple
sesquiquarta
,
2
4/4
dupla
superquadripartiens
,
&c.
And
thus
much
for
instruction
sake
.
For
I
know
that
these
wordes
are
very
straunge
to
vs
,
and
not
heard
of
,
nor
vsed
among
the
best
writers
.
For
they
when
they
would
speake
of
the
proportions
of
things
,
were
wont
to
expresse
them
by
two
numbers
.
As
Archimedes
in
expressing
the
proportion
of
the
circumference
of
a
circle
to
the
Diameter
,
sayd
not
that
it
was
triple
sesquitertia
,
but
as
22
was
to
7.
Moreouer
the
quotients
teach
vs
the
way
not
onely
how
to
expresse
the
proportions
giuen
,
but
also
how
to
set
them
downe
in
the
least
termes
.
For
if
you
set
an
vnitie
vnderneath
,
or
ouer
against
the
quotients
being
whole
numbers
,
you
shall
find
out
the
least
termes
of
the
multiplex
which
you
seeke
for
.
As
in
the
multiplices
,
if
you
set
an
vnitie
right
against
3
,
4
,
5
,
the
proportion
in
the
least
termes
will
be
triple
,
quadruple
,
quintuple
.
Againe
,
if
you
reduce
the
quotients
being
mixt
numbers
,
by
the
seconde
reduction
of
mixt
numbers
vnto
partes
,
you
shall
in
like
manner
finde
out
the
least
termes
in
the
other
kind
of
proportions
:
as
in
Superparticular
proportion
,
if
you
seeke
for
the
least
termes
of
1
1/2
sesquialtera
,
1
1/3
sesquitertia
,
1
1/8
sesquioctaua
,
then
the
quotients
reduced
after
the
foresaide
manner
,
the
least
termes
will
be
these
.
The
like
is
in
all
the
other
kinds
,
as
if
you
would
find
out
the
least
termes
in
duple
superpartiens
fiue
sixe
parts
,
then
shall
you
of
2
5/6
make
the
antecedent
17
,
and
the
consequent
6
,
and
they
shall
be
the
least
termes
in
the
proportion
assigned
,
and
so
foorth
in
the
rest
.
Thus
much
for
notation
.
Numeration
is
the
addition
or
subtraction
of
proportions
.
Addition
,
is
that
which
taketh
the
products
made
by
multiplying
the
like
termes
together
,
(
that
is
,
by
multiplying
the
antecedents
by
the
antecedents
,
and
the
consequents
by
the
consequents
)
for
the
termes
of
the
compound
portion
.
The
adding
of
proportions
is
set
downe
in
the
fifth
def.
of
the
sixth
booke
thus
.
A
Proportion
is
said
to
be
made
of
proportions
,
when
the
quantities
of
the
proportions
multiplyed
together
produce
another
proportion
.
By
the
quantities
of
the
proportions
he
meaneth
the
quotients
whereof
the
proportions
take
their
names
.
As
if
you
adde
a
sesquialtera
to
an
equall
proportion
,
it
maketh
a
sesquialtera
.
Adde
a
sesquiquarta
to
a
duple
superpartiens
two
third
parts
,
the
product
will
be
a
sesquitertia
:
as
you
see
in
these
examples
.
|
Simple
proper
.
|
Compound
proport
.
|
Simple
.
|
Compound
.
|
Anteced
.
|
3
|
4
|
12
|
1
|
8
|
40
|
Conseq
.
|
2
|
4
|
8
|
4
|
3
|
12
|
That
is
,
adde
|
1
1/2
vnto
0
|
1
1/2
|
1
1/4
vnto
2
2/3
|
3
1/3
|
That
the
products
made
of
multiplying
the
termes
together
,
haue
a
compound
proportion
appeareth
by
the
5
prop.
of
the
eight
booke
of
Euclide
,
which
saith
,
that
Plaine
or
superficiall
numbers
,
are
in
that
proportion
one
to
the
other
which
is
composed
of
their
sides
.
As
3
and
12
being
plaine
numbers
,
haue
proportion
one
to
another
,
according
to
their
sides
,
as
you
see
.
The
sides
.
|
The
plaine
numbers
.
|
3
|
4
|
12
|
1
|
3
|
3
|
If
diuerse
proportions
be
to
be
added
together
,
first
adde
two
of
them
,
then
adde
the
compound
to
that
which
remaineth
.
As
if
the
proportions
giuen
were
,
First
two
of
them
added
together
,
make
18/5
to
which
the
third
proportion
being
added
,
maketh
a
Sextuple
.
120/22
:
6.
Thus
euery
proportion
may
be
tripled
or
quadrupled
:
if
the
termes
of
the
proportion
giuen
being
set
downe
three
or
foure
times
be
added
together
after
the
foresaide
manner
.
So
a
double
proportion
being
tripled
,
maketh
an
octuple
.
And
a
sesquialtera
being
doubled
foure
times
,
maketh
a
quintuple
sesquidecima
,
as
you
see
here
.
Subduction
of
proportions
,
is
that
which
taketh
the
products
made
of
the
opposite
termes
of
the
proportion
giuen
,
for
the
termes
of
the
proportion
remaining
.
Which
proportion
is
least
,
and
which
may
be
subduced
one
from
the
other
in
Multiplices
may
be
easily
knowne
by
the
denomination
onely
.
But
in
the
other
(
the
termes
being
compared
together
as
partes
)
it
may
be
gathered
by
that
which
hath
bene
taught
before
in
the
partes
.
Subduction
seemeth
to
be
the
consequence
of
that
which
was
set
downe
in
the
fifth
def
of
the
sixt
booke
.
For
if
the
addition
of
proportions
be
wrought
by
multiplycation
,
it
followeth
then
that
subduction
must
be
wrought
by
diuision
.
Therefore
whether
you
diuide
the
quotients
one
by
the
other
,
or
multiply
the
opposite
termes
of
the
partes
giuen
one
by
the
other
(
for
in
partes
the
multiplying
of
the
termes
crossewayes
is
as
much
as
diuision
)
the
termes
produced
will
haue
a
proportion
left
like
vnto
the
former
.
As
take
a
sesquialtera
from
a
double
proportion
,
the
remainder
will
be
a
sesquitertia
.
Item
,
take
a
sesquiquarta
from
a
double
sesquialtera
,
there
remaineth
a
double
,
as
in
these
examples
.
|
The
prop.
giuen
.
|
The
rem
.
|
The
propor
.
giuen
.
|
The
remainder
.
|
Anteced
.
|
2
|
3
|
4
|
5
|
5
|
20
|
2
|
|
|
|
|
|
|
or
|
Conseq
.
|
1
|
2
|
3
|
2
|
4
|
10
|
1
|
|
2
from
1
1/2
|
1
1/3
|
2
1/2
from
1
1/4
|
2
|
There
is
no
multiplication
in
proportions
properly
as
in
numbers
,
vnlesse
you
take
the
continuing
of
the
proportions
for
their
multiplication
.
Neither
is
there
any
diuision
required
in
them
beside
the
finding
out
of
one
or
more
proportionals
.
Wherefore
we
will
proceede
to
the
rest
.
CHAP.
XIII
.
Of
proportion
and
the
kinds
thereof
,
but
especially
of
Arithmeticall
proportion
.
F.
The
comparison
of
reasons
,
is
-
Either
of
vnequall
reasons
,
and
is
called
improportion
,
whereo●
in
the
8
d.
5.
with
the
which
Arithmeticke
doth
not
meddle
.
-
Or
of
equall
reasons
,
and
is
called
proportion
,
consisting
-
Either
in
the
equalitie
of
the
differences
:
and
is
called
Arithmeticall
proportion
,
which
is
-
Either
disiunct
,
-
Or
continuall
,
and
this
againe
is
,
-
Either
simple
,
-
Or
multiplex
,
&
is
called
progression
:
wherin
we
are
to
obserue
,
-
Either
the
order
of
the
termes
,
-
Continuall
,
as
they
follow
one
another
.
-
Or
-
Interrupted
,
when
we
seeke
but
for
certaine
termes
.
-
Or
the
summe
of
the
termes
.
-
Or
in
the
equalitie
of
the
quotients
:
and
is
called
Geometricall
proportion
.
G.
NOw
that
we
haue
spoken
of
comparing
the
termes
together
it
remayneth
to
speake
of
the
comparison
of
reasons
,
wherein
first
we
haue
to
deale
with
proportion
.
Proportion
is
the
equalitie
of
reasons
.
Euclide
in
the
fourth
definition
of
the
fifth
booke
,
defineth
proportion
to
bee
a
similitude
of
reasons
.
Aristotle
in
the
fifth
booke
and
3.
Chapter
of
the
Ethickes
,
calleth
it
an
equalitie
of
reasons
.
If
wee
shoulde
define
it
to
be
the
Identitie
of
reasons
,
we
should
not
varie
much
from
the
meaning
of
Euclide
,
who
in
the
sixth
definition
of
the
fifth
booke
speaketh
after
the
same
manner
,
calling
them
proportionall
quantities
which
haue
the
same
reason
one
to
another
.
And
albeit
that
these
definitions
doe
properly
agree
to
Geometricall
proportion
,
yet
if
you
take
the
word
reason
in
the
largest
sense
,
for
any
kinde
of
respect
it
shall
be
generall
,
so
that
as
reason
or
proportionalitie
is
a
mutuall
respect
of
two
termes
one
to
another
,
so
proportion
shall
be
a
respect
of
like
termes
one
to
another
.
Proportion
is
either
Arithmeticall
or
Geometricall
.
Arithmeticall
proportion
is
an
equalitie
of
differences
.
As
in
8
,
7
,
6
,
and
in
12
,
9
,
6
,
3.
Here
are
two
proportions
,
wherein
you
may
see
an
equalitie
of
differences
,
or
an
equall
excesse
of
the
antecedents
.
In
the
first
,
the
difference
is
1
,
in
the
latter
it
is
3.
So
likewise
in
15
,
11
:
8
,
4
,
as
15
exceede
11
by
4
,
so
8
exceede
4
by
4.
Arithmeticall
proportion
is
double
,
for
either
it
is
continued
in
the
termes
,
or
it
is
seuered
,
whereupon
it
is
called
continuall
,
or
disiunct
proportion
.
These
differences
are
common
both
to
Arithmeticall
,
and
to
Geometricall
proportion
.
That
proportion
is
continuall
wherein
not
onely
the
extremes
are
continued
together
by
one
meane
,
but
generally
euerie
proportion
,
wherein
many
and
sundrie
middle
termes
haue
a
continuall
respect
one
to
another
,
so
that
any
of
them
may
be
eyther
the
antecedent
or
the
consequent
.
In
so
much
,
that
not
onely
the
proportion
betwene
3
,
2
,
1
is
continuall
:
where
2
is
the
consequent
of
the
former
,
and
the
antecedent
of
the
latter
proportion
:
but
this
also
5
,
7
,
9
,
11
,
where
the
same
difference
is
continued
from
the
first
to
the
last
.
And
any
of
the
middle
numbers
either
7
or
9
,
haue
the
same
nature
in
respect
of
them
that
goe
before
and
follow
after
,
that
2
had
in
the
former
example
.
Proportion
is
seuered
in
the
termes
when
the
first
onely
hath
the
same
respect
to
the
second
,
that
the
third
hath
to
the
fourth
,
but
the
second
agreeth
not
proportionablie
with
the
thirde
:
that
is
,
when
all
the
termes
in
order
as
they
followe
are
not
of
like
proportion
one
to
the
other
.
As
10
,
8
,
4
,
2
,
are
seuerally
proportionall
because
2
the
first
difference
is
not
continued
betwene
8
and
4.
So
that
here
is
a
seuering
not
onely
of
the
termes
,
but
also
of
the
proportions
.
Leauing
therefore
in
this
place
to
speake
of
disiunct
proportion
,
wee
will
handle
that
which
is
continuall
.
Continuall
proportion
is
either
Simple
,
where
the
extreames
haue
but
one
middle
terme
:
or
manifold
,
wherein
many
middle
termes
go
on
in
order
continually
:
whereupon
it
is
called
progression
.
Progression
is
a
continuall
enlarging
of
the
termes
of
the
proportion
giuen
:
wherein
there
are
two
extreames
giuen
,
and
manie
other
numbers
in
the
middle
betwene
them
.
In
Arithmeticall
progression
we
are
to
finde
out
two
things
,
either
the
number
of
the
termes
as
they
stand
in
order
,
or
else
their
summe
.
Sometimes
the
termes
be
all
sought
for
as
they
follow
in
order
,
and
are
found
out
by
the
continuall
adding
of
the
difference
to
the
last
terme
in
the
progression
.
As
if
you
would
know
the
termes
of
this
progression
1
,
5
,
9
,
13
,
as
they
follow
in
order
,
you
must
adde
4
,
which
is
the
difference
vnto
13
,
then
the
fifth
terme
will
be
17
,
to
which
if
you
adde
4
againe
,
it
maketh
21
,
which
is
the
sixth
terme
,
and
so
foorth
infinitely
.
Sometime
breaking
off
their
naturall
order
,
we
seeke
for
some
one
terme
in
the
progression
,
which
may
be
found
out
by
the
Theoreme
following
.
The
progression
increasing
,
if
you
take
an
vnitie
from
the
name
of
the
terme
which
you
seeke
for
,
&
adde
the
product
made
of
the
remainder
,
and
the
common
difference
of
the
termes
,
vnto
the
first
terme
,
the
totall
will
be
that
which
you
desire
.
The
progression
decreasing
,
if
that
product
be
taken
from
the
first
terme
the
remainder
is
the
terme
sought
for
.
As
in
this
progression
4
,
7
,
10
,
where
the
difference
is
3
,
I
desire
to
know
the
12
terme
beginning
at
4.
Here
12
is
the
name
of
the
terme
which
I
seeke
for
,
from
whence
I
take
1
,
and
multiply
11
which
remaine
by
3
the
difference
,
the
product
is
33
,
whereto
I
adde
4
the
first
terme
.
So
the
totall
is
37
the
terme
which
I
sought
for
.
Likewise
the
30
terme
is
91
,
and
so
foorth
.
The
termes
|
4
|
7
|
10
|
37
|
91
|
The
names
or
number
of
the
termes
.
|
1
|
2
|
3
|
12
|
30
|
Thus
much
for
the
finding
out
of
the
termes
.
The
finding
out
of
the
summe
is
the
halfe
of
the
product
made
by
multiplying
the
two
extremes
added
together
,
and
the
name
of
the
last
terme
,
as
it
standeth
in
order
.
As
in
this
example
,
1
,
5
,
9
,
13
,
17
,
21
,
both
the
extreames
added
together
make
22
,
the
name
of
the
last
terme
is
6
,
for
it
is
the
sixt
in
order
.
Whereby
if
you
multiply
22
,
the
product
will
be
132
,
the
halfe
whereof
(
that
is
)
66
,
is
the
summe
sought
for
.
This
rule
may
be
proued
by
the
41
prop.
of
the
first
booke
of
Euclide
,
which
saith
,
If
a
parallelogramme
and
a
triangle
haue
one
and
the
selfe
same
base
,
and
be
in
the
same
altitude
,
the
parallelogramme
shall
be
double
to
the
triangle
.
The
numbers
then
of
the
progression
represent
the
triangle
:
whereof
one
side
is
the
name
of
the
last
terme
,
or
(
as
they
commonly
terme
it
)
the
number
of
the
places
:
the
other
is
the
extremes
added
together
.
But
for
so
much
as
the
product
made
of
these
two
sides
maketh
not
a
triangle
,
but
a
parallelogramme
,
therefore
in
taking
the
one
halfe
I
gather
by
it
how
bigge
the
platforme
of
the
triangle
is
,
which
is
the
summe
of
the
progression
.
The
example
of
this
progression
2
,
4
,
6
,
8
,
10
,
is
set
downe
here
as
you
see
:
5
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
0
|
0
|
The
whole
parallelogram
is
60
,
wherfore
30
which
is
the
halfe
,
is
the
summe
of
the
progression
.
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
0
|
0
|
0
|
0
|
1
|
1
|
1
|
1
|
1
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
1
|
1
|
1
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
1
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
|
12
|
|
To
set
forth
the
profite
and
vse
of
this
in
resoluing
certaine
questions
,
let
this
be
the
first
example
.
For
so
much
as
a
naturall
day
hath
24
houres
,
I
demaund
how
many
strokes
the
clocke
will
strike
in
a
day
.
In
this
example
we
seeke
for
the
number
of
the
strokes
beginning
at
one
,
and
adding
one
till
you
come
to
the
24
terme
.
After
this
manner
the
clockes
in
Italy
and
Norinberge
are
made
.
Both
the
extremes
added
together
make
25
,
which
being
multiplyed
by
24
,
the
name
of
the
last
place
,
make
600
,
whose
halfe
are
300
strokes
,
which
is
the
summe
sought
for
.
Another
.
There
be
30
egges
laide
in
a
rancke
,
euery
one
three
foote
from
another
.
I
must
put
them
in
a
basket
which
standeth
13
foote
from
the
first
egge
:
yet
so
that
at
euery
time
I
goe
to
and
fro
I
must
take
vp
but
one
.
The
question
is
,
how
farre
a
man
may
go
before
I
haue
gathered
them
all
.
This
was
wont
to
be
a
pastime
vsed
amongst
young
men
,
when
one
gathered
the
egges
and
put
them
in
a
basket
without
breaking
them
,
and
another
went
to
a
certaine
place
and
came
againe
,
so
he
which
first
had
done
his
taske
,
got
the
game
.
Here
the
question
is
about
the
summe
of
the
Arithmeticall
progression
.
13
|
16
|
19
|
40
|
64
|
100
|
1
|
2
|
3
|
10
|
18
|
30
|
Both
the
extremes
together
are
113
,
th●
product
made
of
this
summe
,
and
the
name
of
the
last
terme
is
3390
,
whose
halfe
is
1693
feete
,
which
make
339
Geometricall
paces
.
This
space
he
goeth
and
commeth
,
wherefore
the
double
of
it
which
was
first
found
out
,
that
is
3390
feete
,
or
678
paces
is
the
sum
which
we
sought
for
.
CHAP.
XIIII
.
Of
Geometricall
proportion
,
and
of
the
golden
rule
belonging
thereunto
.
G.
Geometricall
proportion
is
two
fold
.
-
Disiunct
:
out
of
whose
proprietie
a
riseth
the
golden
rule
:
and
that
is
,
-
Simple
:
simple
proportions
is
-
Direct
:
before
which
there
goeth
-
Either
no
kind
of
Arithmeticall
account
Chap.
14.
-
Or
some
Arithmeticall
account
,
and
that
-
Either
simple
,
as
the
numeration
of
-
Whole
numers
.
Chap.
15.
-
Parts
or
fractions
.
Chap.
16.
-
Mixt
numbers
.
-
Chap.
17.
Or
,
compared
as
the
proportion
-
Arithmeticall
.
-
Geometricall
.
-
Or
Reciprocall
.
Chap.
18.
-
Or
Manifold
.
H.
-
Coniunct
or
Continuall
.
L.
THus
much
of
Arithmeticall
proportion
,
Geometricall
followeth
,
which
is
the
equalitie
of
reasons
.
Here
this
word
Reason
is
not
taken
generally
,
but
specially
,
for
the
equalitie
of
the
quotients
arising
of
the
numbers
compared
together
.
As
in
2
,
6
,
4
,
12
,
what
reason
or
proportionalitie
is
betweene
2
and
6
,
the
same
is
betweene
4
and
12
,
that
is
,
subtriple
.
And
if
foure
numbers
be
directly
proportional
,
they
shall
be
also
proportionall
backwarde
,
and
crossewayes
in
this
manner
.
2
|
6
|
4
|
12
|
Direct
.
|
6
|
2
|
12
|
4
|
Backward
.
|
2
|
4
|
6
|
12
|
Crossewaies
.
|
Geometricall
proportion
is
double
:
Disiunct
,
or
Continuall
.
Disiunct
is
that
,
that
agreeth
neither
in
termes
nor
proportionalitie
,
or
reason
.
As
in
6
,
3
,
16
,
8
,
for
here
are
two
double
proportions
distinquished
both
in
termes
and
in
reason
,
because
that
betweene
3
and
16
there
is
not
the
same
reason
obserued
,
that
is
betwene
6
and
3
and
betweene
16
and
8.
Disiunct
proportion
consisteth
of
foure
termes
at
the
least
.
It
hath
this
propertie
that
the
product
of
the
extremes
,
is
equall
to
the
product
made
of
the
middle
numbers
,
and
contrariwise
.
16.
p.
6.
&
19.
p.
7.
Among
the
properties
of
Geometricall
proportion
set
foorth
by
Euclide
,
this
is
most
excellent
and
necessarie
to
be
knowne
,
because
the
force
and
fruitfulnesse
thereof
is
such
,
that
it
may
very
well
be
called
the
head
of
cyphering
,
that
is
,
the
foundation
and
off-spring
of
all
rules
,
as
we
shall
see
hereafter
.
It
hath
this
meaning
,
that
the
number
made
by
multiplying
of
the
first
into
the
fourth
,
is
equall
to
the
number
made
by
multiplying
the
seconde
into
the
thirde
,
as
in
6
,
4
,
18
,
12
,
the
product
made
of
6
,
multiplyed
by
12
is
72
,
the
which
number
ariseth
by
multiplying
the
middle
numbers
4
and
18
one
by
another
.
And
this
thing
is
proper
to
continuall
,
and
generally
to
all
Geometricall
proportion
,
as
wee
shall
see
hereafter
.
This
propertie
ministreth
vnto
vs
the
golden
Rule
,
which
three
proportionall
numbers
being
giuen
,
diuiding
the
product
made
by
multiplying
the
third
into
the
second
by
that
which
remaineth
,
findeth
out
the
fourth
proportionall
number
.
Out
of
the
fore named
propertie
,
as
out
of
a
most
plentifull
spring
is
drawen
this
Rule
,
wherein
consisteth
the
chiefest
partes
of
casting
accounts
.
The
which
some
for
the
excellencie
of
it
doe
call
the
Rule
,
others
the
rule
of
Proportion
,
other
the
rule
of
three
,
and
all
men
all
most
in
generall
,
for
the
excellent
vse
of
it
,
doe
call
it
the
golden
Rule
.
The
drift
and
end
of
it
,
is
the
finding
out
of
a
fourth
terme
,
or
number
,
that
shall
haue
the
same
proportion
with
the
three
numbers
giuen
:
the
which
thing
it
bringeth
to
passe
by
multiplying
two
termes
of
the
proportions
giuen
one
by
the
other
,
that
is
,
the
antecedent
of
the
one
,
and
the
consequent
of
the
other
,
and
diuiding
the
product
by
the
other
which
remaineth
:
for
the
quotient
will
be
the
fourth
number
of
the
proportion
giuen
.
As
in
3
,
2
,
9
,
I
desire
to
knowe
the
fourth
that
shall
be
to
9
,
as
2
to
3
,
multiply
therefore
2
by
9
,
and
diuide
18
the
product
by
3
which
remaineth
,
the
quotient
will
be
6
,
the
proportionall
number
which
was
sought
for
.
The
cause
of
this
consequence
is
manifest
,
because
hauing
the
product
of
the
middle
numbers
,
it
is
all
one
as
if
you
should
say
,
that
you
had
the
product
made
of
the
first
and
fourth
numbers
,
and
to
proue
that
,
or
to
bring
to
light
the
fourth
number
lying
hidde
in
the
product
,
you
must
diuide
the
product
by
the
first
number
,
and
say
,
that
the
quotient
is
the
other
number
vnknowen
which
made
the
diuidend
:
because
that
euerie
number
is
diuided
by
those
numbers
by
which
it
is
made
,
and
euery
product
diuided
by
one
of
the
multiplicators
bringeth
forth
the
other
.
As
in
9
,
6
,
12
,
I
desire
to
finde
the
fourth
,
multiply
therefore
6
by
12
,
that
is
the
middle
number
,
then
I
say
,
the
product
72
is
the
product
arising
of
9
,
and
the
fourth
number
vnknowen
.
The
product
then
being
had
,
and
one
of
the
multipliers
knowen
,
the
other
cannot
any
longer
be
hid
.
Wherefore
I
diuide
72
by
9
the
quotient
will
be
8
,
which
is
the
other
multiplier
,
and
also
the
fourth
number
in
proportion
:
thus
,
9
,
6
,
12
,
8.
This
rule
is
vsed
in
proportion
simple
or
manifolde
.
Simple
proportion
is
that
which
consisteth
onely
of
foure
termes
or
numbers
.
The
forme
whereof
is
double
,
for
either
the
proportion
proceedeth
directly
forward
,
or
else
backewarde
,
wherevpon
it
is
called
Direct
,
or
Reciprocall
.
I
call
that
Direct
proportion
,
wherein
as
the
first
terme
is
to
the
second
,
so
the
third
is
to
the
fourth
,
and
crossewayes
.
Which
is
Direct
proportion
,
may
be
easily
gathered
by
this
,
that
by
howe
much
the
third
number
is
greater
or
lesse
,
by
so
much
the
fourth
in
order-shall
be
greater
or
lesse
,
whether
the
numbers
be
giuen
ordinarily
,
or
crossewaies
,
or
backwards
.
And
it
is
handled
two
manner
of
wayes
,
for
sometimes
besides
the
ordering
of
the
terms
there
is
nothing
else
required
:
sometimes
it
requireth
some
former
kind
of
numeration
.
The
ordering
or
disposing
of
the
termes
to
conclude
the
proportion
attributeth
to
euery
one
that
is
giuen
his
owne
place
:
that
is
,
to
the
thing
which
is
called
into
question
,
it
giueth
the
third
place
:
the
first
place
to
the
thing
of
the
same
kind
:
the
middle
place
to
that
which
remaineth
.
The
lawfull
order
of
the
termes
is
that
in
distinguishing
the
reasons
,
we
ioyne
the
consequent
of
euery
reason
to
his
owne
antecedent
of
the
same
kind
.
As
if
I
should
say
,
seeing
Iupiter
in
foure
moneths
moueth
tenne
degrees
in
the
Zodiacke
:
therefore
in
sixe
monethes
it
moueth
15
degrees
.
Here
the
proportion
ordinarily
shall
stand
thus
,
4
,
6
,
10
,
15.
For
as
4
monethes
is
to
6
monethes
,
so
is
10
degrees
vnto
15
degrees
.
For
here
also
the
product
made
of
the
extreames
is
equall
to
the
product
made
of
the
middle
numbers
.
But
the
termes
in
the
Rule
of
proportions
is
otherwise
ordered
,
because
in
things
of
one
kind
the
question
is
not
so
moued
,
that
they
should
be
ioyned
together
,
but
crossewayes
.
For
we
doe
not
moue
the
question
customably
from
the
antecedent
of
the
second
reason
to
his
consequent
thus
,
if
4
moneths
are
6
monethes
,
what
are
tenne
degrees
?
But
the
question
is
moued
from
the
consequent
of
the
first
reason
to
the
consequent
of
the
second
crossewayes
:
if
4
monethes
giue
10
degrees
,
then
6
monethes
shall
giue
15
degrees
.
Therefore
seeing
the
proportion
is
moued
crossewayes
:
and
thereby
may
be
handled
more
conueniently
,
and
may
be
better
conceiued
by
this
meanes
of
the
learner
,
therefore
in
the
ordering
of
the
termes
,
we
place
them
which
be
of
like
sort
crosse-wayes
in
sundrie
places
,
not
ioyntly
together
.
And
hereupon
the
question
being
moued
somewhat
confusedly
,
we
must
set
that
terme
that
hath
the
question
adioyned
vnto
it
in
the
third
place
,
and
the
terme
of
the
same
kinde
,
that
is
the
terme
which
betokeneth
the
same
thing
,
and
is
called
by
the
same
name
in
the
first
place
:
and
the
other
in
the
middest
:
then
must
we
followe
the
Methode
and
direction
of
the
rule
,
multiplying
the
thirde
by
the
second
,
or
contrariwise
,
the
second
by
the
third
,
and
diuiding
the
product
by
the
first
.
The
examples
.
A
traueller
is
to
goe
180
miles
,
whereof
he
goeth
9
in
two
dayes
,
the
question
is
when
he
shall
end
his
iourney
.
Here
the
question
beeing
propounded
somewhat
confusedly
,
you
set
in
order
the
termes
.
And
for
so
much
as
the
question
is
moued
about
the
180
miles
,
in
what
time
they
may
be
gone
thorough
,
you
shall
set
that
number
in
the
third
place
,
the
9
miles
being
a
terme
of
the
same
sort
,
shall
stande
in
the
first
place
,
the
two
dayes
in
the
middle
.
Then
multiply
2
by
180
,
and
diuide
the
product
360
by
9
,
the
quotient
wil
be
40
,
the
fourth
number
in
proportion
answering
the
question
.
The
example
is
thus
:
If
9
require
2
dayes
,
ergo
180
,
40
daies
.
If
a
pole
sixe
foote
long
standing
vpright
vppon
the
ground
casteth
a
shadowe
tenne
foote
long
,
I
demaund
how
high
that
Tower
is
,
which
at
the
same
time
casteth
a
shadowe
125
foote
long
.
Here
I
see
that
the
numbers
giuen
are
proportionall
.
For
as
the
height
of
the
pole
being
knowen
is
vnto
the
shadowe
,
so
is
the
hight
of
the
tower
vnknowen
vnto
his
shadow
.
The
termes
therefore
being
set
duely
in
order
,
and
handled
according
vnto
the
rule
,
the
example
shall
be
thus
.
If
10
giue
6
:
ergo
125
shall
giue
75.
After
the
same
manner
in
this
rule
may
the
parts
be
ordered
and
handled
,
as
may
be
perceiued
by
the
examples
following
:
If
the
sixt
part
of
the
Moone
encreaseth
when
she
is
distant
from
the
Sunne
the
twelfth
part
of
a
circle
:
how
bigge
shall
she
be
when
she
is
distant
from
him
halfe
a
circle
.
Here
the
first
terme
shall
be
1/12
of
a
circle
,
the
second
1/6
of
the
Moone
,
which
in
that
distance
is
lightened
with
the
light
of
the
Sun
:
the
third
1/2
,
the
worke
being
wrought
there
ariseth
1
,
which
signifieth
that
the
Moone
is
at
the
full
when
she
is
halfe
a
circle
off
from
the
sunne
.
The
termes
are
thus
:
1/12
1/6
ergo
1/2
1.
I
spend
in
the
fourth
part
of
a
moneth
5/6
of
a
crowne
,
what
doe
I
spend
in
7/10
of
a
moneth
?
Here
the
termes
are
giuen
in
order
,
wherfore
I
multiply
the
third
by
the
second
,
the
facit
will
be
7/12
,
which
being
diuided
by
1/4
after
the
manner
of
parts
or
fractions
,
the
quotient
will
be
2
1/3
,
so
that
the
example
shall
be
thus
:
1/4
5/6
:
ergo
7/10
2
1/3.
But
here
if
you
desire
to
knowe
the
value
of
1/3
of
a
crowne
or
of
any
other
fraction
what
so
euer
,
you
shall
search
it
out
in
like
manner
by
the
rule
.
For
that
they
teach
commonly
,
that
the
number
made
by
multiplying
of
the
numerator
,
and
the
partes
of
the
whole
being
knowen
,
must
be
diuided
by
the
denominator
,
whereuppon
proceedeth
the
value
of
the
parts
giuen
,
it
is
nothing
else
but
to
practise
the
rule
of
proportion
.
For
it
is
all
one
,
as
if
you
should
set
the
denominator
of
the
parts
in
the
first
place
,
the
numerator
in
the
third
,
and
the
parts
of
the
whole
being
knowen
in
the
middlemost
place
,
and
so
worke
accordingly
.
As
,
put
case
three
parts
of
a
crowne
be
45
pence
,
you
shall
find
out
the
value
of
1/3
part
thus
:
If
3
be
45
pence
:
I
shall
be
15
pence
.
If
one
or
more
of
the
termes
be
mixt
,
you
must
vse
the
same
methode
which
before
was
vsed
in
whole
and
broken
numbers
:
sauing
that
both
the
whole
and
broken
numbers
must
first
be
reduced
into
fractions
.
As
the
question
being
moued
,
two
yardes
and
1/6
of
cloth
are
bought
for
fiue
crownes
,
what
shall
I
pay
for
5
3/4
of
a
yarde
:
you
shall
reduce
therfore
the
first
terme
vnto
13/6
,
the
second
vnto
5/13
the
third
vnto
23/4
,
whereof
there
amount
13
crownes
&
7/26
,
which
is
the
number
required
.
Certaine
abridgements
.
In
whole
numbers
it
shall
not
be
amisse
to
haue
regarde
vnto
the
abridgement
following
.
If
of
the
proportionall
numbers
giuen
,
the
first
and
second
,
or
the
first
and
thirde
be
compound
one
to
another
,
then
they
being
diuided
by
the
greatest
common
diuisor
:
the
proportion
shall
be
more
briefly
concluded
by
the
prime
numbers
.
The
reason
of
this
abridgement
is
this
,
because
the
termes
being
chaunged
after
this
manner
,
yet
the
proportion
is
not
chaunged
by
the
15
prop
of
the
fifth
booke
,
where
it
is
saide
,
that
Like
parts
of
multiplices
,
and
also
their
multiplices
compared
together
haue
one
and
the
same
proportion
.
As
in
12
,
32
,
15
,
because
12
and
32
are
compound
one
vnto
the
other
,
therfore
they
being
diuided
by
4
,
I
take
their
quotients
3
and
8
in
stead
of
the
numbers
gluen
.
Thus
,
if
3
giue
8
,
therefore
15
shall
giue
40.
Likewise
seeing
the
first
and
the
third
that
is
to
say
12
and
15
are
also
compound
the
on●
vnto
the
other
:
taking
their
quotient
4
and
5
,
being
prime
numbers
found
out
by
their
greatest
common
Diuisor
,
for
the
numbers
giuen
,
the
proportion
may
be
easily
concluded
in
this
manner
.
If
4
be
32
:
therefore
5
shall
be
40.
In
the
fractions
it
seemeth
a
briefe
kind
of
working
to
dispatch
the
matter
by
multiplication
onely
:
to
wit
,
if
you
take
the
number
made
by
multiplying
of
the
denominator
of
the
first
terme
,
by
the
numerator
of
the
second
terme
,
and
then
by
the
numerator
of
the
third
terme
,
for
the
numerator
of
the
fourth
:
and
the
facit
of
the
numerator
of
the
first
terme
,
by
the
denominators
of
the
second
,
and
third
for
the
denominator
of
the
4.
The
draught
of
them
is
thus
:
As
if
1/4
giue
2/3
,
what
shall
5/6
giue
?
Multiply
4
by
2
,
there
ariseth
8
,
then
by
5
,
the
product
is
40
,
which
is
the
numerator
of
the
number
which
we
seeke
for
.
Furthermore
multiply
1
by
3
,
and
then
by
6
,
the
facit
will
be
18
,
the
denominator
of
the
fourth
terme
.
The
example
is
thus
:
1/4
2/3
ergo
5/6
40/18
,
or
2
2/9.
CHAP.
XV.
Of
examples
of
the
rule
of
proportion
,
requiring
some
kinde
of
simple
numeration
before
them
.
IN
the
former
examples
,
the
rule
of
proportion
was
simple
.
But
oftentimes
it
requireth
some
certaine
kind
of
numeration
to
goe
before
it
:
the
which
numeration
serueth
to
the
ordering
of
the
termes
of
the
principall
proportion
,
according
to
the
nature
and
lawes
of
the
question
propounded
,
that
therby
the
proportionall
number
,
which
we
desire
to
know
,
might
be
with
more
easily
found
out
.
And
this
numeration
is
drawen
out
of
both
the
partes
of
Arithmeticke
:
Sometime
out
of
absolute
or
simple
Arithmeticke
,
and
is
either
simple
or
mixt
.
Great
is
the
varietie
of
those
questions
which
may
be
resolued
by
Geometricall
proportion
.
Wherefore
here
wit
is
to
be
required
,
that
by
the
former
conditions
and
nature
of
the
question
offered
,
wee
may
be
able
to
perceiue
what
is
meete
for
the
vnknitting
and
conclusion
of
the
proportion
,
and
howe
also
(
as
it
were
a
certaine
preparatiue
to
the
future
demonstration
)
the
termes
of
the
proportion
lying
hid
,
are
to
be
vnfolded
,
orderred
,
set
downe
,
and
to
be
applyed
to
the
rule
it selfe
.
First
therefore
in
this
example
addition
of
whole
numbers
is
required
.
A
Vintener
buying
at
Ansa
in
Italie
120
hogesheads
of
wine
for
2400
ss.
laid
out
for
the
cariage
of
them
to
Basill
1200
ss.
and
to
the
waggeners
for
lading
and
vnlading
them
30
ss.
for
custome
in
diuerse
places
he
paide
50
:
his
expenses
in
his
iourney
came
to
64.
He
desireth
to
gaine
800.
I
demaund
for
what
he
shall
sell
an
hogshed
.
In
this
example
Addition
ordereth
one
of
the
proportionall
termes
:
for
adde
the
price
to
his
expences
,
and
to
the
gaine
,
the
totall
will
be
4544.
Whereby
I
perceiue
that
the
number
of
the
hogsheds
assigned
is
vnto
this
totall
summe
as
one
hogshed
is
to
the
price
which
we
seeke
for
.
Wherefore
the
example
shall
be
thus
.
If
120
Hog
.
be
worth
4544
sss
.
:
ergo
1
Hog
.
giues
37
1●/1●
sss
.
Another
.
Two
trauellers
going
from
Basill
vnto
two
contrarie
places
,
doe
trauaile
a
diuerse
pase
one
faster
then
the
other
:
the
one
goeth
euerie
day
fiue
,
the
other
but
three
miles
:
I
demaund
how
many
miles
they
shall
be
a
sunder
at
nine
dayes
end
.
Here
also
Addition
helpeth
to
the
ordering
of
one
of
the
termes
.
For
adde
their
daily
iourneyes
5
and
3
,
the
totall
will
be
8
,
which
is
the
distance
for
one
day
,
whereupon
the
rule
concludeth
,
1
day
.
giueth
8
miles
.
:
therefore
9
dayes
72.
miles
.
Examples
of
subduction
going
before
.
A
pipe
of
water
voydeth
into
a
cesterne
contayning
250
firkins
,
euery
houre
24
firkins
,
now
there
runne
out
at
another
pipe
euerie
houre
16
firkins
,
I
demaund
in
what
time
the
Cesterne
will
be
filled
.
Here
you
must
find
out
by
subduction
,
one
of
the
termes
of
the
proportion
to
be
ordered
according
vnto
the
rule
.
For
reason
telleth
me
that
I
must
seeke
out
the
excesse
of
the
filling
aboue
the
emptying
.
Subducting
therfore
16
firkins
from
24
,
there
remaines
8
firkins
,
for
so
many
euery
houre
are
left
in
the
cesterne
.
Hereby
the
proportion
may
easily
be
concluded
:
8
Remaine
in
1
:
ergo
250
31
1/4
A
certaine
man
selling
an
hundred
pound
of
certaine
kinde
of
marchandise
for
15
crownes
,
found
at
the
length
that
in
an
hundred
crowns
he
had
lost
so
much
as
he
paide
for
euerie
hundred
waight
:
I
demaund
what
he
paid
for
an
hundred
.
Here
you
shall
subduct
the
price
of
an
hundred
pound
,
to
wit
,
15
crownes
,
(
which
number
also
representeth
the
losse
)
out
of
an
hundred
crownes
,
the
remainder
will
be
85
,
the
first
terme
of
the
proportion
.
For
euen
as
85
crownes
which
is
the
stocke
diminished
,
is
vnto
an
100
,
which
is
the
whole
stocke
:
euen
so
15
the
price
of
an
hundred
waight
which
was
lost
,
must
be
vnto
the
stocke
whereof
it
was
raysed
:
whereupon
the
termes
shall
stand
thus
:
If
there
remaine
85
Crownes
.
of
100
:
ergo
15
de
17
11/17.
Examples
wherein
Multiplication
goeth
before
.
A
Polonian
going
vnto
the
Vniuersitie
of
Basill
to
studie
,
deliuereth
vnto
a
Marchant
126
golden
Ducats
(
euery
one
being
valued
at
28
vrsati
,
to
receiue
againe
of
him
for
them
at
Basill
siluerlings
(
which
commonly
are
called
Thaleri
)
being
worth
18
vrsati
.
The
question
is
how
many
siluerlings
he
shall
receiue
.
Here
you
shall
reduce
by
multiplication
126
ducats
vnto
vrsati
,
accounting
to
euerie
ducate
28
,
whereof
the
whole
summe
amounteth
vnto
3528
vrsati
.
Then
looke
what
proportion
there
is
betweene
18
vrsati
,
and
one
siluerling
,
the
same
there
is
betweene
the
whole
number
of
the
vrsati
,
and
betwene
the
siluerlings
which
we
seeke
for
.
Wherefore
the
termes
shall
be
thus
:
18
Vrsati
are
worth
1
Thaler
.
:
ergo
3528
,
196
This
and
such
like
examples
may
be
also
resolued
by
the
rule
of
reciprocall
proportion
,
as
shall
appeare
hereafter
.
He
that
hath
34
crownes
for
his
yearely
wages
,
what
hath
he
for
eight
dayes
?
Here
I
see
there
is
a
question
of
proportion
moued
,
but
for
so
much
as
the
first
and
third
termes
be
of
a
diuerse
denomination
,
I
reduce
them
by
multiplication
to
one
kind
of
denomination
.
And
so
generally
the
terms
of
diuerse
kindes
are
by
a
former
multiplication
to
be
drawen
to
the
least
denomination
.
Therefore
I
take
for
a
yeare
365
dayes
,
and
that
the
account
might
be
the
more
exact
,
I
diuide
the
34
crownes
into
510
vrsati
(
reckoning
15
vrsati
to
a
crowne
)
or
if
it
please
you
to
bring
them
into
a
smaller
coyne
,
you
may
resolue
them
into
pence
,
and
then
the
example
shall
be
thus
:
365
510
:
ergo
8
11
13/73
Sometimes
Multiplication
goeth
before
with
Addition
or
Subduction
.
The
examples
of
the
first
.
He
that
desireth
in
sixe
crownes
to
gaine
10
vrsati
,
how
shall
he
sell
a
pound
of
that
ware
which
cost
20
vrsati
?
For
so
much
as
the
money
here
set
downe
is
of
a
diuerse
denomination
:
first
of
all
I
reduce
the
6
crownes
by
multiplication
vnto
vrsati
:
valuing
euery
crowne
at
15
vrsati
,
the
facit
is
90.
Whereunto
if
I
adde
the
gaine
,
the
whole
summe
of
the
vrsati
,
to
wit
,
100
shall
be
the
middle
terme
of
the
proportion
after
this
manner
:
If
of
90
there
arise
100
:
then
of
20
22
2/9
In
the
least
termes
thus
:
9
10
:
ergo
20
22
2/9
A
Butcher
deliuering
vnto
his
seruant
216
crownes
,
sent
him
to
market
willing
him
to
buy
as
many
Oxen
,
Calues
,
Weathers
,
and
Hogges
of
euery
one
an
euen
number
,
as
that
money
would
afford
him
to
buy
:
I
demaund
how
many
beasts
he
bought
.
First
let
vs
appoint
the
prices
:
an
Oxe
at
11
crownes
,
a
Calfe
at
two
,
a
Weather
at
one
crowne
and
5
vrsati
,
an
Hogge
at
3
crownes
and
tenne
vrsati
.
Here
then
is
there
a
double
multiplication
with
addition
required
,
to
the
ordering
out
of
the
principall
proportion
.
For
here
you
must
resolue
the
216
crownes
by
15
(
for
at
so
many
vrsati
doe
Germaines
commonly
value
their
crownes
)
into
3240
vrsari
:
and
likewise
reduce
the
seuerall
prices
of
euerie
beast
vnto
one
denomination
:
to
wit
,
the
prices
of
the
Oxen
vnto
165
,
of
the
Calues
vnto
30
,
of
the
Weathers
vnto
20
,
of
the
Hogges
vnto
55
vrsati
,
and
then
adde
together
the
prices
of
the
same
denomination
,
the
totall
270
shall
be
the
first
terme
of
the
proportion
.
For
as
this
number
is
to
one
of
each
kind
of
beasts
,
so
the
whole
summo
of
money
is
vnto
the
number
of
the
beastes
vnknowen
.
The
example
is
thus
:
270
giue
1
:
ergo
3240
,
12
The
example
of
the
latter
.
A
theefe
hauing
by
stealth
conuayed
away
certaine
siluer
vessels
,
in
his
flight
goeth
euerie
day
128
furlongs
.
Foure
dayes
after
the
owner
following
the
theefe
on
horsebacke
,
rideth
euerie
day
174
furlongs
.
I
demaund
when
he
shall
ouertake
him
.
For
the
resoluing
of
this
question
,
two
termes
of
the
principall
proportion
must
be
set
downe
,
the
one
by
multiplication
,
the
other
by
subduction
.
For
first
we
must
see
how
farre
the
theefe
was
gone
before
the
owner
espying
the
matter
entred
his
iorney
.
Wherfore
I
multiply
128
furlongs
by
4
dayes
,
the
facit
is
512
furlongs
.
Secondly
,
I
consider
how
much
the
latter
outgoeth
the
former
euery
day
.
Subducting
therefore
128
from
174
the
remainder
is
46.
Then
looke
what
proportion
the
dayly
excesse
hath
vnto
a
day
,
the
same
proportion
shall
there
be
betweene
the
former
flight
and
all
those
dayes
wherein
the
latter
ouertaketh
the
former
.
So
that
the
termes
of
the
proportion
shall
be
thus
:
46
giue
1
:
ergo
512
11
4/23.
CHAP.
XVI
.
Of
examples
of
the
golden
Rule
,
wherein
there
is
required
some
numeration
of
broken
or
mixt
numbers
.
IN
the
former
examples
before
the
proportions
could
be
concluded
,
there
was
required
a
certaine
numeration
of
whole
numbers
:
in
those
which
follow
,
the
numeration
is
of
broken
or
mixt
numbers
.
Example
.
1.
A
certaine
man
selling
his
house
for
75
crownes
,
findeth
he
hath
gotten
the
fourth
part
of
the
money
which
he
paide
for
him
,
I
demaunde
what
he
paide
for
him
?
Here
adde
vnto
the
money
which
he
gaue
(
which
is
supposed
to
be
4/4
)
the
games
,
to
wit
1/4
,
the
totall
will
be
5/4
:
for
he
that
of
his
stocke
laide
out
raiseth
1/4
,
he
by
4
gaineth
5.
Wherefore
the
principall
proportion
shal
be
thus
:
5
75
:
ergo
4
,
60
A
Farmer
hauing
gotten
by
one
yeares
haruest
certaine
sackes
of
corne
,
payeth
to
his
Landlords
for
his
yeares
rent
,
to
one
2/5
,
to
another
1/4
,
and
hath
remaining
35
sackes
:
I
demaund
how
many
he
had
at
the
first
.
In
this
and
such
like
examples
of
broken
numbers
hauing
diuers
denominations
,
out
of
the
least
number
which
may
commonly
be
diuided
by
the
denominators
of
the
fractions
giuen
,
must
be
drawen
whole
numbers
being
vnto
the
number
diuided
as
the
parts
assigned
do
require
.
As
here
the
least
number
which
may
be
diuided
by
the
denominators
set
downe
which
are
5
&
4
,
is
20
:
wherof
2/5
are
8
,
&
1/4
is
5
,
which
being
added
together
,
&
the
common
denominator
set
vnderneath
do
make
13/20.
The
farmer
then
of
his
whole
heap
of
corn
which
is
supposed
to
be
20
,
paid
13
,
wherefore
13/20
being
subducted
out
of
20/20
,
there
remaine
7/20
:
wherupon
the
denominators
being
omitted
,
the
termes
shall
thus
be
set
together
:
7
are
35
:
ergo
13
were
65
A
certaine
man
selling
his
horse
lost
the
sixth
part
and
2/15
of
the
money
which
he
paid
for
it
.
and
had
remaining
243
crownes
:
the
question
is
what
the
house
cost
him
.
This
example
is
like
to
the
former
.
for
you
must
take
the
least
number
which
may
commonly
be
diuided
by
the
denominators
of
the
parts
giuen
,
6
and
15
,
to
wit
,
30
:
whose
parts
being
of
the
same
denomination
,
to
wit
,
1/6
is
5
,
and
2/15
are
4.
The
partes
added
make
9/30
,
which
being
subducted
from
20/30
leaue
21/30
where
vpon
the
proportionall
numbers
according
to
the
rule
shall
be
:
21
are
243
:
ergo
30
347
1/3
So
much
was
the
principall
stocke
wherfore
he
lost
104
crownes
1/7.
Sometimes
not
onely
simple
,
but
also
mixt
numeration
helpeth
oftentimes
to
the
ordering
of
proportion
.
For
example
.
Of
three
siluer
pots
,
the
first
wayeth
12
halfe
ounces
and
2/3
,
the
second
16
1/2
,
the
third
23
1/6.
The
halfe
ounce
is
worth
11
vrsati
1/2
:
what
then
is
the
price
of
the
pots
?
Here
of
necessity
must
addition
of
mixt
numbers
goe
before
,
although
the
principall
proportion
also
consisteth
of
mixt
numbers
.
For
the
waight
of
all
the
pots
being
added
together
,
you
shal
haue
halfe
ounces
52
1/3
:
wherupon
the
proportion
shall
be
thus
framed
.
If
1
cost
11
1/2
,
therefore
52
1/3
601
5/6
A
Vintener
hauing
bought
16
hogsheds
of
wine
for
27
3/4
crownes
,
the
price
of
the
wine
falling
,
could
not
sell
them
but
for
crownes
19
5/6
:
I
demaund
what
he
should
haue
lost
if
there
had
bene
84
hogsheds
?
Here
subduction
of
mixt
numbers
must
go
before
.
For
the
price
of
the
thing
sold
for
lesse
then
it
cost
being
subducted
out
of
the
price
which
it
was
bought
for
,
will
bring
out
the
second
terme
of
the
proportion
,
so
that
you
may
easily
answere
the
question
.
Therefore
19
5/7
being
subducted
from
27
3/4
there
remaines
7
11/12
,
the
losse
in
16
hogsheads
of
wine
.
Wherefore
the
termes
of
proportion
shall
thus
be
set
downe
:
In
16
hogs
.
there
are
lost
7
11/12
:
ergo
84
,
41
9/16
Mixt
numeration
is
more
common
a
great
deale
then
any
other
,
because
the
measures
,
prices
,
&c.
of
things
are
seldome
contayned
in
whole
numbers
:
wherefore
the
studious
may
here
haue
plentifull
matter
to
exercise
him selfe
in
.
CHAP.
XVII
.
Of
examples
of
the
golden
Rule
requiring
proportion
before
them
.
HItherto
numeration
vsed
in
simple
and
absoulte
Arithmeticke
serued
to
the
finding
out
of
the
conclusion
in
Geometricall
proportion
.
Yet
oftentimes
some
part
of
compared
Arithmetick
is
required
therunto
:
as
for
example
,
proportion
either
Arithmeticall
or
Geometricall
.
First
then
let
these
be
examples
wherein
Arithmeticall
proportion
must
go
before
.
An
Espie
enformeth
his
Captaine
of
the
distance
of
his
enemies
campe
after
this
manner
.
If
we
take
our
iourney
hence
,
and
goe
the
first
day
30
,
the
second
28
,
the
third
26
furlongs
,
diminishing
after
that
order
euery
day
two
furlongs
,
we
shall
meete
with
them
at
fifteene
dayes
end
.
Now
the
enemie
knowing
of
their
comming
,
made
toward
them
,
so
that
at
nine
dayes
end
they
met
together
.
I
demaund
how
farre
the
Captaine
went
before
he
met
with
the
enemie
.
Here
first
and
formost
we
must
gather
the
sum
of
the
proportion
Arithmeticall
(
which
commonly
we
call
progression
)
whose
termes
are
15
,
and
the
extremes
30
,
and
2
:
the
whole
summe
,
then
is
240
furlongs
,
which
is
the
second
terme
seruing
to
conclude
the
Geometricall
proportion
on
this
manner
.
15
240
:
ergo
9
144
So
farre
therefore
they
trauelled
,
the
rest
,
to
wit
,
96
furlongs
the
enemies
went
.
A
labourer
was
hyred
for
this
wages
,
to
receiue
the
first
day
a
pennie
,
the
second
three
,
the
third
fiue
,
and
after
the
same
order
his
wages
should
be
continued
for
the
dayes
following
:
at
length
being
dismissed
,
he
receiued
1500
pence
,
I
demaund
how
many
dayes
he
serued
?
Here
if
you
gather
together
all
the
termes
of
the
Arithmeticall
progression
into
one
summe
,
you
shall
make
a
way
to
the
Geometricall
progression
.
As
for
example
,
if
you
adde
together
his
wages
for
30
dayes
,
the
first
terme
of
the
progression
being
an
vnitie
,
and
the
difference
2
,
the
whole
summe
shall
be
900
pence
:
whereupon
I
may
conclude
:
If
900
require
30
:
ergo
1500
,
50.
In
the
examples
following
,
Geometricall
proportion
must
goe
before
to
informe
the
principall
proportion
.
When
a
web
of
cloth
contayning
40
yardes
is
bought
for
50
crownes
,
for
how
much
shall
he
sell
the
yard
,
that
in
an
hundred
crownes
,
desireth
to
gaine
12
?
Before
you
can
conclude
the
question
by
Geometricall
proportion
,
being
as
it
were
the
decider
of
the
whole
controuersie
,
you
must
seeke
out
the
price
of
one
yarde
.
If
40
yardes
are
bought
for
50
crownes
:
then
one
yarde
doth
cost
11
1/4
of
a
crowne
.
This
shall
be
the
third
terme
of
the
principall
proportion
,
in
this
manner
.
If
of
an
100
crownes
there
arise
112
:
then
of
1
1/4
there
arise
1
2/5
A
Marchant
buying
at
Venice
an
hundred
yardes
of
silke
for
a
crowne
a
yearde
,
spent
in
his
iourney
and
for
the
carriage
of
them
30
crownes
.
At
Lipsia
where
the
yarde
is
more
by
a
quarter
then
the
yarde
of
Venice
,
he
solde
euery
yard
for
3
crownes
.
The
question
is
what
he
gayned
?
In
this
example
addition
of
whole
numbers
,
together
with
proportion
goeth
before
.
For
his
stocke
laide
out
,
and
his
charges
being
added
,
the
whole
is
130.
Besides
,
to
know
how
many
yardes
of
Lipsia
aunswere
the
Venice
measure
,
you
shall
worke
by
proportion
on
this
manner
.
If
5/4
of
Lipsia
measure
,
make
a
yarde
of
Venice
,
then
80
yardes
of
Lipsia
shall
make
an
hundred
Venice
yardes
.
So
that
you
may
readily
inferre
the
principall
proportion
thus
:
If
1
is
solde
for
3
:
ergo
80
240
Whereupon
130
crownes
being
the
stocke
and
charges
subducted
out
of
240
,
I
see
he
gayned
110.
Another
marchant
bought
three
hogsheads
of
honie
.
The
first
wayed
349
pound
,
the
second
286
pound
,
the
third
300
,
out
of
euerie
hundred
are
subducted
tenne
pound
because
of
the
vessell
,
and
a
pound
of
honnie
is
solde
for
5
pence
:
I
demaund
what
he
is
to
paie
.
For
so
much
as
the
price
of
one
pound
of
honie
is
set
downe
,
therefore
by
proportion
seeke
out
the
price
of
an
hundred
,
1
pound
is
solde
for
fiue
pence
:
ergo
100
,
for
500
pence
,
then
adde
the
waight
of
the
vessels
,
the
whole
is
110
,
adde
also
the
waight
of
the
three
hogsheds
,
the
totall
is
935
:
whereupon
you
shall
conclude
:
110
are
sold
at
500
pence
,
ergo
935
at
4250
,
that
is
,
170
florentines
.
Another
buying
two
yardes
of
cloth
at
three
crownes
,
solde
sixe
yardes
afterwardes
for
11
crownes
.
Hauing
got
by
this
meanes
46
crownes
,
there
ariseth
a
question
howe
many
yardes
he
bought
,
and
how
many
he
sold
.
Before
you
can
come
vnto
the
principall
proportion
,
you
must
order
the
termes
therof
by
a
former
proportion
,
and
by
subduction
.
First
therefore
you
must
consider
howe
much
the
sixe
yardes
of
cloth
which
afterwardes
were
solde
at
an
higher
price
cost
at
the
first
,
the
which
may
easily
be
inferred
by
that
which
is
presupposed
in
the
question
:
for
if
two
yardes
cost
three
crownes
,
then
six
yardes
cost
three
crownes
,
then
six
yardes
were
sold
at
nine
crownes
.
This
price
I
subduct
from
11
,
and
I
see
that
sixe
yardes
were
sold
for
two
crownes
more
then
they
cost
.
So
that
it
is
easie
to
resolue
the
doubt
thus
:
If
2
come
of
6
,
then
46
,
138
A
Vintener
bought
1.
24
hogsheds
of
wine
on
this
condition
,
that
subducting
out
of
euerie
two
hogsheds
because
of
the
leese
fiue
gallons
,
he
should
pay
for
euery
hogshed
26
crowines
:
Tell
me
what
he
is
to
pay
.
Proportion
going
before
with
multiplication
and
subduction
,
will
set
in
order
the
principall
proportion
.
First
I
seeke
howe
much
is
subducted
out
of
al
the
hogsheds
because
of
the
dragges
.
Out
of
two
hogsheds
are
drawen
5
gallons
,
ergo
out
of
120
hoggesheds
are
drawen
310.
Then
I
reduce
all
the
foresaid
hogsheds
vnto
gallons
,
multiplying
them
by
32
,
for
so
many
gallons
doth
euerie
hogshed
hold
,
the
facit
is
3968.
Out
of
these
I
subduct
according
to
couenant
310
measures
:
the
remainder
is
3658
gallons
,
whereupon
the
proportion
shall
be
thus
:
32
giue
26
:
ergo
3658
2972
1/8
Of
two
Carpenters
,
the
one
alone
would
frame
an
house
in
30
daies
,
the
other
in
40.
Now
taking
another
vnto
them
,
they
finished
the
worke
in
15
dayes
.
In
what
time
then
would
the
third
man
haue
made
it
?
Here
must
you
vse
proportion
with
addition
,
and
subduction
of
fractions
.
The
first
would
haue
done
the
worke
in
30
dayes
,
ergo
in
15
daies
he
finished
15/30
or
1/2
of
the
worke
,
the
second
would
haue
wrought
it
in
40
daies
ergo
in
15
dayes
he
wrought
15/40
or
3/8.
The
partes
added
together
are
7/8
of
the
worke
:
which
being
subducted
out
of
8/8
,
that
is
,
out
of
the
whole
worke
,
there
remaineth
1/8
,
the
third
mans
worke
.
In
so
much
that
the
proportion
may
thus
be
concluded
:
1/8
is
framed
in
15
:
ergo
1/1
120
Put
case
14
pound
of
some
ware
were
bought
for
6
crownes
,
and
euery
hundred
were
sold
for
47
crownes
:
subduct
so
many
pound
out
of
the
whole
as
will
in
the
sale
amount
to
80
crownes
gaine
.
How
many
pound
then
were
there
sold
?
To
vnknit
this
knot
you
shall
vse
proportion
with
subduction
of
mixt
numbers
.
First
I
seeke
what
he
paide
for
an
hundred
pound
.
If
14
pound
be
bought
for
6
crownes
,
then
100
pound
giue
42
6/7.
Moreouer
for
so
much
as
an
hundred
pound
were
solde
for
47
crownes
,
therefore
42
6/7
being
subducted
out
of
47
,
the
remainder
is
4
1/7
the
gaine
of
an
hundred
:
whereupon
I
say
,
If
4
1/7
come
of
100
,
then
80
of
1931
1/29.
Sometimes
many
and
sundrie
kindes
of
numeration
,
as
well
out
of
simple
as
compared
Arithmeticke
,
goe
before
the
principall
proportion
,
wherein
the
diligent
young
practitioner
may
exercise
himselfe
.
CHAP.
XVIII
.
Of
reciprocall
proportion
.
HItherto
of
simple
direct
proportion
:
now
must
we
come
to
Reciprocall
proportion
.
Reciprocall
proportion
is
that
wherein
the
termes
being
set
downe
crossewaies
,
as
the
third
is
vnto
the
second
,
so
reciprocally
the
first
is
to
the
fourth
.
A
mong
the
vulgar
people
we
call
Reciprocation
the
going
backewarde
the
same
way
we
begunne
:
which
is
euidently
seene
in
reciprocall
proportion
being
compared
with
direct
proportion
.
For
that
way
that
direct
proportion
goeth
on
forwarde
,
the
same
way
reciprocall
proportion
returneth
backwarde
.
For
in
direct
proportion
the
termes
are
proportionall
,
not
onely
as
they
stand
in
order
,
but
also
crossewayes
:
that
is
to
say
,
as
the
first
is
to
the
second
,
so
the
third
is
to
the
fourth
,
and
also
as
the
first
is
to
the
third
,
so
the
seconde
is
to
the
fourth
.
But
in
reciprocall
proportion
the
termes
being
taken
first
orderly
,
are
altogether
without
proportion
,
and
crossewayes
they
differ
in
all
respects
in
that
one
is
of
greater
inequalitie
,
the
other
alwayes
of
lesse
:
yea
rather
as
the
third
is
to
the
second
,
so
backwardes
,
the
first
is
to
the
fourth
,
in
so
much
that
that
which
in
direct
proportion
was
the
first
terme
,
is
the
thirde
in
reciprocall
proportion
:
and
againe
that
which
is
the
third
in
direct
proportion
,
is
the
first
in
reciprocall
proportion
.
For
example
sake
,
if
twelue
hyred
seruantes
doe
their
worke
in
8
dayes
,
then
24
shall
doe
it
in
foure
dayes
.
Here
are
set
downe
foure
termes
of
reciprocall
proportion
,
which
being
placed
directly
shall
stande
thus
:
stion
I
must
vse
the
rule
of
reciprocall
proportion
on
this
manner
:
If
40
require
7
:
then
70
how
many
?
Multiply
40
by
7
,
the
facit
shall
be
280
,
which
being
diuided
by
70
,
giueth
4
or
two
ounces
,
which
is
the
waight
of
the
loafe
.
Three
milles
in
two
dayes
grinde
20
bushels
of
corne
,
in
what
time
then
shall
fiue
milles
grind
so
many
bushels
?
Here
againe
by
how
many
more
milles
there
be
,
in
so
much
lesse
time
they
shal
grind
the
corne
.
Wherefore
the
proportion
shall
be
concluded
reciprocally
in
this
manner
.
3
grinde
2
:
ergo
5
1
1/5.
A
certaine
man
borrowing
66
crownes
of
his
friend
,
and
repaying
them
at
seuen
monethes
end
,
promiseth
to
pleasure
his
credit
or
in
as
great
a
matter
.
Whereupon
the
creditor
afterward
desired
to
borrow
of
him
112
crowns
,
but
they
doubt
how
long
time
he
should
keepe
the
money
,
that
the
lending
of
his
money
might
be
correspondent
.
For
so
much
as
the
latter
man
desired
to
borrow
the
greater
summe
,
therefore
he
is
bound
to
restore
it
sooner
then
the
first
shold
,
whose
summe
was
lesse
:
whereupon
the
reciprocation
shall
be
concluded
thus
:
66
giue
7
:
ergo
112
4
1/8.
So
likewise
the
former
example
of
reducing
126
Hungarie
ducats
(
each
of
them
being
valued
at
28
vrsati
)
vnto
Thaleri
being
worth
18
vrsati
,
may
be
handled
and
resolued
after
this
fashion
.
28
126
:
18
196.
CHAP.
XIX
.
Of
manifold
proportion
,
and
first
of
that
which
is
compound
by
Addition
commonly
called
the
rule
of
Fellowshippe
.
Manifold
proportion
is
-
Either
compounded
,
and
is
two
fold
,
-
Prime
,
wherein
there
is
required
-
Either
one
kind
of
numeration
only
,
as
-
Addition
.
Chap.
19.
-
Multiplication
.
Chap.
20.
-
Or
many
kinds
of
numeration
:
as
that
which
is
compounded
by
multiplication
,
and
addition
together
.
Chap.
21.
-
Or
Second
,
and
is
called
Alligation
.
-
Or
continued
.
Chap.
27.
HItherto
hath
bene
handled
simple
proportion
,
as
well
direct
as
reciprocall
,
being
set
forth
by
diuerse
examples
:
now
followeth
manifold
proportion
.
Manifold
proportion
is
that
which
consisteth
of
more
thenfoure
termes
.
Simple
proportion
disiunct
,
consisteth
of
foure
termes
,
but
manifold
proportion
consisteth
of
more
,
as
when
in
one
and
the
same
proportion
there
be
manie
consequents
to
one
antecedent
,
or
contrariwise
,
or
else
when
there
be
two
rowes
of
numbers
proportionall
in
two
numbers
,
or
some
such
like
manner
.
Manifolde
proportion
is
double
:
for
either
it
is
compounded
in
the
termes
,
or
els
continued
.
The
first
ioyneth
together
the
termes
giuen
,
either
by
one
onely
kinde
of
numeration
,
to
wit
,
either
by
addition
or
multiplication
,
or
else
by
more
,
as
by
addition
and
multiplication
together
.
Compound
proportionalitie
or
proportion
is
said
to
be
that
wherein
the
termes
are
vnited
together
.
But
the
things
so
vnited
and
ioyned
together
,
are
of
many
brought
into
one
,
and
heaped
together
two
maner
of
waies
,
either
by
addition
or
multiplication
alone
,
or
else
both
waies
together
.
By
Addition
,
as
when
many
as
well
antecedents
as
consequents
are
taken
and
compared
,
as
one
antecedent
to
one
consequent
.
Although
proportions
may
be
compounded
many
wayes
by
Addition
,
yet
of
them
all
this
is
the
most
vsuall
:
wherefore
this
speciall
definition
seruing
most
for
this
purpose
is
taken
in
stead
of
that
which
is
more
generall
.
For
example
sake
,
if
3
vnto
2
,
be
as
9
vnto
6
,
then
by
addition
3
and
9
that
is
12
,
shall
be
in
like
manner
vnto
2
and
6
,
that
is
8
,
so
that
the
termes
of
the
proportion
may
stand
as
you
see
,
12
vnto
8
:
are
as
3
to
2
and
9
to
6
In
this
kind
of
manifold
proportion
,
the
rule
following
is
to
be
vsed
,
If
in
the
termes
of
manifold
proportion
the
consequent
of
the
one
proportionality
be
giuen
added
together
,
and
the
antecedents
seuerally
by
themselues
,
then
as
the
antecedents
knit
together
by
addition
,
are
vnto
the
consequents
added
,
so
shall
the
seuerall
antecedents
be
vnto
their
seuerall
consequents
,
and
so
contrariwise
.
This
is
the
12
p.
7
,
sauing
that
it
is
reuersed
and
applyed
some
what
more
fitly
to
this
purpose
.
For
the
words
of
the
proposition
be
these
:
If
there
be
a
multitude
of
numbers
,
how
many
so
euer
proportionall
,
as
one
of
the
antecedents
is
to
one
of
the
consequents
,
so
are
all
the
antecedents
vnto
all
the
consequents
.
And
the
same
backeward
is
true
also
.
By
which
conuersion
it
appeareth
how
the
examples
which
commonly
are
referred
vnto
the
rule
of
fellowshippe
,
hauing
many
proportions
coupled
by
addition
,
are
to
be
handled
and
to
be
resolued
by
the
rule
of
proportion
,
in
which
examples
the
antecedents
of
the
one
proportion
are
set
downe
seuerally
,
to
wit
,
the
contributions
of
the
partners
,
and
the
consequents
are
added
together
,
to
wit
,
the
common
gaine
or
losse
.
So
that
the
contributions
and
common
stocke
of
the
partners
,
or
else
generally
all
those
numbers
into
which
the
other
is
to
bee
distributed
proportionally
must
be
added
together
,
and
the
totall
must
be
set
in
the
first
place
in
stead
of
the
antecedent
of
the
first
proportion
,
whose
consequent
must
be
the
foresaide
number
which
is
taken
to
be
distributed
proportionally
.
In
the
third
place
the
termes
compounded
by
addition
must
be
placed
seuerally
,
&
the
rule
must
be
repeated
as
often
as
there
be
seuerall
numbers
,
whereupon
I
may
conclude
the
proportion
.
As
in
the
examples
following
.
Three
men
being
partners
laide
their
money
together
,
the
first
60
,
the
second
100
,
the
third
135
crownes
.
Nowe
they
gayned
after
a
while
45
crownes
.
The
question
is
howe
much
of
the
gaines
each
of
them
by
duetie
ought
to
haue
.
In
this
and
such
like
examples
equitie
and
reason
requireth
,
that
that
which
is
gotten
by
the
common
stocke
should
also
be
common
,
and
should
be
so
distributed
,
that
euery
mans
portion
should
haue
the
same
proportion
to
the
whole
gaine
,
that
the
mony
which
he
laide
downe
had
vnto
the
whole
stocke
:
and
crossewayes
as
the
whole
stocke
is
vnto
the
whole
gaine
,
so
then
euery
mans
portion
laide
together
,
should
be
vnto
his
part
of
the
gaines
.
Wherefore
here
this
question
of
manifold
proportion
is
manifest
.
Nowe
the
termes
giuen
are
these
:
The
first
propor
.
|
The
second
propor
.
|
I
|
II
|
III
|
IIII
|
|
|
60
|
|
|
45
|
100
|
|
|
|
135
|
|
The
consequents
of
the
second
proportion
are
set
down
but
ioyntly
together
,
to
wit
,
45
,
the
antecedents
are
seuerall
.
Wherefore
to
conclude
the
seuerall
consequents
,
I
set
the
whole
stocke
gathered
by
adding
of
60
,
100
,
135
,
to
wit
,
295
in
the
first
place
,
which
is
vnto
the
whole
gaine
,
as
the
seuerally
termes
in
the
third
place
are
vnto
the
seuerall
termes
in
the
fourth
place
,
which
we
desire
to
know
.
The
example
is
thus
:
Three
partners
in
a
common
stocke
of
500
crownes
,
lost
124
,
to
the
repayring
whereof
the
first
was
bound
to
be
contributarie
20
crownes
,
the
second
43
:
I
demaunde
howe
much
euery
one
put
into
the
common
purse
.
Here
wee
haue
of
the
latter
proportion
three
antecedents
,
two
are
assigned
in
expresse
words
,
but
the
third
is
set
downe
somwhat
obscurely
.
But
for
so
much
as
the
totall
of
the
three
added
together
is
set
downe
,
to
wit
,
124
,
the
third
may
easily
be
knowen
by
subduction
.
Adde
therefore
20
and
43
,
the
totall
is
63
,
which
being
subducted
out
of
124
there
remaines
61
,
which
is
the
third
mans
portion
in
the
losse
.
Wherefore
as
in
the
former
example
we
gathered
euery
mans
gaine
by
euery
ones
contribution
,
so
likewise
in
this
place
by
the
partes
of
the
losse
we
may
gather
each
mans
stocke
.
The
example
is
thus
:
Cicero
in
his
Oration
for
Cecinna
sayeth
:
A
woman
making
her
will
departed
this
life
,
she
made
Cecinna
heire
of
eleuen
ounces
and
an
halfe
of
her
goods
,
and
Marcus
Fulcinus
the
free
man
of
her
former
husbande
heire
of
two
fixt
parts
of
an
ounce
.
Vnto
Eutius
she
gaue
one
sixt
part
.
How
much
then
had
euerie
one
for
his
portion
.
In
this
example
the
whole
patrimonie
is
diuided
into
eleuen
ounces
,
halfe
an
ounce
,
and
three
sixt
partes
of
an
ounce
:
the
which
parts
are
the
antecedents
of
the
late
proportion
.
But
in
that
they
be
of
diuerse
denominations
,
I
resolue
them
all
into
the
least
denomination
,
to
wit
,
into
sixts
.
If
then
you
allow
to
an
ounce
sixe
sixths
,
and
to
the
pound
twelue
ounces
,
there
shall
be
in
a
pound
72
sixths
.
Whereof
11
ounces
and
an
:
halfe
are
worth
69
,
and
two
sixths
2
,
one
1.
Whereupon
supposing
the
whole
patrimonie
to
be
2500
crownes
,
the
termes
shall
be
thus
:
Three
couenanted
betweene
themselues
,
that
of
their
gaines
the
first
should
haue
twice
as
much
as
the
second
.
Now
they
had
laid
downe
24
crownes
,
and
the
third
36.
But
when
they
had
gotten
12
crownes
they
cannot
tell
howe
to
diuide
them
lawfully
according
to
their
couenant
.
Here
that
which
the
first
man
laide
down
being
couertly
expressed
,
must
be
gathered
out
of
the
couenant
.
For
in
that
he
was
to
receiue
twice
so
much
as
the
second
,
it
is
manifest
that
that
which
he
laide
out
was
twice
as
much
as
that
whereof
the
second
was
contributary
,
whether
it
were
money
,
or
whether
it
were
his
labour
,
or
whether
it
were
valued
in
both
.
Wherefore
if
for
the
first
you
take
48
crownes
the
double
of
24
,
then
they
being
added
together
,
the
whole
doubt
shall
be
remooued
by
the
rule
on
this
manner
.
The
good
man
of
an
house
bequeatheth
vnto
foure
men
584
crownes
,
to
be
diuided
after
this
rate
,
that
A
should
receiue
1/2
,
B
2/3
,
C
3/4
,
D
5/6
,
what
was
euery
ones
portion
?
Here
the
proportion
of
the
parts
vnto
the
goods
which
were
left
is
expressed
:
yet
the
parts
themselues
in
the
goods
doe
not
yet
appeare
manifest
.
The
least
number
therefore
that
may
be
diuided
by
the
denominators
of
the
fractions
,
which
is
12
,
will
declare
the
proportion
.
For
1/2
of
12
,
are
6
,
2/3
are
8
,
3/4
are
9
,
5/6
are
10.
These
being
added
,
the
example
shall
be
thus
:
But
if
he
willed
to
make
such
a
diuision
among
them
that
were
to
receiue
the
legacie
that
the
first
should
haue
1/2
and
1/5
wanting
20
crownes
,
the
second
1/3
and
30
crownes
,
the
third
3/4
and
10
crownes
,
the
fourth
1/6
wanting
12
crownes
:
then
what
should
their
portions
be
?
In
this
case
that
which
is
more
then
their
portions
I
take
it
from
the
whole
substance
,
that
which
wanteth
I
adde
vnto
it
:
as
30
and
50
are
80
,
which
being
taken
from
584
,
there
remaines
504.
Hereunto
adde
20
and
12
,
or
32
,
the
totall
is
536.
Then
I
seeke
out
the
least
number
that
may
be
commonly
diuided
by
the
denominators
,
to
wit
,
60
,
the
partes
of
the
same
denomination
are
for
the
first
30
and
12
,
for
the
second
20
,
for
the
third
45
,
for
the
fourth
10
:
and
then
the
portion
of
the
first
being
first
added
together
,
and
afterward
all
the
rest
,
the
example
will
be
thus
:
Then
that
the
testator
may
be
satisfied
,
from
the
first
mans
portion
I
take
20
crowns
there
remaines
172
48/117
,
vnto
the
second
adde
30
,
they
make
121
73/117
,
vnto
the
third
adde
50
,
his
whole
portion
will
be
256
18/117
,
lastly
from
the
fourth
take
12
,
there
remaine
vnto
him
33
95/117
,
all
which
partes
are
answerable
vnto
the
584
crownes
,
which
were
to
be
diuided
.
Thre
partners
laide
together
three
equall
portions
,
but
they
left
them
there
,
some
for
longer
time
and
some
for
shorter
:
the
first
left
his
money
in
the
stocke
two
monethes
and
an
halfe
,
the
seconde
three
monethes
and
2/5
,
the
third
foure
monethes
and
2/3.
Put
case
they
gained
12
crownes
,
I
demaund
what
was
euery
mans
portion
.
This
example
being
of
mixt
numeration
,
requireth
no
other
worke
then
the
former
.
For
the
seuerall
times
being
added
together
,
wherein
they
left
their
stocke
for
the
common
trafficke
,
the
totall
will
be
317/30
:
So
that
the
example
will
be
.
CHAP.
XX.
Of
multiplied
proportion
compound
by
multiplication
,
which
commonly
is
called
the
double
Rule
.
THe
former
examples
appertained
vnto
proportion
compound
by
addition
:
now
followeth
that
which
is
compound
by
multiplicacion
,
which
in
stead
of
two
termes
of
the
proportion
,
vseth
the
facit
made
of
them
.
This
is
the
right
compounding
of
proportion
,
which
gathereth
them
together
by
multiplication
as
was
said
before
,
and
appeareth
by
the
fift
definition
of
the
sixth
booke
of
Euclide
,
because
it
doth
not
only
ioyne
together
the
termes
themselues
,
but
also
maketh
the
proportion
of
the
compound
termes
to
comprehend
the
proportion
of
the
simple
.
Wherefore
hitherto
are
to
be
referred
,
and
by
this
rule
of
proportion
are
to
be
handled
these
examples
,
wherein
two
termes
by
multiplication
must
be
reduced
vnto
one
:
for
there
proportion
as
appeareth
by
the
fifth
prop.
of
the
eighth
booke
of
Euclide
,
is
gathered
of
the
proportion
of
the
termes
multiplied
one
by
the
other
.
Of
this
sort
are
all
those
examples
which
commonly
are
referred
to
the
double
rule
,
wherein
two
termes
are
set
in
the
same
places
,
that
is
,
the
principall
proportion
hath
a
certaine
circumstance
adioyned
vnto
it
,
in
so
much
that
here
the
proportion
should
be
twice
concluded
,
were
it
not
that
for
those
two
termes
the
products
made
of
them
are
taken
as
one
.
The
termes
propounded
by
multiplication
,
are
either
of
the
same
place
,
as
in
direct
proportion
,
or
of
diuerse
places
,
as
in
reciprocall
proportion
.
The
examples
of
the
first
are
these
.
Foure
Students
spend
in
3
monethes
19
crownes
:
how
much
therefore
shall
eight
Students
spend
in
9
monethes
.
You
see
here
the
principall
termes
haue
annexed
vnto
them
a
circumstance
of
time
,
&
therfore
(
the
maner
of
ordering
the
termes
in
the
golden
rule
being
obserued
)
the
first
and
the
third
are
double
,
on
this
fashion
:
Wherefore
for
those
two
I
take
the
compounds
by
multiplication
.
As
for
4
and
3
I
take
12
,
for
8
and
9
,
72
,
to
wit
their
products
:
So
that
the
compound
multiplication
shall
be
concluded
thus
:
Here
betwene
the
expences
of
foure
students
in
three
monethes
,
there
is
obserued
the
same
proportion
which
is
betweene
the
expences
of
8
students
in
9
moneths
,
to
wit
,
the
proportion
compound
of
the
two
,
to
wit
,
of
4
vnto
8
,
and
3
vnto
9
,
or
to
speake
more
plainely
,
the
expences
are
one
to
another
,
as
thrice
foure
vnto
nine
times
eight
.
The
which
may
easily
be
proued
,
if
you
put
in
the
meane
proportionall
number
to
whom
the
termes
of
the
antecedents
and
consequents
may
belong
.
So
that
if
you
should
set
it
downe
that
foure
student
in
9
monethes
should
make
their
expence
,
the
example
will
be
thus
:
Here
then
if
foure
students
in
three
monethes
dispende
euery
one
but
one
,
then
so
many
students
in
nine
monethes
shall
spende
thrice
as
much
.
Wherefore
the
proportion
of
the
expences
of
A
vnto
B
,
shall
be
the
same
that
is
betwene
three
and
nine
.
Likewise
if
foure
students
in
nine
moneths
spend
euerie
one
but
one
,
then
8
students
in
as
manie
monethes
will
spende
twise
as
much
.
Whereupon
there
will
be
the
same
reason
betweene
B
and
C
,
as
is
betweene
4
and
8.
Furthermore
if
you
multiply
3
and
9
,
the
termes
of
the
first
reason
by
one
and
the
same
number
,
namely
by
4
,
the
productes
,
namely
12
and
36
,
shall
keepe
the
same
reason
by
the
17
proposition
of
the
7
of
Euclide
:
so
shall
36
and
72
being
made
of
4
and
8
multiplyed
by
9.
Whereby
it
is
gathered
that
as
the
charges
of
A
is
vnto
B
,
so
is
12
vnto
36.
And
in
like
manner
as
B
is
vnto
C
,
so
is
36
vnto
72.
To
conclude
,
seeing
there
are
here
three
proportionall
numbers
,
A
,
B
,
C
,
and
as
many
more
in
number
answerable
vnto
them
12
,
36
,
72
,
therefore
by
the
14
of
the
7
(
taking
the
extremes
the
meane
proportionals
being
withdrawen
)
the
charges
of
A
shall
be
all
one
to
the
charges
of
C
,
as
12
is
vnto
72
,
whereby
the
goodnesse
of
the
working
doth
appeare
.
An other
example
.
Two
Printers
in
foure
dayes
print
16
formes
,
ho●e
many
shall
seuen
Printers
print
in
14
dayes
?
Here
the
placing
of
the
termes
prescribed
in
the
rule
of
proportion
being
obserued
,
they
shall
be
set
in
this
manner
:
Printers
.
|
2
|
16
|
7
|
Dayes
.
|
4
|
|
14
|
Then
,
the
first
termes
and
the
last
being
multiplied
one
by
another
,
the
proportion
is
concluded
thus
:
8
16
:
therefore
98
196
Eight
yardes
of
cloth
,
4
yardes
and
1/4
broad
are
bought
for
11
crownes
:
therefore
how
shall
15
yardes
of
cloth
be
bought
1
yarde
3/4
broad
?
This
example
of
mixt
numeration
varieth
nothing
from
the
former
,
therefore
it
is
thus
dispatched
:
Three
men
trading
together
by
ill
lucke
lost
52
crownes
.
The
first
put
in
110
crownes
for
5
monethes
,
the
second
put
in
84
crownes
,
the
third
65
,
I
know
not
for
what
time
.
Their
traficke
being
ended
,
the
first
found
that
he
had
lost
22
crownes
,
the
second
found
he
had
lost
18
,
the
third
12.
How
long
therefore
was
the
money
of
the
last
two
in
the
common
stocke
?
I
thinke
that
this
example
(
though
it
seeme
to
belong
to
the
Rule
of
Fellowship
)
may
well
be
referred
to
the
rule
of
proportion
compounded
by
multiplication
,
yet
it
is
vnlike
to
the
other
,
because
of
the
numbers
giuen
,
the
middlemost
onely
is
compounded
by
multiplication
,
the
other
are
sought
out
by
the
rule
of
three
,
proceeding
from
a
terme
known
to
those
which
are
vnknowne
.
Therefore
the
first
mans
22
crownes
being
multiplied
by
his
5
moneths
,
the
facit
is
550.
Whereupon
I
conclude
,
if
his
money
,
that
lost
22
crownes
,
multiplied
by
the
time
doth
make
550
,
what
shall
the
losse
of
18
crownes
?
and
what
shall
the
losse
of
12
crownes
make
?
The
termes
shall
stand
thus
:
-
22
550
:
therefore
18
450
-
22
550
:
therefore
12
300
The
fourth
numbers
inferred
by
the
rule
of
three
are
compounded
of
ech
mans
stocke
multiplyed
by
the
time
for
which
it
was
laid
out
.
Therefore
ech
of
them
seuerally
diuided
by
his
owne
stocke
,
the
time
is
seuerally
inferred
,
so
that
the
second
mans
money
was
out
of
his
handes
fiue
monethes
and
5/14
,
the
third
mans
money
was
out
foure
and
8/13
of
a
moneth
.
An
abridgement
of
the
former
worke
.
If
the
termes
in
the
first
and
third
place
fall
out
to
be
equall
,
then
they
being
taken
away
,
the
rest
of
the
termes
shall
inferre
the
proportion
.
The
reason
of
this
abridgement
ariseth
out
of
the
17
proposition
of
the
7
,
for
the
numbers
giuen
keepe
the
same
proportionalitie
which
they
would
haue
being
multiplied
by
one
and
the
same
number
.
As
:
If
the
gaine
of
25
crownes
in
foure
yeare
be
eight
crownes
,
what
shall
100
crownes
yeeld
in
foure
yeare
?
Here
the
termes
are
giuen
as
you
see
.
Stocke
|
25
|
8
|
100
|
Yeares
|
4
|
|
4
|
Forsomuch
therefore
as
25
hath
the
same
reason
to
100
which
they
would
haue
being
multiplyed
by
4
,
as
it
appeareth
by
the
17
prop.
7
,
therefore
the
equall
numbers
which
are
in
the
first
and
thirde
place
,
being
cast
away
,
I
conclude
the
proportion
in
simple
termes
thus
:
25
8
therefore
100
32
Likewise
the
question
being
thus
propounded
,
the
gaine
of
25
crownes
in
4
yeare
,
is
8
crownes
,
therefore
what
shall
100
yeeld
in
25
yeares
?
The
termes
shall
stand
thus
:
Here
againe
the
alternate
termes
of
the
first
and
third
place
being
omitted
,
the
rest
shall
conclude
the
proportion
thus
:
4
8
:
therefore
100
200
If
the
proportion
chance
to
be
reciprocall
,
then
are
the
termes
of
the
first
and
third
place
compounded
by
alternate
or
crosse
multiplication
,
and
the
question
afterward
is
concluded
directly
.
Some
transpose
the
principall
termes
setting
the
first
in
the
third
and
contrariwise
the
third
in
the
first
place
,
&
then
bring
the
twofold
termes
into
one
by
multiplication
.
The
example
:
Eight
horses
in
12
daies
eate
9
bushels
of
oats
,
in
how
many
dayes
shall
18
horse
eate
24
bushes
?
I
easily
perceiue
that
there
is
here
a
reciprocation
,
for
18
horses
will
spend
in
a
great
deale
lesser
time
that
heape
of
oates
,
which
8
horse
will
consume
in
12
daies
.
Therefore
by
how
much
the
third
terme
is
the
greater
,
by
so
much
the
fourth
terme
is
the
lesser
.
Therefore
I
must
take
heede
that
disposing
the
termes
as
I
did
before
,
I
conclude
not
directly
the
reciprocall
proportion
by
multiplying
of
the
first
and
last
termes
one
by
another
.
The
termes
in
the
example
are
thus
:
Here
therefore
I
multiply
9
by
18
and
make
162
the
first
terme
,
likewise
I
multiply
8
by
24
and
make
192
the
third
terme
,
whereby
the
lawfull
induction
of
the
proportion
is
thus
dispatched
:
162
12
:
therefore
192
14
2/9
If
100
crownes
doe
giue
5
crownes
interest
euerie
yeare
,
in
what
time
will
56
crownes
giue
.
12
?
In
this
question
for
so
much
as
a
greater
interest
is
sought
for
by
a
lesser
stocke
of
money
,
the
reciprocation
is
manifest
,
the
which
interest
craueth
so
much
the
more
time
by
how
much
the
more
the
stocke
is
lesser
.
Therefore
the
termes
being
set
downe
as
it
is
conuenient
,
and
multiplied
crossewayes
,
the
doubt
is
answered
by
the
rule
of
three
.
CHAP.
XXI
.
Of
manifold
proportion
compounded
by
multiplication
and
addition
.
THus
much
concerning
the
examples
of
proportion
compounded
by
addition
or
multiplication
onely
:
now
must
we
proceede
to
that
proportion
which
ioyneth
both
kindes
of
numeration
together
.
Therefore
proportion
compounded
by
multiplication
and
addition
,
is
that
which
first
of
all
multiplieth
the
manifold
termes
giuen
one
with
another
,
and
then
addeth
their
productes
together
.
This
kinde
of
proportion
by
a
two
folde
,
yea
sometimes
by
a
fourefolde
composition
,
bringeth
many
termes
into
one
.
Vnto
this
appertaine
the
examples
of
the
Rule
of
second
fellowship
,
as
they
commonly
cal
them
,
wherein
the
time
is
annexed
to
the
stocke
,
or
some
other
circumstance
to
the
principall
terme
.
The
Theoreme
following
serueth
for
the
working
of
those
examples
.
If
among
the
termes
of
the
manifold
proportion
,
the
consequents
of
one
reason
be
giuen
added
together
,
but
some
antecedents
be
giuen
seuerally
and
double
:
then
as
the
antecedents
compounded
by
multiplication
and
addition
are
vnto
the
consequents
added
together
,
so
shall
the
seuerall
antecedents
compounded
by
multiplication
be
vnto
the
seuerall
consequents
.
As
thus
for
example
.
Two
men
in
common
trafficke
got
60
crownes
,
the
first
brought
50
crownes
for
two
yeare
,
the
second
15
for
fiue
yeare
:
what
shall
be
ech
man
his
share
.
In
this
question
containing
a
partnership
of
equall
contributions
in
regard
of
the
diuerse
times
,
I
see
the
diuision
of
the
gaine
must
be
made
according
to
the
proportion
of
the
time
.
That
this
diuision
may
be
made
the
consequents
of
the
latter
reason
are
giuen
added
together
,
namely
60
,
but
the
antecedents
are
giuen
not
only
seuerally
,
but
also
double
,
as
you
see
here
.
The
first
reason
.
|
The
second
reason
.
|
I
|
II
|
III
|
IIII
|
|
|
50
2
|
|
|
60
|
|
|
|
|
15
5
|
|
Therefore
first
of
all
I
compound
the
double
termes
by
multiplication
,
so
that
after
a
sort
I
reduce
the
vnequall
times
to
an
equalitie
,
the
products
are
100
and
75
,
which
are
all
one
as
if
you
should
say
,
that
the
first
man
should
haue
as
much
gaine
by
100
crownes
in
one
yeare
,
as
he
should
haue
for
50
crownes
in
2
yeare
.
In
like
manner
the
second
man
should
get
by
75
crownes
in
one
yeare
,
as
much
as
he
should
get
by
15
crownes
in
fiue
yeare
.
Moreouer
these
two
products
100
and
75
set
downe
seuerally
,
shall
possesse
the
third
place
of
the
proportion
:
but
added
together
they
shall
stand
in
the
first
place
:
nowe
the
rule
of
three
twice
repeated
,
shall
deliuer
the
numbers
sought
for
in
this
manner
:
-
175
60
:
therefore
100
34
2/7
-
175
60
:
therefore
75
25
5/7
I
doe
of
set
purpose
omit
the
demonstration
of
this
composition
of
the
termes
,
because
it
is
not
much
vnlike
the
former
.
A
hundred
and
sixtie
footmen
and
fortie
horsemen
got
a
bootie
of
138
crownes
to
be
diuided
betweene
them
so
,
that
as
often
as
the
footmen
receiued
one
,
the
horsemen
shoulde
receiue
three
.
How
much
were
the
footemen
,
howe
much
were
the
horsemen
to
haue
?
Here
likewise
I
place
the
consequents
of
the
second
reason
added
together
in
the
second
place
,
namely
,
138
:
in
the
thirde
place
I
set
the
products
made
by
multiplying
the
numbers
signifying
how
often
ech
one
shold
take
his
share
by
the
number
of
the
souldiers
:
namely
,
160
and
120
,
these
added
together
and
making
280
,
shall
haue
the
first
place
.
The
example
is
thus
:
-
280
138
:
therefore
160
78
6/7
-
280
138
:
therefore
120
59
1/7
Three
Butchers
hyred
a
meddow
together
,
promising
to
pay
yearely
rent
for
it
30
crownes
,
the
first
fedde
in
it
20
Oxen
70
daies
,
the
second
fedde
46
Oxen
56
dayes
,
the
third
fedde
32
Oxen
60
dayes
:
How
much
of
the
rent
shall
each
partner
pay
?
Againe
therefore
I
multiply
each
mans
heard
by
the
seueral
times
,
the
products
1400
,
2576
,
1920
,
shall
possesse
the
third
place
:
and
being
added
together
,
they
shall
haue
the
first
place
,
the
yearely
rent
shall
be
in
the
middest
thus
:
-
5896
30
:
therefore
1400
7
91/737
-
5896
30
:
therefore
2576
13
79/737
-
5896
30
:
therefore
1920
9
567/737
Three
marchants
were
partners
for
a
yeare
.
The
first
in
the
beginning
brought
250
crownes
,
but
after
3
moneths
he
withdrew
100.
The
second
after
2
moneths
brought
180
crownes
,
but
after
6
moneths
of
their
partnership
were
expired
,
he
tooke
away
50.
The
third
after
three
moneths
brought
in
235
crownes
,
and
after
fiue
moneths
added
45
crownes
.
Now
hauing
gotten
68
crownes
,
what
is
each
mans
share
?
These
kind
of
examples
being
in
outward
shew
most
intricate
,
require
a
litle
more
labor
,
otherwise
they
are
to
be
handled
by
the
same
art
.
They
wil
soone
be
dispatched
,
if
you
indeuor
but
to
seuer
euerie
mans
time
accordingly
as
he
changed
his
stocke
,
and
make
as
many
multiplications
as
there
were
changes
.
As
in
this
present
example
:
Because
the
first
man
left
in
the
partnership
250
crownes
for
three
moneths
,
let
them
be
multiplied
together
,
the
product
will
be
750.
Then
withdrawing
100
crownes
,
he
left
but
150
for
the
nine
moneths
remaining
.
Wherfore
these
being
againe
multiplied
by
their
time
,
make
1350.
Which
numbers
being
added
,
the
totall
summe
for
the
first
partner
is
2100.
Likewise
the
second
man
his
180
crownes
being
multiplied
by
the
4
moneths
wherein
he
left
them
in
the
common
trafficke
,
doe
make
720.
Moreouer
50
crownes
being
subducted
from
180
(
for
so
manie
he
fetcht
away
from
the
principall
after
the
sixth
moneth
of
their
partner
ship
)
there
remaine
130
crownes
,
which
if
you
multiply
by
the
sixe
moneths
remaining
,
the
product
will
be
780
,
which
added
to
the
former
,
maketh
1500.
So
also
multiplie
the
third
man
his
235
crownes
by
9
,
they
make
2115.
Item
45
by
7
,
they
make
315
,
which
added
to
the
former
,
make
2430.
These
termes
being
thus
compounded
,
if
you
proceed
as
you
did
before
,
the
proportion
will
be
inferred
,
which
here
you
see
set
downe
,
with
ordering
of
the
whole
example
.
|
The
stock
.
|
Moneth
.
|
Numbers
compounded
first
by
multiplication
.
|
Numbers
compounded
by
addition
.
|
|
I.
|
250.
|
3.
|
750.
|
2100.
|
|
150.
|
9.
|
1350.
|
II.
|
180.
|
4.
|
720.
|
1500.
|
130.
|
6.
|
780.
|
III.
|
235.
|
9.
|
2115.
|
2430.
|
45.
|
7.
|
315.
|
6030.
|
68.
|
2100.
|
23
137/201
|
1500.
|
16
184/201
|
2430.
|
27
81/201
|
CHAP.
XXII
.
A
treatise
of
Alligation
,
whereof
B.
SALIGNACVS
was
the
Author
.
l.
In
Aligation
we
are
to
consider
,
-
The
proprietie
which
is
this
:
that
in
alligation
,
-
The
measures
should
be
like
.
-
The
price
of
the
mixt
measure
should
be
meane
in
quantitie
betweene
the
prices
of
the
simple
measures
.
-
The
kinds
which
are
two
-
Prime
:
which
coūterchaungeth
the
differences
betweene
the
extremes
&
the
mean
:
&
herein
we
are
to
consider
the
-
Chap.
23.
Propriety
,
which
is
this
,
that
if
the
extremes
are
in
number
-
Euen
:
ech
of
them
seuerally
must
be
compared
once
only
with
the
meane
.
-
Odde
:
then
-
Euery
extreame
of
the
greatest
number
must
be
compared
with
the
meane
once
onely
.
-
Euery
extreame
of
the
least
number
must
be
compared
diuerse
times
with
the
meane
.
-
KIndes
.
which
are
two
,
-
The
first
kind
,
some
part
of
the
numbers
sought
for
being
giuen
,
inferreth
the
rest
by
simple
proportion
.
Chap.
24.
-
The
second
kind
,
the
totall
summe
made
of
the
numbers
sought
for
being
giuen
,
inferreth
all
the
particulars
by
proportion
compounded
by
addition
.
Chap.
25.
-
Second
:
which
inferreth
the
meane
by
proportion
compounded
by
multiplication
and
addition
.
Chap.
26.
Of
the
Definition
and
proprieties
of
Alligation
.
Chap.
1.
HItherto
we
haue
spoken
of
the
first
kinde
of
compound
proportion
:
the
second
followeth
,
and
is
called
Alligation
.
Alligation
is
an
art
,
which
(
by
the
meanes
of
certaine
things
giuen
)
maketh
equall
the
totall
made
of
the
prices
of
the
measures
mingled
together
,
vnto
the
totall
made
of
prices
of
the
measures
taken
seuerally
.
There
is
100
pound
waight
of
siluer
worth
17
pound
to
be
mingled
with
other
siluer
worth
24
pound
,
so
that
the
totall
made
of
the
prices
of
the
waightes
mingled
together
may
be
equall
to
the
totall
made
of
the
prices
of
the
waights
taken
seuerally
.
Here
all
the
prices
are
giuen
,
the
art
therefore
which
by
the
meanes
of
these
prices
giuen
maketh
equall
the
totall
numbers
is
Alligation
.
Item
,
Of
two
kinds
of
corne
mingled
together
suppose
there
were
10
bushels
worth
16
shillings
,
and
18
bushels
worth
12
shillings
:
foure
bushels
of
this
mingled
corne
are
to
be
sold
,
so
that
the
totall
summe
made
of
the
prices
of
the
28
bushels
mingled
together
be
equall
to
the
totall
made
of
the
prices
of
the
10
and
18
bushels
seuerally
taken
.
Here
both
the
number
and
the
price
of
the
simple
measures
are
giuen
,
but
there
is
but
only
a
certain
part
giuen
of
the
number
of
the
measures
mingled
together
.
The
art
therefore
which
by
the
meanes
of
the
thinges
giuen
maketh
equall
the
totals
propounded
in
the
question
,
is
called
Alligation
.
The
propertie
of
Alligation
is
this
,
that
the
measures
in
it
be
alike
:
but
the
price
of
the
mingled
measure
must
be
in
quantitie
meane
between
the
prices
of
the
measures
seuerally
giuen
.
A
bushell
of
corne
worth
16
shillings
is
supposed
to
bee
mingled
with
a
bushell
of
corne
worth
18
shillings
.
Here
by
this
property
first
of
all
the
bushels
are
like
measures
.
Then
the
price
of
a
mingled
bushell
must
be
in
a
quantitie
meane
betweene
16
and
18.
that
is
to
say
,
it
must
be
greater
then
16
,
and
lesse
then
18.
So
that
the
price
of
a
simple
measure
is
called
the
Extreme
,
but
the
price
of
a
mingled
measure
is
the
Meane
.
CHAP.
XXIII
.
Of
Prime
Alligation
,
and
the
propertie
thereof
.
ALLigation
is
two
fold
.
Prime
,
which
doeth
counterchange
the
differences
of
the
termes
from
the
meane
.
By
the
definition
of
Alligation
generally
taken
it
appeareth
,
that
in
all
alligation
there
are
certaine
things
giuen
.
Therfore
in
Prime
alligation
generally
let
all
prices
be
vnderstood
to
be
giuen
.
Moreouer
to
counterchange
the
differences
of
the
extremes
,
is
nothing
else
but
to
attribute
the
difference
of
the
lesser
extreme
to
the
greater
extreme
,
and
contrariwise
to
attribute
the
difference
of
the
greater
extreame
to
the
lesser
extreame
.
Nowe
this
counterchaunge
is
therfore
vsed
,
that
we
may
thereby
make
equall
the
totall
made
of
the
prices
of
the
measures
seuerally
taken
vnto
the
totall
of
the
prices
of
the
measures
mingled
together
.
That
this
may
be
the
better
demonstrated
,
we
must
first
of
all
set
downe
two
premisses
.
The
first
is
this
:
If
one
and
the
same
number
doe
multiply
certaine
numbers
seuerally
,
and
the
totall
made
of
them
being
added
together
,
the
product
made
of
the
totall
shall
be
equall
to
the
totall
made
of
the
products
of
the
parts
.
As
here
:
let
10
be
the
totall
number
made
of
4
and
6
,
let
the
number
multiplying
them
all
,
be
2
,
the
products
made
of
4
and
6
,
are
8
and
12
,
and
the
totall
made
of
them
is
20.
Therefore
the
product
made
of
10
by
2
shall
be
20.
The
second
premisse
is
this
:
If
three
vnequall
numbers
being
giuen
,
you
multiply
any
one
of
them
by
the
other
two
remayning
,
and
augment
the
selfe
same
number
by
one
of
them
which
remaine
,
and
diminish
it
by
the
other
:
and
then
multiply
the
number
augmented
by
that
which
was
taken
away
,
and
the
number
diminished
by
that
which
was
added
to
:
the
totall
number
made
of
the
latter
products
shall
be
equall
to
the
totall
made
of
the
former
products
.
As
here
:
Let
there
be
three
vnequall
numbers
4
,
7
,
9
,
multiply
9
by
4
and
7
,
and
let
the
products
be
36
and
63
,
then
augment
and
diminish
the
selfe
same
number
9
as
is
aforesaid
,
adde
therefore
7
vnto
it
,
and
let
the
totall
be
16
,
also
from
9
take
4
,
and
let
the
remainder
be
5
,
then
multiply
16
by
4
,
and
5
by
7
,
and
let
the
products
be
64
and
35
:
here
the
totall
made
of
64
and
35
,
shall
be
equall
to
the
totall
made
of
36
and
63.
These
things
being
thus
set
downe
:
nowe
of
two
kindes
of
wine
,
let
one
be
worth
14
pence
,
another
worth
11
pence
,
let
the
difference
be
2
and
1
from
the
meane
,
which
is
12
pence
.
The
totall
made
of
the
differences
is
3.
The
meane
price
is
12
,
the
extreames
are
14
and
11.
Therefore
multiply
the
meane
which
is
12
,
by
the
totall
made
of
the
differences
,
namely
by
3
,
the
product
is
36
,
that
is
to
say
,
three
pottles
worth
12
pence
a
peece
shall
be
worth
36
pence
,
I
say
therefore
that
one
pottle
worth
14
pence
,
and
two
pottles
worth
11
pence
a
peece
shall
be
worth
36
pence
.
That
is
:
I
say
that
the
totall
made
of
1
and
2
,
by
14
and
11
,
is
36.
For
here
3
is
the
totall
made
of
1
and
2
as
was
aforesaid
.
Therfore
by
the
first
premisse
if
you
multiply
12
by
2
and
1
,
the
totall
made
of
the
productes
24
and
12
shall
be
equall
to
the
product
made
of
12
and
3.
But
12
multiplied
by
3
maketh
36
,
therefore
the
totall
of
the
products
made
of
12
multiplyed
by
2
and
1
shall
be
36.
Againe
these
three
numbers
11
,
12
,
and
14
are
vnequall
,
the
differences
of
the
extremes
14
and
11
from
12
,
which
is
the
meane
,
are
2
and
1.
The
greater
extreame
14
is
12
more
by
2
,
the
lesser
extreme
11
is
12
lesse
by
1.
Therefore
by
the
second
premisse
if
you
multiply
12
,
augmented
by
1
,
which
was
taken
from
it
,
and
12
diminished
by
2
,
which
was
added
to
it
:
that
is
to
say
,
if
you
multiply
14
by
1
,
and
11
by
2
,
the
totall
made
of
their
products
shall
be
equall
to
the
totall
made
of
the
products
of
12
multiplyed
by
1
&
2.
But
the
total
of
the
products
made
of
12
by
1
and
2
is
36
,
as
it
appeared
before
:
therefore
the
totall
of
the
products
made
of
1
and
2
by
14
and
11
,
shall
be
also
36
,
which
was
the
thing
to
be
demonstrated
.
Therfore
the
reason
why
with
vse
this
counterchaunging
is
as
you
see
.
Here
this
also
is
to
be
noted
:
If
ech
number
of
the
extreames
be
a
number
of
multitude
,
the
differences
may
be
counterchaunged
diuersely
:
and
therefore
then
in
one
and
the
same
example
a
diuerse
alligation
may
be
made
.
But
all
Alligation
maketh
equall
the
totall
made
of
the
prices
of
the
mixt
measures
to
the
totall
made
of
the
measures
seuerally
taken
,
as
appeareth
by
the
definition
of
generall
alligation
:
wherefore
in
the
alligation
intended
when
each
number
of
the
extremes
is
a
number
of
multitude
,
the
counterchanging
of
the
differences
is
at
our
owne
choise
.
Hereafter
followe
two
proprieties
,
and
as
many
kindes
of
prime
alligation
.
The
first
proprietie
is
thus
:
If
the
extremes
be
equall
in
number
,
each
of
them
is
compared
with
the
meane
but
once
onely
.
The
second
proprietie
is
thus
:
If
the
extremes
be
vnequall
in
number
,
then
each
of
the
extremes
whose
number
is
the
greater
,
are
compared
with
the
meane
once
onely
:
but
touching
the
lesser
number
of
extremes
,
if
the
extreme
be
but
one
onely
,
then
is
that
one
extreme
to
be
cempared
with
the
meane
so
often
as
there
is
vnities
in
the
greater
number
of
the
extremes
.
As
here
:
11
|
18
|
3
|
14
|
3
|
8
|
7
,
3
10
|
In
this
example
the
greater
number
of
the
extremes
is
2
,
and
therefore
the
extreme
of
the
lesser
number
is
twise
compared
with
the
meane
.
Where
also
you
shall
note
that
the
totall
made
of
the
manifold
differences
attributed
to
one
and
the
same
extreme
,
is
taken
for
one
difference
:
for
,
for
7
and
3
wee
take
10.
But
if
the
extremes
of
the
lesser
number
be
many
,
if
you
compare
more
then
one
of
them
oftentimes
with
the
meane
,
the
number
arising
of
the
comparison
is
without
arte
:
but
if
you
compare
but
one
onely
with
the
meane
,
that
shall
be
compared
with
it
so
often
as
the
difference
of
the
vnequall
numbers
is
being
added
to
an
vnitie
,
but
the
rest
shall
be
compared
with
the
meane
but
once
onely
.
As
here
:
15
|
4
|
17
|
5
|
17
|
6
|
17
|
7
|
13
|
28
|
8
|
32
|
11
,
10
,
9
30
|
In
this
example
the
greater
number
of
the
extremes
is
4
,
the
lesser
is
2
,
the
difference
of
these
vnequall
numbers
is
2
,
vnto
which
if
you
adde
1
,
the
totall
will
be
3.
Now
I
compare
onely
one
extreame
of
the
lesser
number
,
namely
32
,
oftentimes
with
the
meane
15
,
and
I
compare
it
three
times
,
but
I
compare
the
extreme
which
remaineth
but
once
onely
with
the
meane
.
And
then
last
of
all
.
I
take
the
totall
made
of
the
manifold
differences
attributed
to
one
extreme
,
for
one
difference
as
I
did
before
.
Thus
much
concerning
the
two
proprieties
of
prime
alligation
:
now
follow
the
two
kindes
for
the
vnderstanding
whereof
we
are
to
note
that
the
numbers
sought
for
are
the
numbers
of
the
simple
measures
.
CHAP.
XXIIII
.
Concerning
the
first
kinde
of
prime
Alligation
.
PRime
alligation
of
the
first
kind
,
is
when
some
part
of
the
numbers
sought
for
being
giuen
,
we
inferre
the
rest
by
simple
proportion
.
Here
we
take
the
difference
attributed
to
that
extreme
,
whose
measures
are
by
number
giuen
,
for
the
first
terme
of
proportion
,
and
for
the
third
we
take
the
number
of
the
measures
giuen
.
The
which
kind
of
working
shall
be
euidently
seene
in
the
examples
following
.
My
neighbour
mingled
12
bushels
of
fine
wheat
worth
14
pence
,
with
other
corne
,
namely
,
wheate
,
barlie
,
and
oates
:
the
bushell
of
wheat
was
worth
18
pence
,
the
barlie
11
pence
,
the
oates
9
pence
,
the
bushell
of
corne
mingled
together
was
worth
10
pence
.
The
question
is
,
how
much
wheat
?
how
much
barlie
?
how
much
oates
was
mingled
together
?
Here
a
part
of
the
numbers
sought
for
is
giuen
namely
12
bushels
,
therefore
first
of
all
I
counterchaunge
the
differences
of
the
extremes
as
you
see
:
10
|
18
|
Wheate
|
1
|
14
|
Fine
wheate
|
1
|
11
|
Barley
|
1
|
9
|
Oates
|
8
,
4
,
1
13
|
Then
for
the
first
terme
of
the
proportion
I
take
the
difference
attributed
to
that
extreme
whose
measures
are
by
number
giuen
,
namely
one
bushell
worth
14
pence
,
but
for
the
third
terme
I
take
the
number
of
the
measures
giuen
,
namely
12
bushels
worth
14
pence
a
bushell
.
And
so
conclude
the
question
propounded
by
two
simple
proportions
.
The
first
proportion
is
thus
:
1
Bushell
worth
14
pence
requireth
13
bushels
worth
9
pence
the
bushell
:
therefore
12
bushels
at
14
pence
the
bushell
require
156
bushels
at
nine
pence
the
bushell
.
The
second
is
thus
:
1
bushell
worth
14
pence
requireth
1
bushell
of
each
other
kinde
:
therefore
12
bushels
at
14
pence
the
bushell
require
12
bushels
of
ech
other
kind
.
Wherefore
with
12
bushels
at
14
pence
the
bushell
were
mingled
156
bushels
at
9
pence
the
bushell
,
and
12
bushell
of
each
of
the
other
kindes
.
Suppose
there
were
foure
kindes
of
siluer
:
let
a
pound
of
the
one
be
worth
20
pound
,
of
the
other
16
pound
,
of
the
third
14
pound
,
of
the
fourth
12
pound
.
These
foure
kindes
of
siluer
are
to
be
mingled
together
,
so
that
a
pound
of
the
mingled
siluer
be
worth
15
pound
.
Nowe
there
were
taken
33
pound
of
the
siluer
worth
16
pound
,
the
question
is
how
many
pound
of
each
other
kind
is
to
be
taken
?
Here
a
part
of
the
numbers
sought
for
is
giuen
,
namely
33
,
therefore
the
differences
being
counterchaunged
as
you
see
:
For
the
first
terme
of
the
proportion
,
I
take
the
difference
added
to
the
extreme
whose
measures
are
by
number
giuen
,
namely
one
pound
worth
16
pound
,
but
for
the
third
I
take
the
number
of
the
measures
giuen
,
namely
33
pound
at
16
pound
the
pound
.
Then
I
conclude
the
question
propounded
by
three
simple
proportions
,
the
first
proportion
is
thus
:
1
pound
worth
16
pound
,
requireth
one
pound
worth
14
pound
:
therefore
33
pound
worth
16
pound
the
pound
,
require
33
pound
worth
14
pound
the
pound
.
The
second
is
thus
:
1
pound
worth
16
pound
,
requireth
three
pound
worth
20
pound
the
pound
:
therfore
33
pound
worth
16
pound
the
pound
require
99
pound
worth
20
pound
the
pound
.
The
third
is
thus
:
1
pound
worth
16
pound
,
requireth
fiue
pound
worth
12
pound
the
pound
:
therfore
33
pound
worth
16
pound
the
pound
require
165
pound
worth
12
pound
the
pound
.
Therefore
if
there
were
33
pound
taken
of
the
siluer
worth
16
pound
the
pound
,
there
might
be
taken
33
pound
of
the
siluer
worth
14
pound
the
pound
,
and
99
pound
of
the
siluer
worth
20
pound
the
pound
,
and
165
pound
worth
12
pound
the
pound
,
I
say
they
may
be
taken
,
but
it
is
not
needefull
they
should
be
taken
.
For
seeing
that
here
each
number
of
the
extremes
is
a
number
of
multitude
,
therefore
the
alligation
may
be
manifolde
,
according
as
I
noted
it
before
in
the
definition
of
prime
alligation
.
The
prices
of
the
same
kinds
of
siluer
being
kept
,
if
there
had
bene
6
pound
taken
of
the
siluer
worth
16
pound
the
pound
,
and
8
pound
of
the
siluer
worth
12
pound
the
pound
,
that
a
pound
of
the
mingled
siluer
might
be
worth
15
pound
,
you
shall
say
thus
:
1
pound
worth
16
pound
requireth
5
pound
worth
12
pound
:
therefore
6
pound
worth
16
pound
the
pound
requireth
30
worth
12
pound
the
pound
.
And
therfore
vnto
those
8
pound
worth
12
pound
the
pound
,
there
are
to
bee
added
22
pound
of
the
same
price
.
The
rest
is
easie
by
that
which
hath
beene
said
before
.
CHAP.
XXV
.
Of
the
second
kind
of
prime
Alligation
.
WE
haue
spoken
of
the
first
kind
of
prime
alligation
.
Prime
alligation
of
the
second
kind
is
that
,
which
by
the
meanes
of
the
totall
made
of
the
numbers
sought
for
inferreth
all
the
particulars
by
the
helpe
of
proportion
compounded
by
addition
.
This
compound
proportion
is
commonly
called
the
rule
of
Fellowship
.
In
it
for
the
first
terme
of
proportion
we
take
the
totall
made
of
the
differences
added
together
:
for
the
second
,
we
take
the
totall
made
of
the
number
of
the
measures
,
and
for
the
antecedents
of
the
reasons
remaining
,
the
counterchannged
differences
of
the
extremes
from
the
meane
.
Mine
hoast
hath
two
sorts
of
wine
,
one
worth
sixe
pence
a
quart
,
another
worth
12
pence
a
quart
.
Of
these
two
he
purposeth
to
drawe
sixe
quarts
at
10
pence
:
how
manie
quartes
therefore
shall
hee
drawe
of
each
seuerall
kinde
.
Here
the
whole
number
giuen
made
of
the
numbers
of
the
simple
measures
is
sixe
.
Therefore
counterchaunging
the
differences
as
you
see
:
For
the
first
terme
I
take
the
totall
made
of
the
differences
which
is
6.
For
the
second
terme
I
take
6
,
and
for
the
antecedents
of
the
reasons
remayning
I
take
2
and
4
,
and
then
I
make
a
compounded
proportion
in
this
manner
:
-
6
6
therefore
2
2
-
6
6
therefore
4
4
Here
therefore
there
must
be
drawen
two
quarts
at
6
pence
the
quart
,
&
4
at
12
pence
the
quart
.
Hereby
it
appeareth
that
if
the
totall
made
of
the
numbers
of
the
simple
measures
be
equall
to
the
totall
made
of
the
numbers
of
the
differences
,
the
numbers
of
the
differences
are
the
numbers
of
the
simple
measures
.
Hiero
King
of
Syracusa
voweth
to
the
goddes
for
the
prosperous
successe
of
his
affaires
a
crowne
of
gold
.
Let
vs
suppose
that
Hiero
for
the
making
of
this
crowne
gaue
vnto
a
workeman
500
pound
of
gold
:
and
that
the
workeman
made
in
deede
a
crowne
of
iust
waight
,
but
mingled
some
siluer
in
the
crowne
.
The
king
demaundeth
of
Archimedes
how
much
golde
,
how
much
siluer
was
in
the
mingled
crowne
.
For
the
resoluing
of
this
question
Archimedes
tooke
two
lumpes
of
mettall
of
the
same
waight
with
the
crown
,
but
of
the
same
kind
with
the
gold
and
siluer
which
was
in
the
crowne
.
I
say
they
must
be
of
the
same
kind
.
For
let
there
be
two
lumps
of
siluer
of
equall
waight
one
to
an
other
,
but
of
vneqall
finenesse
,
the
finer
lumpe
shall
fill
a
lesser
place
then
the
other
which
is
not
so
fine
.
The
same
is
to
be
said
of
the
lumpe
of
gold
.
So
that
here
will
be
a
most
manifest
error
,
if
the
mettals
be
they
either
golde
or
siluer
,
be
not
both
of
one
kind
.
I
say
therefore
that
the
lumpe
both
of
golde
and
siluer
was
of
the
same
kinde
with
the
golde
and
siluer
in
the
crowne
.
Now
let
there
be
3
such
bodies
chosen
:
let
the
one
be
a
lumpe
of
gold
,
the
other
a
lump
of
siluer
,
the
third
the
crowne
mingled
of
gold
and
siluer
.
And
let
them
be
hanged
one
after
another
seuerally
in
a
vessell
full
of
water
,
and
let
the
waight
of
the
first
be
968
,
of
the
second
952
,
of
the
third
964.
Then
let
the
waights
be
taken
for
the
value
of
the
bodies
themselues
:
the
value
therefore
of
the
lumpe
of
gold
shall
be
968
,
of
the
siluer
952
,
of
the
crowne
964.
Now
in
the
mingled
crowne
by
supposition
there
are
500
pound
waight
:
therefore
here
the
totall
example
made
of
the
numbers
of
the
measures
shall
be
500
,
so
that
the
differences
being
counterchaunged
as
you
see
:
For
the
first
terme
I
take
the
totall
made
of
the
differences
,
namely
16
,
for
the
second
I
take
the
totall
made
of
the
numbers
of
the
measures
sought
for
,
namely
500
,
for
the
antecedents
of
the
reasons
remaining
.
I
take
12
and
4
,
and
thereof
frame
a
compound
proportion
in
this
manner
:
-
16
500
:
therefore
4
125
-
16
500
:
therefore
12
375
Whereby
it
is
inferred
that
there
was
in
the
mingled
crowne
125
pound
of
siluer
,
and
375
of
gold
.
Archimedes
in
Vitruuius
is
saide
to
haue
found
out
the
mixture
of
the
golde
,
grounding
his
reason
on
the
differences
of
the
water
running
out
of
the
vessell
.
But
that
the
art
may
stand
sure
,
I
construe
his
manner
of
reason
thus
:
that
is
to
say
,
that
by
the
knowen
differences
of
the
water
running
out
of
the
vessell
Archimedes
attained
vnto
the
other
differences
of
the
water
vnknowen
,
and
then
at
length
answered
the
question
by
prime
alligation
of
the
second
kind
.
Otherwise
suppose
an
vnequall
running
out
of
the
water
,
that
is
to
say
,
when
the
gold
was
put
in
,
say
that
there
ranne
out
20
,
when
the
siluer
was
put
in
36
,
and
when
the
crowne
was
put
in
24.
These
seuerall
quantities
of
water
running
out
are
to
be
taken
for
the
values
of
the
bodies
as
they
were
put
in
,
according
to
the
intent
of
him
which
gainesayeth
mine
assertion
.
Therefore
the
value
of
a
lumpe
of
gold
of
500
pound
,
shall
be
lesser
then
the
value
of
a
masse
of
siluer
of
the
same
waight
,
which
is
against
reason
.
Therefore
Archimedes
by
the
knowen
differences
of
the
water
running
out
,
attained
to
the
differences
of
the
water
remaining
in
this
manner
.
Suppose
that
the
vessell
into
the
which
the
crownes
were
put
were
488
pintes
.
If
therefore
the
water
which
ranne
out
when
the
lumpe
of
golde
was
put
in
were
20
pintes
,
the
rest
was
468
,
the
like
we
may
iudge
of
the
rest
.
My
hoast
mingled
foure
sortes
of
wine
to
the
quantitie
of
300
quartes
:
A
quart
of
the
one
wine
was
worth
12
pence
,
of
another
10
pence
,
of
the
third
9
pence
,
of
the
fourth
7
pence
:
He
solde
a
quart
of
the
mingled
wine
for
11
pence
,
how
many
quartes
of
the
first
?
how
many
of
the
second
?
how
many
of
the
third
?
and
how
many
of
the
fourth
kinde
did
he
mingle
together
?
Here
the
totall
number
giuen
made
of
the
number
of
the
simple
measures
is
300.
Therefore
counterchaunging
the
differences
as
you
see
:
11
|
7
|
1
|
9
|
1
|
10
|
1
|
12
|
4
,
2
,
1
7
|
For
the
first
terme
I
take
the
totall
made
of
the
differences
,
namely
10
,
for
the
second
I
take
300
,
for
the
antecedents
of
the
reasons
remayning
I
take
1
,
1
,
1
and
7
,
and
then
I
frame
a
compounded
proportion
thus
:
-
10
300
therefore
7
210
-
10
300
therefore
1
30
-
10
300
therefore
1
30
-
10
300
therefore
1
30
Whereupon
I
conclude
that
the
quartes
herein
mingled
,
were
first
210
at
12
pence
a
quart
,
then
of
each
of
the
rest
there
were
30
quartes
.
An
Apothecary
was
to
mingle
pepper
,
sugar
,
cinnamon
and
ginger
to
the
quantitie
of
fiftie
ounces
.
An
ounce
of
pepper
was
worth
25
,
sugar
24
,
cinnamon
22
,
ginger
18.
An
ounce
of
the
spice
mingled
together
was
worth
23
:
How
many
ounces
of
pepper
,
how
many
of
sugar
,
how
many
of
each
other
kinde
are
to
be
mingled
together
?
Here
first
of
all
the
totall
giuen
being
made
of
the
numbers
of
the
simple
measures
is
500.
Therefore
the
differences
being
counterchaunged
as
you
see
:
For
the
first
terme
of
the
proportion
I
take
the
totall
made
of
the
differences
,
namely
9
,
for
the
second
I
take
500
,
for
the
antecedents
of
the
reasons
remaining
I
take
5
,
1
,
1
,
and
2
,
and
thereof
I
frame
a
compounded
proportion
in
this
manner
:
-
9
500
therefore
5
277
7/9
-
9
500
therefore
1
55
5/9
-
9
500
therefore
1
55
5/9
-
9
500
therefore
2
111
1/9
Here
therefore
may
be
mingled
of
pepper
277
ounces
and
7/9
,
of
sugar
55
5/9
,
and
so
much
cinnamon
,
of
ginger
111
and
1/9.
I
say
they
may
be
mingled
,
but
they
shall
not
be
mingled
.
For
in
this
question
each
number
of
the
extremes
is
a
number
of
multitude
,
and
therfore
a
manifold
alligation
may
be
made
therein
,
of
which
manifold
alligation
I
gaue
a
note
before
in
the
first
kind
of
prime
alligation
.
CHAP.
XXVI
.
Of
second
Alligation
.
HItherto
of
prime
alligation
:
Second
alligation
followeth
which
by
proportion
compounded
by
multiplication
and
addition
inferreth
the
meane
.
Here
both
the
numbers
and
prices
are
giuen
of
each
seuerall
measure
,
but
there
is
but
some
part
onely
of
the
mingled
measures
giuen
.
Therefore
here
for
the
first
terme
I
take
the
totall
made
of
the
numbers
of
the
simple
measures
,
and
for
the
second
terme
,
I
take
the
totall
of
the
products
made
by
multiplying
those
numbers
by
their
extremes
,
and
last
of
all
for
the
third
terme
of
the
proportion
,
I
take
the
part
giuen
of
the
number
of
the
mixt
measures
.
Of
two
kinds
of
corne
there
were
mingled
10
bushels
at
16
pence
a
bushell
,
with
18
bushels
at
12
pence
a
bushell
:
what
is
the
bushell
of
mingled
corne
worth
?
Here
first
of
all
I
adde
together
the
numbers
of
the
simple
measures
10
and
18
,
the
totall
is
28.
Then
I
multiply
10
by
16
,
and
18
by
12
,
and
make
160
and
216
,
the
totall
of
them
is
376.
The
totall
therefore
of
the
prices
of
the
simple
measures
is
376
pence
.
But
alligation
maketh
the
totall
of
the
prices
of
the
mixt
measures
equall
to
the
totall
of
the
simple
measures
as
appeareth
by
the
definition
thereof
.
Therefore
these
28
mingled
bushels
shall
be
worth
376.
Therfore
I
conclude
the
question
thus
:
28
mingled
bushels
are
worth
376.
therefore
one
bushell
of
mingled
corne
is
worth
13
3/7.
P.
Ramus
the
most
famous
Philosopher
of
our
time
,
calleth
this
alligation
,
alligation
of
the
meane
sought
for
:
and
defineth
it
to
be
that
,
which
(
two
extremes
being
giuen
)
seeketh
out
the
meane
by
diuiding
the
extrems
added
together
by
their
number
.
He
addeth
these
wordes
as
if
there
be
two
extremes
the
diuisor
must
be
2
,
if
there
be
three
extreames
the
diuisor
must
be
3
,
and
so
foorth
.
Therefore
by
this
definition
if
corne
at
16
pence
and
12
pence
a
bushell
be
mingled
together
to
the
quantitie
of
28
bushels
as
is
aforesaide
,
the
price
of
a
bushell
of
mingled
corne
shall
be
14
pence
,
for
here
the
number
of
the
extremes
is
2
,
and
the
extremes
16
and
12
being
added
make
28
,
so
that
if
thou
diuide
28
by
2
,
the
quotient
is
14.
But
let
vs
see
how
true
this
is
:
Here
by
the
rule
of
prime
alligation
the
counterchaunging
of
the
differences
is
in
this
manner
:
By
the
which
counterchange
it
is
insinuated
,
that
when
there
are
taken
two
bushels
of
corne
at
16
pence
,
then
must
there
be
two
also
taken
of
12
pence
the
bushell
.
Therefore
when
there
shall
be
10
taken
at
16
pence
the
bushell
,
then
shall
there
also
be
10
taken
at
12
pence
the
bushell
.
The
numbers
of
the
simple
measures
10
and
18
being
added
together
make
28.
Therefore
I
reason
thus
:
If
when
10
bushels
at
16
pence
the
bushell
are
mingled
with
18
bushels
at
12
pence
the
bushell
,
the
meane
is
then
14
:
therefore
when
28
bushels
of
these
simple
seueral
cornes
are
so
mingled
,
that
the
price
of
a
mingled
bushell
is
14
pence
,
then
shall
10
bushels
at
16
pence
the
bushell
,
be
mingled
with
18
bushels
at
12
pence
the
bushell
.
(
For
it
is
all
one
way
from
Newhause
to
Heidelberge
,
and
from
Heidelberge
to
Newhause
.
)
But
this
second
assertion
is
false
(
for
here
when
10
bushels
at
16
pence
the
bushell
are
taken
,
then
also
there
are
taken
10
bushels
at
12
pence
the
bushell
as
appeared
before
)
therefore
the
first
assertion
is
also
false
.
Whether
with
18
,
or
with
10
bushels
at
12
pence
a
bushell
,
you
mingle
10
bushels
at
16
pence
a
bushell
,
the
extremes
added
together
in
both
are
28
,
and
their
number
is
2.
Therefore
whether
you
mingle
10
bushels
at
16
pence
the
bushell
,
with
18
,
or
10
bushels
at
12
pence
a
bushell
,
the
price
of
a
bushell
of
mingled
corne
shall
be
all
one
,
and
so
the
price
of
the
cheaper
wheate
shall
be
the
price
of
the
dearer
wheate
,
which
is
absurde
.
I
alwaies
reuerenced
my
master
while
he
was
aliue
as
my
duetie
required
,
and
much
more
doe
I
now
imbrace
the
writings
of
that
most
holy
Martyr
being
dead
.
But
it
is
incident
to
a
man
to
erre
:
and
therefore
he
being
a
Philosopher
of
most
holy
memorie
,
esteemed
those
men
to
be
madde
that
thought
it
an
hurtfull
thing
for
a
common
wealth
to
haue
their
faults
amended
:
therefore
following
his
owne
decree
by
my
definition
,
I
correct
his
in
this
place
.
For
he
himselfe
either
did
correct
it
before
his
death
,
or
if
he
had
liued
any
longer
,
I
doubt
not
but
he
would
willingly
haue
amended
it
:
if
he
did
correct
it
,
truely
it
was
neuer
my
chance
as
yet
to
see
his
correction
.
Let
6
ounces
of
cloues
at
36
,
and
8
ounces
of
cinnamon
at
16
,
be
mingled
with
4
ounces
of
pepper
at
15
:
how
shall
4
ounces
of
the
mingled
spice
be
solde
?
Here
the
numbers
of
the
simple
measures
are
6
,
8
,
and
4
,
whereof
the
totall
is
18.
The
products
of
36
,
16
,
and
15
,
made
by
multiplying
them
by
6
,
8
,
and
4
are
216
,
128
,
and
60.
The
totall
of
them
is
404.
Therefore
I
will
say
thus
:
18
ounces
of
mingled
spice
are
worth
404
therefore
4
ounces
are
worth
89
7/9.
Let
the
prices
of
the
simple
spices
be
as
they
were
before
:
and
let
1/2
of
an
ounce
of
cloues
,
and
1/8
of
an
ounce
of
cinnamon
be
mingled
with
1/4
of
an
ounce
of
pepper
:
what
shall
an
ounce
of
the
mingled
spice
be
worth
?
Here
first
of
all
the
vnlike
measures
which
are
giuen
1/2
,
1/3
,
1/4
must
be
made
like
.
For
the
propertie
of
generall
alligation
requireth
that
the
measures
should
be
like
.
Therefore
by
reduction
of
fractions
to
one
denomination
,
for
1/2
,
1/3
and
1/4
,
I
finde
out
6/12
,
4/12
,
and
3/12.
Then
that
the
worke
may
be
more
easie
for
the
parts
found
out
,
by
reduction
of
fractions
to
whole
numbers
,
I
take
6
,
4
,
and
3.
The
totall
therefore
of
the
numbers
of
the
measures
shall
be
13.
Then
multiply
36
,
16
,
&
15
,
by
the
numbers
of
their
measures
6
,
4
,
and
3
,
you
shall
produce
216
,
64
,
and
45
:
whereof
the
totall
is
325.
Then
shall
you
conclude
the
question
thus
:
13
ounces
of
mingled
spice
are
worth
325
,
therefore
one
ounce
is
worth
25
3/13.
CHAP.
XXVII
.
Of
Manifolde
proportion
continued
in
the
termes
.
MAnifolde
proportion
compounded
in
the
termes
hath
beene
handled
hitherto
:
that
remayneth
which
is
continued
in
the
termes
.
Manifold
proportion
continued
in
the
termes
is
,
when
vnto
the
disiunct
termes
of
the
reasons
giuen
,
other
proportionall
termes
doe
aunswere
,
whereof
each
middlemost
terme
ioyneth
the
antecedent
reason
with
the
consequent
.
As
in
the
examples
following
For
the
reasons
giuen
in
the
vppermost
rowe
of
numbers
(
namely
a
subsesquialtera
,
and
a
subsesquiquarta
)
are
in
the
nethermost
row
of
numbers
so
continued
and
knit
one
to
an
other
,
that
of
the
two
middlemost
termes
there
is
made
one
,
which
according
to
the
reasons
giuen
in
the
first
ranke
of
numbers
is
both
the
consequent
of
the
former
,
and
the
antecedent
of
the
latter
.
For
as
2
is
vnto
3
,
so
is
8
vnto
12
:
and
as
4
is
vnto
5
,
so
is
12
vnto
15.
In
this
kind
of
proportion
therefore
there
is
intended
an
inuention
or
finding
out
of
the
least
termes
continually
proportionall
with
the
reasons
giuen
how
many
so
euer
.
This
inuention
is
set
foorth
in
the
4
prop.
8
,
where
it
is
demonstrated
by
what
meanes
,
how
many
reasons
so
euer
being
giuen
in
the
least
termes
,
you
may
finde
out
other
least
termes
continually
proportionall
,
keeping
the
reasons
giuen
.
Let
that
probleme
therefore
be
turned
into
two
theoremes
:
whereof
let
the
one
concerne
two
reasons
giuen
,
and
let
the
other
concerne
many
reasons
giuen
in
this
manner
:
I
If
the
middlemost
termes
being
prime
one
to
another
of
the
two
reasons
giuen
,
be
set
crossewayes
,
and
then
the
former
of
the
two
multiply
the
termes
of
the
first
reason
,
and
the
latter
the
termes
of
the
consequent
:
the
three
products
shall
be
the
least
termes
continually
proportionall
as
the
reasons
giuen
are
.
Therefore
we
are
to
note
,
that
in
the
reasons
giuen
the
middlemost
termes
are
either
prime
numbers
or
compound
,
if
they
be
compound
,
they
must
be
reduced
to
prime
numbers
.
As
in
3
,
2
,
7
,
4
,
the
middlemost
numbers
are
prime
numbers
in
respect
one
of
an
other
.
Wherefore
being
set
crossewise
one
vnder
another
,
if
you
multiply
7
by
2
and
3
,
which
are
the
termes
of
the
former
reason
:
and
then
multiply
2
by
7
and
4
,
which
are
the
termes
of
the
reason
following
,
the
products
21
,
14
,
8
,
shall
be
the
least
numbers
continually
proportionall
,
as
the
numbers
giuen
are
,
as
you
see
here
:
For
as
3
is
vnto
2
,
so
is
21
to
14
:
and
as
7
is
to
4
,
so
is
14
to
8.
Let
these
two
reasons
or
proportions
be
giuen
like
one
to
an
other
3
,
2
,
6
,
4.
Here
2
and
6
are
compound
numbers
in
respect
one
of
another
,
wherfore
I
reduce
them
to
prime
or
the
least
numbers
,
namely
to
1
and
3
,
and
place
them
crossewise
,
and
multiply
3
into
3
and
2
the
antecedents
,
and
likewise
1
into
6
and
4
the
consequents
,
the
products
are
9
,
6
,
4
,
which
are
the
least
termes
continually
proportionall
,
as
the
reasons
giuen
,
according
as
you
see
them
here
written
.
II.
Any
three
reasons
being
giuen
,
if
two
terms
prime
in
respect
one
of
another
,
(
whereof
the
one
is
the
last
of
the
least
continuall
proportionall
terms
of
the
former
reasons
found
out
,
the
other
is
the
antecedent
of
the
thirde
reason
)
be
set
crossewise
,
and
then
all
the
numbers
found
out
be
multiplyed
by
the
first
,
&
the
numbers
following
by
the
last
:
the
products
shall
be
the
least
numbers
continually
proportionall
as
the
reasons
giuen
are
:
and
so
forth
if
there
be
more
giuen
.
For
example
sake
,
let
there
be
three
reasons
giuen
4
vnto
3
,
2
vnto
1
,
5
vnto
6
,
to
be
continued
in
the
least
or
prime
proportionall
termes
.
First
therefore
I
dispatch
the
former
two
reasons
,
and
find
8
,
6
,
3.
And
because
3
(
the
last
of
them
that
are
found
out
)
is
a
prime
number
to
5
the
antecedent
of
the
reason
remaining
:
therefore
they
being
set
crossewise
,
let
5
multiply
8
and
6
,
the
products
are
40
and
30
:
but
let
3
multiply
the
numbers
following
5
and
6
,
the
products
are
15
,
30
,
so
that
we
haue
now
foure
termes
in
the
least
numbers
continually
proportionall
as
the
three
reasons
giuen
were
.
The
example
is
in
this
manner
:
Let
these
three
be
giuen
to
be
continued
after
the
maner
aforesaid
,
3
2
,
—
5
4
,
—
4
3.
The
same
kind
of
working
must
be
followed
,
sauing
that
after
that
2
reasons
are
dispatched
,
in
steede
of
8
and
4
which
are
compound
numbers
in
respect
one
of
another
,
I
take
2
and
1
,
which
are
prime
numbers
as
you
see
in
this
example
:
In
the
example
following
there
are
foure
reasons
continued
together
,
and
after
the
same
manner
there
may
be
as
many
as
you
please
continued
together
in
the
least
termes
.
Therefore
the
questions
of
proportion
wherein
any
number
offereth
it selfe
to
be
parted
according
vnto
manie
reasons
in
diuerse
termes
so
linked
together
,
that
the
former
terme
hath
respect
vnto
that
which
followeth
,
are
to
be
answered
by
the
rule
of
proportion
compounded
by
addition
,
the
least
termes
being
first
found
out
continually
proportionall
as
the
reasons
giuen
are
.
The
example
.
Two
hundred
crownes
are
to
be
diuided
on
that
condition
that
the
first
man
shall
haue
three
times
so
much
as
the
second
:
the
second
shall
haue
foure
times
as
much
as
the
third
.
How
much
shall
euery
man
haue
▪
Here
there
are
two
reasons
propounded
,
the
one
is
triple
,
the
other
is
quadruple
,
which
are
knit
together
in
this
manner
:
That
is
to
say
,
let
a
be
triple
to
c
,
but
let
c
be
quadruple
vnto
e.
These
are
knit
together
by
two
middle
termes
1
and
4
,
which
being
so
lincked
together
,
that
one
and
the
same
terme
according
to
the
reasons
giuen
,
may
be
both
consequent
to
a
,
and
antecedent
vnto
e
,
I
shall
easily
resolue
the
doubt
.
Now
this
will
be
done
,
if
I
finde
out
the
least
termes
continually
proportionall
as
the
termes
are
which
are
giuen
,
namely
3
,
1
,
4
,
1
,
the
which
termes
found
out
will
be
these
,
12
,
4
,
1.
For
then
they
being
added
together
,
the
proportion
will
be
inferred
thus
:
-
17
200
therefore
12
141
3/17
-
17
200
therefore
4
47
1/17
-
17
200
therefore
1
11
11/17
Let
there
be
24
crownes
so
to
be
parted
,
that
as
often
as
the
first
hath
3
,
so
often
the
second
hath
4
,
and
as
often
as
he
hath
two
,
so
often
the
third
hath
3
,
and
as
often
as
the
third
hath
6
,
so
often
the
fourth
hath
1.
What
shall
each
mans
portion
be
?
Here
the
termes
of
the
three
reasons
3
,
4
:
2
,
3
,
6
,
1
:
are
to
be
reduced
(
by
the
last
theoreme
)
into
foure
termes
continually
proportionall
as
the
reasons
giuen
are
,
the
which
foure
termes
are
these
,
3
,
4
,
6
,
1.
The
totall
made
of
these
termes
added
together
,
shall
be
the
first
terme
of
the
proportion
,
and
being
seuerally
set
downe
,
they
shall
be
the
third
terme
.
The
number
which
is
to
be
diuided
shall
be
the
middlemost
,
and
then
the
proportion
shall
be
concluded
thus
:
-
14
24
:
therefore
3
5
1/7
-
14
24
:
therefore
4
6
6/7
-
14
24
:
therefore
6
10
2/7
-
14
24
:
therefore
1
1
5/7
Foure
other
men
were
to
diuide
120
crownes
gotten
in
partnershippe
,
so
that
as
often
as
the
second
had
5
,
so
often
the
third
had
9
,
and
as
often
as
the
third
had
7
,
so
often
the
fourth
had
11
,
and
as
often
as
the
fourth
had
9
,
so
often
the
first
had
13
,
what
was
each
man
his
portion
.
This
question
is
somewhat
differing
in
shewe
,
but
the
working
is
all
one
.
For
the
matter
is
dispatched
by
proportion
of
addition
,
the
least
numbers
being
found
out
continually
proportionall
as
the
reasons
giuen
are
.
As
you
see
here
:
-
340
120
therefore
35
12
6/17
-
340
120
therefore
63
22
4/17
-
340
120
therefore
99
34
16/17
-
340
120
therefore
143
50
8/17
CHAP.
XXVIII
.
Of
continuall
proportion
.
L
Coniunct
or
continuall
proportion
,
is
either
-
Simple
.
-
Or
-
Manifold
:
and
is
called
progression
,
wherein
we
are
to
consider
-
The
order
of
the
termes
,
-
Entring
into
the
progression
.
-
Or
-
Making
a
new
progression
.
-
Or
-
The
sum
of
the
termes
.
HItherto
of
disiunct
proportion
:
now
followeth
continuall
.
Continuall
proportion
is
that
whose
middle
termes
doe
all
supply
the
place
of
an
antecedent
,
and
a
consequent
.
In
continuall
proportion
the
first
extreme
is
onely
an
antecedent
,
and
the
last
is
onely
a
consequent
:
of
the
other
termes
,
which
are
in
the
middest
betweene
the
extremes
any
of
them
in
respect
of
that
which
went
before
it
is
a
consequent
,
in
respect
of
that
which
followeth
it
is
an
antecedent
,
as
this
is
in
three
termes
9
,
6
,
4
,
in
foure
termes
,
8
,
12
,
18
,
27
,
in
fiue
termes
32
,
16
,
8
,
4
,
2
&c.
Continuall
proportion
is
either
simple
,
or
manifold
,
simple
continuall
proportion
,
is
that
wherin
there
is
but
one
only
meane
betwene
the
extrems
.
This
is
the
9
d.
5.
which
affirmeth
that
proportion
consisteth
in
three
termes
at
the
least
,
as
in
9
,
6
,
4
,
where
the
extremes
9
and
4
are
lincked
together
by
one
onely
terme
.
This
is
framed
in
the
least
termes
,
if
you
multiply
the
termes
of
the
reason
giuen
being
prime
in
respect
one
of
another
,
both
by
themselues
,
and
one
by
another
,
for
the
products
are
the
termes
of
the
continuall
proportion
,
according
to
the
reason
giuen
.
As
for
example
,
of
2
,
1
,
which
are
the
termes
of
a
double
reason
,
you
shall
make
a
simple
continuall
proportion
if
you
multiply
2
by
it selfe
,
and
then
by
1
,
and
last
of
all
1
by
it selfe
,
for
the
products
will
be
4
,
2
,
1.
Likewise
of
3
,
2
,
the
termes
of
a
sesquialter
reason
,
you
shall
by
the
same
meanes
make
9
,
6
,
4.
The
propertie
of
it
is
this
:
The
product
made
of
the
extremes
,
is
equall
to
the
product
made
of
the
meane
,
and
contrariwise
.
This
is
the
17
p.
6.
concerning
magnitudes
,
and
also
the
20
p.
7.
where
this
propertie
is
also
attributed
vnto
numbers
,
because
in
three
proportionall
numbers
,
the
product
made
of
the
extremes
,
is
equall
vnto
the
square
made
of
the
meane
.
As
in
9
,
6
,
4
,
the
product
made
of
9
and
4
,
is
36
,
which
is
equall
to
the
square
of
the
meane
,
namely
vnto
sixe
times
sixe
,
or
which
is
all
one
to
the
product
made
of
the
meane
multiplied
by
it selfe
.
For
it
is
,
as
if
you
shold
inferre
the
proportion
disiunctiuely
in
foure
termes
after
this
manner
:
9
giue
6
:
therefore
6
giue
4.
Wherefore
it
is
to
be
gathered
,
that
this
propertie
is
generall
to
all
simple
proportion
whether
it
consist
of
three
or
foure
termes
.
Manifold
continuall
proportion
,
is
that
wherin
many
meane
termes
proceede
continually
after
one
and
the
same
reason
.
Hereupon
it
is
called
progression
.
As
1
,
2
,
4
,
8.
Item
16
,
24
,
36
,
54
,
81.
In
progression
we
haue
aneye
to
the
finding
out
of
the
order
of
the
terms
making
the
progression
,
or
else
to
the
summe
of
them
.
The
finding
out
of
the
termes
is
two
fold
:
for
either
the
terms
giuen
enter
into
the
progression
,
or
else
new
termes
are
made
by
their
meanes
.
All
the
termes
of
a
proportion
giuen
,
can
not
be
extended
and
continued
.
In
manifold
proportion
it
may
be
,
but
in
other
reasons
it
can
not
so
well
be
done
.
It
shall
appeare
therefore
by
the
theoreme
following
,
what
termes
giuen
will
admit
a
progression
.
If
the
last
terme
sauing
one
doth
diuide
the
product
made
of
the
last
terme
,
the
quotient
shall
be
the
proportionall
terme
following
,
Or
,
If
the
last
terme
doth
diuide
the
product
made
of
the
last
sauing
one
,
the
quotient
shall
be
the
antecedent
proportionall
.
This
Theoreme
is
gathered
out
of
the
18
p.
9.
It
is
also
deduced
as
it
were
a
consequence
out
of
the
proprietie
of
continuall
proportion
,
whereby
not
onely
proportion
,
but
any
reason
giuen
may
be
continued
.
As
in
3
,
9
,
27
,
wherein
9
diuideth
729
,
the
product
made
of
the
last
number
27
,
the
quotient
is
81
,
which
is
the
fourth
proportionall
number
.
Likewise
let
9
which
is
the
second
number
diuide
9
which
is
the
product
made
of
the
first
number
3
,
the
quotient
1
,
is
the
first
proportionall
number
.
So
that
the
progression
shall
be
in
this
manner
,
1
,
3
,
9
,
27
,
81.
This
continuation
of
the
termes
in
manifold
proportion
may
be
infinitely
inferred
onely
by
the
multiplication
of
the
last
terme
by
the
denominator
of
the
reason
or
proportion
:
Yet
in
those
whose
reason
is
manifold
,
any
terme
sought
for
may
be
found
our
readily
in
this
manner
:
If
you
diuide
the
product
made
of
the
last
number
multiplied
in
it selfe
,
by
any
of
the
numbers
which
went
before
it
:
by
how
many
degrees
the
diuisor
went
before
the
number
multiplied
,
by
so
many
degrees
shall
the
quotient
followe
the
same
number
multiplied
.
For
example
sake
,
let
this
double
proportion
be
giuen
.
2
|
4
|
8
|
16
|
32
|
64
|
128
|
4
|
3
|
2
|
1
|
2
|
3
|
4
|
Antecedents
|
|
Consequents
.
|
And
let
16
be
the
number
to
be
multiplied
by
it selfe
,
th
product
is
256
,
diuide
that
by
8
which
is
the
number
next
before
it
,
the
quotient
32
is
the
number
which
must
followe
next
after
16.
But
if
you
diuide
the
product
256
by
4
which
went
before
16
in
the
thirde
place
,
the
quotient
will
be
64
,
which
must
follow
in
the
third
place
after
16.
If
you
make
the
diuision
by
2
,
which
went
before
16
in
the
fourth
place
,
the
quotient
128
must
follow
in
the
fourth
place
after
16
,
and
so
forth
in
the
rest
.
For
it
is
as
if
you
should
set
the
rule
of
proportion
thus
:
If
2
giue
16
:
then
16
shall
giue
128.
Out
of
the
foresaid
abridgement
,
another
ariseth
,
whereby
we
may
finde
out
any
terme
of
manifold
progression
,
in
this
manner
:
If
some
certaine
termes
of
continuall
progression
be
giuen
,
and
the
numbers
following
one
another
from
an
vnitie
according
to
their
naturall
order
,
doe
aunswere
the
saide
termes
from
the
second
forward
,
the
quotient
of
the
product
made
by
any
of
the
termes
whatsoeuer
diuided
by
the
first
terme
,
shall
be
the
terme
of
the
progression
more
by
one
,
then
both
the
numbers
answering
to
the
numbers
multiplied
,
doe
amount
vnto
being
added
together
.
As
in
this
triple
proportion
,
let
there
be
giuen
some
termes
,
and
let
the
numbers
be
written
vnderneath
them
in
order
as
you
see
:
Then
if
you
multiply
27
by
81
,
and
diuide
the
product
2187
by
3
,
which
is
the
first
number
in
the
progression
,
the
quotient
729
shall
be
the
sixth
terme
of
the
progrestion
which
is
more
by
one
,
then
2
and
3
(
that
is
to
say
5
)
are
.
Likewise
if
you
multiply
243
by
it selfe
,
and
diuide
the
product
by
the
first
,
the
quotient
shall
be
19683
,
to
be
placed
in
the
ninth
place
,
and
so
forth
of
the
rest
.
The
termes
aswell
of
manifolde
reason
as
of
proportion
and
progression
,
may
in
this
manner
be
continued
infinitely
.
But
the
nature
of
numbers
doth
not
beare
it
so
well
in
the
other
kinds
of
reason
or
proportion
.
For
first
of
all
,
if
of
the
reason
giuen
the
termes
be
prime
numbers
one
to
another
,
then
can
not
a
third
,
much
lesse
a
fourth
or
fifth
in
whole
numbers
be
adioyned
to
them
in
continuall
proportion
:
as
it
appeareth
by
the
16
p.
of
the
9.
As
in
the
reason
of
3
to
5
,
which
is
superpartient
two
fifts
,
you
shall
neuer
bring
forth
a
continuall
proportion
,
vnlesse
you
ioyne
to
it
this
mixt
or
surde
number
8
1/3
,
which
cannot
be
expressed
by
an
whole
number
.
Likewise
how
many
proportionall
numbers
soeuer
being
giuen
,
if
the
extremes
be
prime
one
to
another
,
no
such
number
can
be
giuen
at
the
last
,
as
the
second
is
vnto
the
first
,
by
the
17
p.
of
the
9.
As
in
this
subsesquialter
4
,
6
,
9
,
because
4
and
9
are
prime
one
to
another
,
therefore
there
can
be
no
whole
number
in
such
proportionalitie
vnto
9
,
as
4
is
vnto
6.
Therefore
in
such
kind
of
proportions
this
Theoreme
following
taketh
place
.
If
the
termes
of
continuall
simple
proportion
,
not
admitting
an
vsuall
progression
be
multiplied
by
the
antecedent
of
the
reason
whereof
they
consist
,
and
the
last
be
multiplied
by
his
consequent
:
there
shall
be
produced
foure
least
termes
continually
proportionall
according
to
the
numbers
giuen
:
and
so
forth
continually
by
multiplying
the
products
by
the
antecedent
giuen
,
and
the
last
by
the
consequent
giuen
,
you
shall
find
the
least
termes
how
many
so
euer
continually
proportionall
according
to
the
reasons
giuen
:
As
for
example
,
seeing
that
this
sesquitertia
16
,
12
,
9
,
hath
not
a
fourth
number
in
progression
.
Therefore
first
of
all
,
I
multiply
all
the
numbers
giuen
by
4
,
the
antecedent
of
the
reason
giuen
betweene
4
and
3
,
whereof
they
were
made
:
the
products
shall
be
64
,
48
,
36.
Likewise
I
multiply
the
last
number
9
by
3
the
consequent
giuen
,
the
product
is
27
,
the
which
numbers
are
all
continually
proportionall
next
vnto
the
numbers
giuen
.
After
this
manner
many
others
may
be
found
out
.
|
|
|
4
|
|
3
|
|
|
|
The
termes
of
the
reason
giuen
.
|
|
|
16
|
|
12
|
|
9
|
|
|
The
simple
prop.
|
|
64
|
|
48
|
|
36
|
|
27
|
|
The
first
progression
.
|
256
|
|
192
|
|
144
|
|
108
|
|
81
|
|
Behold
also
this
example
following
of
a
manifold
superparticular
reason
.
|
|
|
5
|
|
2
|
|
|
|
|
|
25
|
|
10
|
|
4
|
|
|
|
125
|
|
50
|
|
20
|
|
8
|
|
625
|
|
250
|
|
100
|
|
40
|
|
16
|
Thus
much
concerning
the
ranging
of
the
termes
:
now
followeth
the
summe
of
the
progression
.
The
finding
out
of
the
summe
,
is
that
which
(
the
first
terme
of
the
progression
increasing
being
subducted
from
the
sec̄ond
and
the
last
)
addeth
vnto
the
last
terme
giuen
a
number
,
to
the
which
the
remainder
of
the
last
terme
hath
such
proportion
as
the
remainder
of
the
second
terme
hath
vnto
the
first
.
I
said
(
the
progression
increasing
)
that
ye
may
vnderstande
that
the
first
terme
is
here
taken
for
the
least
,
and
the
last
for
the
greatest
.
Now
the
masterie
of
this
inuention
is
generall
,
being
not
onely
of
force
in
manifold
proportion
,
as
that
rule
is
which
the
common
sort
of
Arithmeticians
do
prescribe
,
but
all
progression
of
what
proportionalitie
so
euer
giuen
:
It
is
taken
out
of
the
33
prop.
9.
which
affirmeth
this
,
of
how
many
proportionall
numbers
soeuer
following
one
another
.
If
from
the
second
and
the
last
terme
there
be
taken
away
numbers
equall
to
the
first
,
the
remainder
of
the
last
terme
shall
be
vnto
all
the
antecedents
going
before
it
,
as
the
remainder
of
the
second
terme
is
vnto
the
first
terme
.
Therefore
these
three
proportionall
termes
being
found
out
,
you
shall
set
the
excesse
or
the
remainder
of
the
last
terme
in
the
third
place
,
and
the
remainder
of
the
second
in
the
first
place
,
and
the
first
or
least
terme
of
the
progression
in
the
middlemost
place
,
and
then
worke
by
the
rule
of
three
,
whereby
you
shall
inferre
a
number
contayning
all
them
which
were
antecedents
to
the
last
,
which
being
therefore
added
to
the
last
shall
containe
the
summe
of
them
all
.
As
in
this
manifold
progression
,
2
,
4
,
8
,
16
,
32
,
if
2
be
taken
from
4
,
and
32
,
there
remaine
2
and
30.
Now
as
2
the
remainder
of
the
second
terme
is
vnto
the
first
terme
2
,
so
is
30
vnto
all
the
antecedents
.
Therefore
for
the
working
of
the
rule
,
the
termes
shall
stand
thus
:
2
giue
2
:
therefore
30
giue
30.
The
reason
is
in
ech
place
alike
and
equall
.
Therefore
seeing
that
the
totall
made
of
the
numbers
going
before
the
last
is
30
,
it
being
added
to
the
last
,
namely
vnto
32
,
declareth
the
summe
of
the
progression
to
be
62.
Likewise
in
this
subtriple
progression
,
2
,
6
,
18
,
54
,
let
2
be
taken
from
6
and
54
,
the
remainders
shall
be
4
and
52.
Now
as
4
the
remainder
of
the
second
terme
is
vnto
the
first
terme
2
,
so
is
52
the
remainder
of
the
last
terme
vnto
all
the
antecedents
.
Wherefore
the
three
termes
knowne
shall
be
thus
ordered
of
the
finding
out
of
the
fourth
terme
.
4
2
:
therefore
52
26
And
for
so
much
as
this
fourth
number
,
is
equall
vnto
all
the
antecedents
,
it
being
added
to
the
last
terme
54
,
the
totall
80
shall
be
the
summe
of
the
progression
.
Likewise
let
there
be
this
subsesquialter
progression
,
16
,
24
,
36
,
54
,
81.
The
remainders
of
the
second
,
and
last
terme
are
8
,
and
65
,
whereupon
by
these
three
proportionall
numbers
8
,
16
,
65
,
the
fourth
number
130
is
inferred
,
which
is
the
totall
made
of
all
the
antecedents
except
the
last
,
it
being
therfore
added
vnto
the
last
terme
81
,
declareth
the
summe
of
the
progression
to
be
211.
Whereby
it
appeareth
that
the
first
,
second
,
and
last
terme
of
the
progression
being
knowen
the
summe
cannot
lie
hid
.
An
Example
.
A
certaine
man
solde
his
house
to
be
payde
for
it
in
wheate
after
such
a
manner
,
that
when
he
that
bought
it
came
in
at
the
first
doore
he
should
giue
him
one
graine
,
at
the
second
two
,
at
the
third
4
,
and
so
foorth
proceeding
continually
by
double
proportion
according
to
the
number
of
the
doores
.
He
that
shrunke
from
his
bargaine
should
pay
to
the
other
12
crowns
for
a
penaltie
.
Now
being
60
doores
,
the
question
is
how
much
wheat
he
was
to
pay
.
A
rich
man
of
Basill
not
ignorant
of
the
huge
increase
of
Geometricall
progression
,
sold
his
house
sometimes
vppon
this
condition
being
among
his
cuppes
.
But
the
condition
of
his
bargaine
being
brought
vnto
an
account
,
the
number
of
the
wheat
was
found
to
be
vnmeasurable
,
so
that
all
the
houses
of
Basill
being
turned
in
garners
,
were
not
sufficient
to
receiue
it
.
I
thought
good
to
set
downe
the
example
.
Let
the
60
terme
of
the
progression
be
compendiously
found
out
.
1
|
2
|
4
|
8
|
16
|
32
|
|
512
|
1024
|
0
|
1
|
2
|
3
|
4
|
5
|
|
9
|
10
|
524288
|
19
|
536870912
|
29
|
288230376151711744
|
58
|
576460752303423488
|
59
|
Wherefore
the
summe
of
all
the
graines
shall
be
,
1152921504606846975
The
which
number
is
so
great
,
that
it
soemeth
sufficient
to
match
the
sand
of
the
Adriatique
sea
.
FINIS
.
A
BRIEFE
TREATISE
ADDED
CONCERNING
THE
RESOlution
of
the
Square
and
Cube
in
numbers
,
whereby
the
side
of
them
may
be
found
.
A
Figured
number
,
is
a
number
made
by
the
multiplying
of
one
number
by
another
.
The
sides
of
the
figured
number
,
are
the
numbers
by
whose
multiplication
it
was
made
.
Albeit
a
number
be
not
a
magnitude
admitting
figure
and
angles
,
yet
seeing
that
the
multiplying
of
numbers
is
like
vnto
the
making
of
right
lined
plane
figures
,
therefore
for
that
resemblance
of
a
Geometricall
thing
,
the
names
of
figures
are
attributed
vnto
numbers
produced
by
multiplication
.
For
euen
as
a
right
line
drawen
first
a
long
one
right
line
comprehendeth
a
plane
figure
,
and
then
a
long
another
,
which
is
the
depth
,
comprehendeth
a
bodie
:
euen
so
the
multiplying
of
one
number
by
another
,
maketh
a
plane
number
(
so
called
because
the
vnities
whereof
i●
consisteth
,
may
be
contriued
into
a
plane
figure
)
and
that
being
multiplyed
by
a
third
number
maketh
a
solide
number
,
receiuing
that
title
because
it
may
be
fashioned
like
a
solide
figure
.
The
numbers
multiplyed
one
by
another
represent
their
sides
.
A
figured
number
is
twofold
,
a
plane
and
a
solide
number
.
A
plane
number
is
that
,
which
is
made
by
multiplying
two
numbers
one
by
another
.
16
d.
7.
Multiplication
(
as
is
aforesaide
)
maketh
a
certaine
thing
like
to
a
right
angled
parallelogramme
.
Wherefore
here
are
excluded
all
other
numbers
,
which
others
call
trigonals
,
pentagonals
,
&c.
made
by
addition
,
whereof
Euclids
Arithmeticke
maketh
no
mention
.
A
plane
number
is
either
made
of
two
equall
numbers
as
a
Square
:
or
of
two
vnequall
numbers
,
as
an
Oblong
.
18
d.
7.
A
square
number
is
made
by
a
number
multiplied
by
it selfe
,
as
foure
times
4
are
16
,
fiue
times
5
are
25
,
ten
times
10
are
100
:
for
these
products
may
by
their
seuerall
vnities
be
so
displayed
in
a
plane
surface
,
that
they
may
represent
plane
squares
,
as
here
you
see
in
the
number
16
:
1
|
1
|
1
|
1
|
16
is
the
platforme
,
4
is
the
side
.
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
wherefore
by
the
making
of
this
Square
,
it
is
euident
that
a
square
may
be
made
of
any
number
giuen
,
and
that
any
number
may
be
the
side
of
a
square
.
But
as
it
is
requisite
that
the
sides
(
being
lesser
then
10
)
of
the
squares
vnder
an
hundred
should
be
gathered
by
the
Table
of
multiplication
:
so
the
sides
of
greater
Squares
are
to
be
sought
out
by
art
.
The
squares
whose
sides
are
simple
numbers
,
are
here
set
downe
as
you
see
:
The
sides
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
The
squares
|
1
|
4
|
9
|
16
|
25
|
36
|
49
|
64
|
81
|
The
Square
is
knowne
by
finding
out
his
side
expressed
by
a
whole
number
.
Albeit
the
finding
out
of
the
side
of
a
square
,
be
applied
to
each
number
giuen
as
to
a
square
,
yet
the
square
numbers
onely
haue
a
side
to
be
expressed
by
a
certaine
number
of
vnities
or
by
rationall
numbers
,
the
other
are
to
be
expressed
but
in
power
only
.
The
sides
are
commonly
called
rootes
by
a
metaphoricall
phrase
.
The
side
of
a
square
is
to
be
found
by
the
Theoreme
following
.
If
the
odde
degrees
of
a
square
number
being
marked
from
the
right
toward
the
left
hande
with
points
,
you
subduct
from
the
number
giuen
,
the
particular
square
of
the
last
period
,
setting
the
side
therof
alon̄e
by
it selfe
,
2
then
going
on
if
you
diuide
the
remainder
(
if
ther
be
any
)
with
the
figure
going
before
it
,
by
the
double
of
the
side
set
alone
by
it selfe
,
3
and
multiply
the
quotient
found
out
(
being
placed
by
the
side
which
was
first
set
alone
by
it
selfe
,
and
also
before
the
doubled
number
on
the
right
hand
)
by
both
the
numbers
(
namely
by
the
doubled
number
,
and
the
figure
set
by
it
)
being
counted
as
one
diuisor
,
subducting
the
products
from
the
number
giuen
,
and
then
renew
this
last
worke
of
diuision
so
many
times
as
there
are
prickes
remaining
,
the
side
of
the
square
shal
be
found
out
.
This
artificall
deuise
is
borowed
out
of
the
4
p.
2.
Where
by
demonstration
it
is
proued
,
that
if
a
right
line
be
cut
into
two
segments
,
how
soeuer
,
the
square
of
the
whole
line
is
equall
to
the
squares
of
the
segments
,
and
to
the
two
right
angled
figures
made
of
the
segments
:
as
in
the
figure
here
annexed
,
the
two
Diagonals
k
g
,
and
b
f
,
are
the
squares
of
the
segments
a
b
,
and
a
c.
Also
the
complements
b
k
and
f
g
,
are
the
right-angled
figures
made
by
multiplying
the
line
a
b
by
b
c.
The
selfe
same
parts
are
to
be
found
in
any
square
number
.
As
for
example
in
the
number
169
,
whose
side
is
13.
This
side
being
diuided
into
two
peeces
10
and
3
,
multiply
ech
peece
by
it selfe
once
,
namely
10
by
10
,
and
3
by
3
,
then
multiply
one
by
another
,
as
10
by
3
,
&
3
by
10
,
so
shall
you
haue
4
plane
numbers
,
whereof
two
are
Squares
as
here
you
see
.
10
|
3
|
10
|
3
|
|
100
|
|
30
|
|
30
|
|
9
|
|
169
|
Therefore
as
the
Square
169
is
made
by
the
adding
together
of
these
foure
plane
numbes
,
so
by
subducting
them
seuerally
it
is
resolued
.
First
therefore
I
marke
each
odde
place
with
points
because
the
particular
squares
are
to
be
found
in
the
odde
places
.
Then
for
so
much
as
the
vnitie
standing
vnder
the
first
point
next
vnto
the
left
hand
,
and
representing
the
last
period
,
is
both
a
square
and
the
side
of
a
square
:
that
figure
therefore
being
set
alone
by
it selfe
in
the
quotient
,
and
being
subducted
from
the
vnity
standing
ouer
the
point
,
there
remaineth
nothing
.
This
vnitie
set
alone
by
it selfe
in
the
quotient
shall
signifie
10
,
when
another
figure
is
set
by
it
representing
the
side
of
some
other
particular
square
.
Whereupon
I
say
,
that
the
greater
Diagonall
kg
,
is
nowe
subducted
from
the
whole
square
,
and
the
side
of
it
k
i
or
a
b
(
for
they
are
equall
one
to
another
)
and
also
the
side
of
one
of
the
complements
is
found
out
.
This
is
the
first
steppe
to
this
resolution
.
Moreouer
I
double
the
figure
found
out
,
because
being
doubled
,
it
is
the
side
of
both
the
complements
taken
ioyntly
together
,
namely
k
i
and
g
i.
Then
setting
2
the
doubled
number
vnder
6
,
I
diuide
6
,
(
which
in
this
place
is
as
much
as
60
,
and
representeth
both
the
complements
)
by
2
,
the
quotient
10
3
,
representing
the
other
side
remaining
of
the
complement
,
namely
,
i
f
,
or
b
c
,
the
which
number
I
set
in
the
quotient
,
and
count
it
for
the
segment
remaining
of
the
right
line
giuen
.
Wherefore
because
this
number
3
is
the
side
of
the
Diagonall
remayning
,
that
is
to
say
,
of
the
lesser
square
b
f
:
therefore
being
set
by
the
diuisor
on
the
right
hand
,
and
multiplyed
both
by
it selfe
,
and
by
the
diuisor
,
it
bringeth
forth
three
plane
numbers
,
namely
,
the
square
b
f
,
and
the
two
complements
a
i
and
i
l
,
which
being
subducted
from
the
numbers
standing
ouer
them
,
there
remaineth
nothing
.
The
example
is
thus
:
which
is
all
one
as
if
you
had
set
down
the
numbers
found
out
thus
:
1̇69̇
|
|
100
|
The
greater
Diagonall
.
|
60
|
The
two
Complements
.
|
9
|
The
lesser
Diagonall
.
|
169
|
|
Take
this
for
an
other
example
to
make
this
deuise
more
plaine
.
Let
the
Square
giuen
be
1764.
This
number
being
marked
with
two
points
telleth
vs
that
the
side
thereof
is
to
be
written
with
two
figures
.
First
therfore
beginning
at
the
point
on
the
left
hande
,
I
seeke
the
side
of
the
last
periode
,
namely
of
17.
But
for
so
much
as
it
is
no
square
number
,
I
take
4
the
side
of
the
next
lesser
square
,
which
I
set
alone
by
it selfe
in
the
quotient
,
and
then
multiply
it
by
it selfe
,
the
product
is
16
,
which
being
subducted
from
17
there
remaineth
1.
Moreouer
,
I
double
the
side
found
out
,
the
product
is
8
,
I
place
this
doubled
number
vnder
6
,
and
by
it
I
diuide
16
standing
aboue
it
,
the
quotient
is
2
,
which
must
be
set
by
4.
This
quotient
2
must
be
set
before
the
diuisor
8
on
the
right
hand
vnder
the
point
,
and
then
must
it
be
multiplied
both
into
it selfe
and
into
8
,
the
product
is
164
,
which
being
subducted
from
the
figures
standing
ouer
them
,
there
remaineth
nothing
,
whereby
I
gather
that
the
number
giuen
is
a
iust
Square
.
The
example
standeth
thus
:
The
same
manner
of
working
is
to
be
followed
in
greater
square
number
giuen
,
sauing
that
the
former
part
of
the
worke
is
to
be
vsed
but
once
,
but
the
latter
part
is
to
be
followed
so
many
times
as
there
are
pointes
remaining
,
excepting
the
last
.
As
in
54756
,
I
say
,
that
the
side
of
the
square
next
vnto
5
is
2
,
therefore
2
being
set
in
the
quotient
,
and
multiplied
by
it selfe
make
4
,
and
taken
from
5
,
the
remainder
is
one
.
Moreouer
I
double
the
quotient
,
the
product
is
4
,
which
I
set
vnder
the
next
figure
toward
the
right
hand
,
and
thereby
diuide
14
,
the
quotient
is
3
,
the
which
3
being
set
both
in
the
quotient
,
and
also
before
the
diuisor
toward
the
right
hand
,
I
multiply
both
the
numbers
by
it
,
the
product
is
129
,
this
being
subducted
from
147
standing
aboue
it
,
the
remainder
is
18.
But
because
there
is
yet
one
point
remaining
with
the
which
I
haue
not
medled
,
therfore
againe
I
double
all
the
whole
quotient
.
For
in
this
case
I
must
take
23
for
the
one
side
of
one
former
square
,
and
generally
in
great
numbers
when
I
light
vppon
more
particular
squares
then
two
,
I
must
esteeme
them
but
as
two
,
and
take
the
sides
which
are
first
found
out
but
as
the
sides
of
one
only
square
.
Therefore
twise
23
are
46
,
by
this
I
diuide
185
,
the
number
to
be
set
in
the
quotient
is
4
,
the
which
number
also
must
bee
set
before
the
diuisor
on
the
right
hande
,
then
must
464
be
multiplied
by
4
,
the
product
is
1856
,
this
product
beeing
subducted
from
the
numbers
standing
ouer
it
,
there
remaineth
nothing
.
The
example
standeth
thus
:
See
also
the
example
following
:
10̇94̇28̇64̇
(
3308
Out
of
this
inuention
this
conclusion
followeth
:
The
number
whose
side
cannot
be
expressed
by
whole
numbers
,
is
no
square
number
.
Such
are
all
prime
numbers
,
&
(
the
squares
themselues
excepted
)
all
other
compound
numbers
.
For
if
in
them
you
desire
to
finde
out
the
square
side
,
you
shall
labour
in
vaine
,
because
they
are
not
squares
for
to
the
whole
numbers
arising
in
the
quotient
,
there
will
be
some
fraction
adioyned
,
whereby
it
commeth
to
passe
,
that
the
number
of
the
side
is
not
to
be
expressed
by
a
true
number
,
and
commonly
it
is
called
a
surde
number
.
Yet
if
you
adioyne
to
the
side
found
out
the
number
remaining
,
taking
his
denomination
from
the
double
of
the
side
augmented
by
an
vnitie
,
you
shall
finde
the
next
side
that
may
be
like
to
the
side
of
a
square
.
As
if
from
40
you
take
the
nearest
square
,
namely
36
,
the
remainder
is
4.
Here
therefore
the
side
sought
for
of
the
square
,
exceedeth
not
the
side
found
out
by
an
vnitie
,
but
either
by
one
,
or
more
parts
of
some
whole
number
:
wherefore
I
double
6
the
side
found
out
,
and
adde
an
vnitie
to
it
being
doubled
,
the
totall
is
13
,
this
number
I
set
vnder
4
the
remainder
,
and
say
that
the
side
of
40
sought
as
neare
as
may
be
,
is
6
4/13
,
the
denominator
of
the
fraction
being
added
to
the
greatest
square
in
the
number
giuen
,
namely
vnto
36
,
maketh
the
next
greatest
square
aboue
it
,
namely
49
,
whose
side
is
7.
But
this
surde
side
namely
6
4/13
,
multiplyed
by
it selfe
maketh
39
133/169
,
which
are
not
iust
equall
vnto
40
the
number
giuen
.
The
like
iudgement
may
be
vsed
in
the
rest
which
are
not
squares
.
Thus
much
concerning
plane
numbers
,
but
especially
such
as
are
square
numbers
.
The
solide
number
followeth
,
which
is
a
number
made
of
three
numbers
.
17
d.
7.
A
solide
number
is
made
by
a
twofold
multiplication
,
euen
as
a
solide
bodie
among
magnitudes
is
made
by
a
twofold
comprehension
:
the
first
is
like
that
which
maketh
a
plane
figure
,
the
second
is
like
that
which
maketh
a
body
.
As
three
times
4
taken
fiue
times
:
for
three
times
4
are
12
,
this
is
a
plane
number
,
fiue
times
12
are
60
,
this
is
a
solide
number
.
A
solide
number
is
made
either
of
3
equall
numbers
as
a
Cube
:
or
of
3
vnequall
numbers
as
an
Oblong
.
19
d.
7.
A
cube
in
numbers
is
made
by
multiplying
the
side
thereof
by
the
square
of
the
side
:
whereby
we
may
gather
,
that
to
multiply
a
number
cubically
,
is
to
multiply
the
number
made
of
it selfe
,
by
himselfe
againe
as
foure
times
4
foure
times
are
64
,
which
is
a
cube
hauing
relation
to
a
Geometrical
thing
,
partly
because
their
making
is
like
,
partly
also
because
the
vnities
of
the
number
set
in
a
certaine
distance
one
from
another
resemble
a
cube
.
The
cube
is
knowne
by
finding
out
the
side
expressed
by
a
whole
number
.
For
euery
Cube
in
numbers
hath
such
a
side
as
may
be
expressed
by
whole
numbers
,
but
in
magnitudes
it
is
not
alwayes
so
,
as
in
deede
in
magnitudes
there
are
many
things
not
to
be
expressed
by
whole
numbers
.
Now
for
so
much
as
the
side
of
any
Cube
vnder
1000
is
a
simple
figure
,
it
is
meete
before
we
vndertake
to
find
out
the
side
of
any
greater
number
,
that
we
should
know
what
Cube
is
made
of
each
simple
figure
,
and
what
is
the
side
of
any
Cube
lesser
then
1000
,
as
I
haue
here
set
them
downe
.
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
1
|
4
|
9
|
16
|
25
|
36
|
49
|
64
|
81
|
1
|
8
|
27
|
64
|
125
|
216
|
343
|
512
|
729
|
But
in
searching
out
of
greater
Cubes
,
we
must
proceede
as
the
Theoreme
following
teacheth
vs
.
If
you
distinguish
with
pointes
as
it
were
into
periods
,
the
Cube
giuen
beginning
at
the
first
figure
on
the
right
hand
,
and
omitting
ech
two
figures
continually
,
and
first
of
all
subduct
the
particular
Cube
of
the
last
period
from
the
number
giuen
,
setting
the
side
therof
in
the
quotient
:
and
then
set
the
triple
of
the
quotient
vnder
the
figure
next
following
the
former
point
on
the
right
hande
,
and
the
square
of
the
quotient
being
tripled
beneath
it
one
degree
more
towarde
the
left
hand
:
and
afterward
diuide
the
number
aboue
written
by
the
triple
of
the
square
,
setting
the
quotient
by
it self
,
and
then
multiply
the
diuisor
by
the
quotient
found
out
,
and
the
tripled
square
by
the
square
of
the
quotient
,
and
the
quotient
cubically
,
subducting
the
products
(
so
orderly
added
together
,
that
ech
figure
may
answere
the
numbers
whereof
it
was
multiplied
)
from
the
number
giuen
:
and
renew
this
last
manner
of
diuision
so
many
times
as
there
are
points
remaining
,
the
side
of
the
Cube
shal
be
found
out
.
The
cube
of
the
greater
segment
10.
The
cube
of
the
lesser
segment
3
,
The
greater
Parallelipipedons
.
The
lesser
parallelipipedons
.
The
cube
therefore
hath
eight
particular
solides
in
numbers
which
are
made
of
the
partes
of
the
number
giuen
,
namely
of
10
and
3
in
this
manner
.
First
let
there
be
foure
plane
numbers
made
,
each
part
being
multiplied
by
it selfe
,
and
one
by
another
:
10
|
3
|
10
|
3
|
|
100
|
|
30
|
|
30
|
|
9
|
|
169
|
If
againe
you
multiply
these
planes
by
the
same
partes
,
there
will
arise
8
solides
,
as
you
see
here
:
9
|
9
|
|
30
|
30
|
|
30
|
30
|
|
100
|
100
|
|
3
|
10
|
|
27
|
90
|
All
these
being
added
together
are
equall
vnto
2197
,
the
cube
of
the
whole
line
.
|
90
|
300
|
90
|
300
|
300
|
1000
|
The
same
way
therefore
that
is
kept
in
making
the
Cube
,
is
also
to
be
followed
in
resoluing
the
Cube
.
As
for
example
,
I
marke
the
cube
giuen
with
points
in
this
manner
,
2197.
Then
I
subduct
the
particular
cube
of
the
number
set
vnder
neath
the
last
point
:
but
for
so
much
as
that
number
is
no
Cube
,
I
take
the
nearest
to
it
,
namely
an
vnitie
,
which
also
I
set
in
the
quotient
.
This
vnitie
in
the
number
giuen
is
1000
,
but
in
the
quotient
it
is
but
10.
The
vnitie
subducted
from
2
,
the
remainder
is
1
,
which
must
be
written
ouer
the
number
giuen
.
So
that
the
greater
cube
A
is
to
be
supposed
to
be
subducted
from
the
number
giuen
.
This
is
the
first
steppe
of
this
worke
:
I
triple
the
quotient
found
out
(
that
is
to
say
,
I
multiply
it
by
3
)
this
triple
representeth
the
three
sides
taken
iointly
together
of
the
three
lesser
solides
marked
with
C
,
I
place
the
tripled
number
vnder
9.
Againe
I
multiply
the
quotient
squarewise
,
and
triple
the
product
which
maketh
likewise
3.
This
product
resembleth
the
three
squared
sides
taken
iointly
together
of
the
three
greater
solides
,
marked
with
D
,
I
place
the
product
one
degree
lower
toward
the
left
hande
vnderneath
1.
With
it
I
diuide
11
,
which
is
written
aboue
it
,
the
quotient
is
3.
This
segment
or
quotient
3
,
being
multiplied
by
3
the
diuisor
,
maketh
9
,
which
in
respect
of
the
place
wherein
it
standeth
is
900
,
and
representeth
the
three
greater
solides
marked
with
D
taken
iointly
together
.
Furthermore
the
same
quotient
being
multiplyed
squarewise
maketh
9
,
and
multiplied
afterwarde
by
the
tripled
number
standing
vnder
9
,
it
maketh
27
,
which
in
respect
of
the
place
wherein
it
standeth
is
270
,
and
representeth
the
three
lesser
solides
marked
with
C.
Last
of
all
,
the
same
quotient
multiplyed
cubically
,
breedeth
the
lesser
Cube
B.
These
three
products
therefore
being
added
together
,
and
the
totall
subducted
from
the
numbers
standing
ouer
it
,
there
remaineth
nothing
,
which
importeth
that
the
number
giuen
is
a
Cube
.
The
example
is
as
you
see
:
Take
these
examples
following
to
make
the
matter
more
plane
.
Let
the
side
be
sought
out
of
this
Cube
,
16387064
:
contriue
it
therfore
(
as
it
were
)
into
certaine
periods
with
points
.
Then
first
of
all
,
seeke
out
the
side
of
the
cube
,
which
you
find
in
the
last
number
ouer
the
point
next
to
the
left
hand
.
But
for
so
much
as
16
is
no
cube
,
take
2
the
side
of
the
next
cube
vnder
it
that
is
to
say
,
of
8
,
and
set
it
in
the
quotient
,
and
subduct
8
the
cube
from
16
,
there
remaine
8.
This
first
worke
is
not
to
be
renewed
throughout
the
whole
number
,
but
the
rules
following
must
be
repeated
as
often
as
there
are
pointes
remaining
.
The
first
steppe
to
finde
out
the
side
is
in
this
manner
.
Moreouer
triple
the
quotient
found
out
,
the
product
is
6
,
which
are
to
be
placed
vnder
8
,
namely
vnder
the
figure
following
the
next
pricke
toward
the
right
hande
.
Then
multiplie
the
quotient
by
this
tripled
number
(
or
which
is
all
to
one
purpose
,
square
the
quotient
,
and
then
triple
the
products
)
it
maketh
12
,
set
that
number
in
a
lower
place
one
degree
nearer
to
the
left
hand
:
and
make
it
the
diuisor
.
Diuide
83
by
12
,
obseruing
this
rule
in
chosing
your
quotient
,
that
it
be
no
greater
,
then
that
the
numbers
afterwarde
produced
by
multiplication
may
not
exceede
the
numbers
standing
ouer
it
.
So
that
here
you
shal
take
1
in
8
,
but
fiue
times
.
Afterward
by
this
number
5
,
multiply
the
diuisor
12
,
and
by
the
square
of
5
multiply
the
trebled
number
6
,
and
last
of
all
,
multiply
5
cubically
,
so
shall
you
produce
three
numbers
,
namely
60
,
150
,
125
,
to
be
described
as
you
see
:
These
numbers
added
together
,
and
subducted
from
8387
,
the
remainder
is
762.
The
seconde
steppe
to
finde
out
the
side
is
in
this
manner
:
And
because
there
is
yet
one
point
remaining
,
this
last
manner
of
diuision
must
be
wrought
againe
.
First
therefore
I
triple
the
quotient
,
the
product
is
75
,
which
must
be
so
placed
,
that
the
first
figure
thereof
,
namely
5
,
may
stand
vnder
6
,
the
second
vnder
the
0.
Againe
multiply
the
quotient
by
this
tripled
number
(
or
which
is
all
one
,
square
the
quotient
,
and
triple
the
product
)
it
maketh
1875
,
which
shall
be
the
diuisor
,
whose
first
figure
namely
5
,
must
be
placed
vnder
7
,
the
last
figure
of
the
tripled
number
.
Then
you
see
that
I
may
be
contayned
in
7
many
times
,
but
I
can
take
it
but
foure
times
,
I
set
4
in
the
quotient
,
and
multiply
the
diuisor
by
4
,
the
product
is
7500
,
afterward
I
square
4
,
it
maketh
16
,
which
I
multiply
by
the
tripled
number
75
,
the
product
is
1200.
Last
of
all
,
I
multiply
4
cubically
it
maketh
64
:
these
products
added
all
together
make
762064
,
this
number
being
subducted
from
the
Cube
,
there
remaineth
nothing
,
whereby
I
gather
that
the
number
giuen
is
a
iust
Cube
.
The
third
steppe
to
finde
out
the
side
is
in
this
manner
:
Behold
also
the
example
following
:
614̇125̇000̇
(
850
Another
manner
of
working
:
Hitherto
the
princely
high
way
to
finde
out
the
side
of
the
Cube
hath
beene
declared
.
But
there
are
other
waies
also
bending
thereto
,
and
leaning
to
the
same
principles
:
whereof
this
is
one
.
Hauing
found
out
in
the
Table
of
the
simple
cubes
,
the
figure
representing
the
side
of
the
cube
contained
in
the
number
standing
vnder
the
first
point
on
the
left
hand
,
set
it
in
the
quotient
,
and
subduct
the
particular
cube
of
that
figure
as
you
did
before
:
then
square
,
that
figure
,
and
triple
the
square
,
the
product
shall
be
the
diuisor
,
whose
first
figure
shall
be
set
vnder
that
figure
which
is
on
the
right
hand
next
of
all
to
the
first
point
aforesaide
.
2
See
how
many
times
the
diuisor
is
contained
in
the
number
written
ouer
it
,
multiply
the
diuisor
by
the
quotient
,
and
subduct
the
product
from
the
diuidend
.
Yet
here
you
must
take
heede
,
that
you
choose
not
a
greater
quotient
,
then
that
the
products
made
afterward
thereby
may
be
subducted
from
the
numbers
giuen
.
3
When
the
subduction
is
done
,
triple
the
first
figure
which
was
set
in
the
quotient
,
and
place
the
last
number
which
was
set
in
the
quotient
on
the
right
hand
of
the
product
.
4
Multiply
this
whole
number
by
the
square
of
the
quotient
last
found
out
,
and
set
downe
the
product
so
,
that
the
first
figure
thereof
toward
the
right
hand
may
stand
vnder
the
point
next
going
before
on
the
same
hand
,
and
then
subduct
it
from
the
number
giuen
.
As
in
804̇357̇
the
particular
Cube
,
namely
,
729
,
being
taken
from
the
number
standing
vnder
the
last
period
vpon
the
left
hand
there
remaine
75357
,
the
side
of
that
particular
cube
is
9
,
which
I
set
in
the
quotient
.
I
square
that
side
,
it
maketh
81
,
and
triple
the
square
,
the
product
243
is
my
diuisor
,
which
I
set
vnder
the
number
giuen
,
so
that
3
may
stand
vnder
3.
With
this
diuisor
diuide
the
number
standing
ouer
it
,
you
shall
finde
2
to
be
contained
in
7
three
times
.
I
set
therefore
3
in
the
quotient
,
and
multiply
the
diuisor
by
it
,
the
product
is
729
,
which
being
subducted
from
753
,
the
remainder
is
24.
As
you
see
in
this
induction
.
Moreouer
I
I
triple
9
,
the
product
is
27
,
by
the
which
on
the
right
hand
I
set
3
the
quotient
last
found
out
,
the
totall
is
273.
This
number
I
multiply
by
9
the
square
of
3
the
quotient
last
found
out
,
the
product
is
2457
,
which
being
subducted
from
the
number
standing
ouer
it
,
there
remaineth
nothing
.
The
induction
is
on
this
manner
:
The
selfe
same
worke
may
be
dispatched
another
way
somewhat
differing
from
the
former
in
this
maner
.
The
figure
in
the
quotient
being
found
out
by
subducting
the
particular
cube
,
and
also
the
second
figure
in
the
quotient
being
found
by
diuision
,
let
the
whole
quotient
be
tripled
,
and
let
the
tripled
number
be
multiplied
by
the
former
figure
in
the
quotient
.
Then
let
the
product
be
multiplied
againe
by
the
latter
figure
found
out
,
and
let
a
cypher
be
set
on
the
right
hande
of
that
product
.
Last
of
all
,
let
the
cube
of
the
latter
figure
found
out
be
added
to
this
product
,
and
let
the
totall
summe
be
subducted
from
the
number
giuen
.
As
in
373248.
The
first
induction
is
in
this
manner
:
Moreouer
I
square
the
side
found
out
,
it
maketh
49
and
triple
the
square
,
the
product
is
147
,
which
shall
be
the
diuisor
,
by
this
I
diuide
302
,
the
number
written
ouer
it
,
the
quotient
is
2.
Now
I
triple
the
whole
quotient
72
,
it
maketh
216
,
and
multiply
this
triple
by
7
,
the
former
figure
in
the
quotient
,
the
product
is
1512.
I
multiply
also
this
product
by
2
the
latter
figure
of
the
quotient
,
and
set
a
cypher
on
the
right
hande
of
it
,
so
that
it
maketh
30240
,
vnto
this
number
I
adde
8
the
cube
of
the
latter
figure
found
out
,
the
totall
is
30248
,
which
being
subducted
from
the
figures
aboue
it
,
there
remaineth
nothing
.
The
induction
is
thus
:
All
the
points
of
the
number
giuen
being
examined
,
if
any
thing
remaine
,
it
signifieth
that
the
number
giuen
is
no
Cube
:
wherefore
the
true
side
of
it
cannot
be
giuen
exactly
in
numbers
.
Yet
if
it
please
you
to
sift
out
the
nearest
side
that
may
be
,
by
the
first
kinde
of
reduction
of
mixt
numbers
,
you
must
reduce
the
number
giuen
into
a
cubicall
fraction
of
a
great
denomination
,
and
afterwarde
seeke
out
the
cubicall
side
of
that
fraction
.
For
example
sake
,
because
120
is
no
Cube
,
therefore
let
it
be
reduced
into
sixtie
cubicall
parts
,
after
this
manner
.
Multiply
60
cubically
in
it selfe
,
it
maketh
216000
,
by
this
being
taken
for
the
denominator
of
the
fraction
,
multiply
120
the
number
giuen
,
the
product
is
25920000
,
whose
cubicall
side
is
295/60
that
is
,
4
11/12
the
nearest
to
the
true
side
that
may
be
.
FINIS
.