MESOLABIVM
ARCHITECTONICVM
.
CHAP.
1.
Of
the
Mesolabe
:
And
of
the
vse
of
it
in
generall
.
1
To
measure
by
this
Rule
,
is
by
two
knowne
lines
,
to
finde
out
the
third
vnknowne
.
THe
Instrument
whose
vse
at
this
time
wee
intend
to
〈◊〉
,
is
no
other
,
in
respect
of
matter
and
forme
,
in
generall
,
but
the
Carpentars
rule
,
by
them
vsed
in
the
measuring
of
Tymber
,
and
Bourd
by
the
Foot
square
:
For
it
is
a
flat
Ruler
,
or
oblong
parallelogram
,
of
two
foote
,
or
a
foot
&
halfe
long
:
Two
inches
and
an
halfe
,
or
there
abouts
,
broad
:
And
of
such
convenient
thicknesse
as
shall
at
cuery
mans
discresion
be
thought
most
fit
.
Againe
,
as
theirs
,
so
this
on
the
one
side
,
contayneth
a
Scale
of
equall
diuisions
,
First
of
〈◊〉
,
Halfe-ynches
,
Quarters
,
Halfe
quarters
,
and
so
forth
:
Then
againe
,
on
the
〈◊〉
side
,
you
hauean
Ynche
diuided
into
Seauen
,
Eleuen
,
Thirteene
,
〈◊〉
,
〈◊〉
,
and
Three
and
twenty
,
and
such
other
〈◊〉
parts
,
as
〈◊〉
man
for
his
〈◊〉
〈◊〉
shall
think
〈◊〉
〈◊〉
,
and
the
〈◊〉
hand
shall
be
able
to
performe
.
More
ouer
,
on
the
other
side
,
as
on
theirs
also
,
you
haue
a
Scale
of
vnequall
diuisions
,
seruing
for
the
measuring
of
Board
and
Tymber
:
But
after
a
farre
different
manner
For
their
diuisions
are
only
markes
or
small
strokes
,
in
one
of
the
limbs
of
that
side
,
determinyng
from
the
Fore-end
of
the
Rular
in
ynches
,
and
partes
of
ynches
,
the
Square
measure
of
solids
or
Tymber
.
Wheras
this
of
ours
consisteth
of
two
for●es
of
straight
lines
,
the
one
Beuelling
o●
sla●ting
,
drawne
as
●●e
from
side
to
side
:
The
other
parallell
that
is
equidistant
one
from
another
running
along
the
Ru●ar
,
from
the
one
end
toward
the
other
:
And
therefore
cutting
those
former
,
and
diuiding
them
into
vnequall
portions
,
whereby
not
onely
their
sayd
Quadrate
or
square
measure
is
performed
:
But
also
all
other
whatsoeuer
,
and
that
with
great
facillity
,
speede
,
and
certainety
.
Lastly
here
,
as
also
there
,
you
must
make
a
distinction
betweene
end
,
and
end
;
For
that
end
we
call
the
For-end
of
the
Rular
,
from
whence
the
diuisions
of
it
into
ynches
,
on
both
sides
are
begun
to
be
reckoned
:
And
that
the
Backer-end
where
they
doe
end
and
determine
:
Or
,
contrarywise
,
the
For-end
is
that
from
whence
the
numbers
asscribed
to
the
Beuelling
linnes
are
lesse
and
lesse
.
But
the
distances
betweene
them
are
greater
and
greater
.
Thus
much
of
the
Ruler
,
and
the
Partes
therof
.
Mensura
,
innuit
Aristoteles
,
in
quolibet
mensurabili
genere
,
est
quippiam
minimum
:
A
measure
,
as
Aristotle
seemeth
to
intimate
,
is
some
small
portion
in
euery
thing
that
is
to
be
measured
:
And
it
is
commonly
termed
of
the
Geometricians
Famosa
mensura
:
Aknowne
,
or
set
measure
generally
agreed
vpon
amongst
all
men
:
As
in
measuring
by
hand-breadths
,
feete
,
and
passes
,
one
hand
breadth
,
on
foot
,
one
passe
.
And
in
deed
it
is
an
old
saying
of
Protagoras
,
as
Aristotle
recordeth
,
That
man
is
the
measure
of
all
things
.
And
true
it
is
,
That
Vi●●●●us
,
and
Hero
the
mechanicke
or
inginer
,
do
shew
,
That
generally
all
measures
are
taken
from
the
partes
of
Mans
body
,
as
a
Finger
,
an
Ynch
(
Pollex
)
an
Hand
,
or
Hands
breadth
,
a
Spanne
,
a
Foot
,
a
Cubite
,
a
Passe
,
an
Elne
,
a
〈◊〉
.
But
who
knoweth
not
,
What
great
difference
there
is
between
man
&
man
?
And
not
only
between
men
of
diuerse
Countreys
and
climats
:
But
eu'n
between
those
of
one
and
the
same
prouince
;
Nay
of
one
and
the
same
family
,
children
of
the
same
parents
?
And
,
the
limmes
of
men
being
proportionall
to
their
bodys
,
what
〈◊〉
must
there
needs
bee
,
betweene
the
measures
taken
from
them
?
And
in
deed
heerupon
it
came
to
passe
,
That
the
Measures
,
not
only
of
diuerse
Nations
:
But
eu'n
of
one
and
the
same
,
are
,
and
alwayes
haue
beene
much
different
,
as
doth
manifestly
appeare
by
the
diligent
comparisons
made
of
them
by
diuerse
and
sundry
learned
men
,
and
especially
by
that
hopefull
Willebrordus
Snellius
,
as
wee
〈◊〉
,
Godwilling
,
shortly
teach
in
Ramus's
Geometry
,
which
wee
purpose
to
set
out
in
English
,
for
the
benesite
of
such
of
our
Countrey
men
,
as
delight
in
these
study's
,
yet
are
ignorant
of
those
languages
where
in
they
are
written
.
This
difference
was
in
this
our
kingdome
complained
of
in
all
ages
:
For
from
hence
arose
many
greeuous
〈◊〉
and
suites
in
the
Law
,
which
our
worthy
Kings
,
and
state
in
their
Parlaments
,
in
all
ages
haue
laboured
to
appease
,
by
reducing
all
to
an
vniformity
:
For
thus
wee
finde
in
our
Statutes
:
It
is
ordeined
,
That
3
grains
of
Barley
,
dry
and
round
,
do
make
an
Ynche
:
Twelue
ynohes
do
make
a
Foot
:
Three
foote
do
make
a
Yard
:
Fiue
yards
and
halfe
do
make
a
〈◊〉
.
And
40
〈◊〉
in
length
,
and
4
in
Breadth
do
make
an
Aker
.
33
of
Edward
the
first
,
De
〈◊〉
mensurandis
Item
,
De
Compositione
〈◊〉
et
Perticarum
.
Againe
in
a
〈◊〉
held
in
the
25
〈◊〉
of
Queene
Elizabeth
,
you
haue
an
Act
,
thus
intituled
:
An
Act
for
the
restrainte
of
New-buildings
,
&c.
in
&
nere
the
citys
of
London
&
Westminster
Be
it
enacted
by
the
authority
afor'said
,
That
a
Mile
shall
be
taken
&
reckoned
in
this
manner
,
&
no
otherwise
:
That
is
to
say
,
a
Mile
to
containe
8
Furlongs
.
And
euery
Furlong
to
containe
40
lugges
or
poales
:
And
euery
Lugge
or
Poale
,
to
containe
〈◊〉
foot
and
an
halfe
.
Although
this
same
our
Rule
may
bee
fitted
for
sundry
other
sortes
of
measures
:
Yet
we
haue
here
nothing
to
do
,
But
with
the
Foote
,
and
his
partes
,
which
are
Ynches
,
Halfe-ynches
,
Quarters
,
Half-quarters
,
and
such
other
sensible
partes
of
the
same
.
2
Things
to
bee
measured
by
this
Rule
,
are
magnitudes
.
3
A
magnitude
is
a
continuall
quantity
.
Amagnitude
,
or
a
bignesse
is
that
which
hath
one
,
or
more
dimensions
:
Now
dimensions
are
in
number
three
,
to
weet
Length
,
Breadth
,
and
Thicknesse
.
4
A
magnitude
is
of
one
〈◊〉
,
or
many
.
5
The
measure
is
of
the
same
nature
with
the
thing
to
be
measured
.
6
A
magnitude
of
one
dimension
is
called
a
Line
.
Aline
,
is
a
magnitude
of
length
onely
.
Or
,
Aline
is
a
magnitude
onely
long
.
Such
are
wayes
,
or
distances
〈◊〉
place
and
place
.
Such
a
magnitude
,
sayth
Proclus
out
of
Apollonius
,
is
conceiued
in
the
measuring
of
iourneys
.
And
by
the
difference
of
a
lightsome
place
,
from
a
darksome
.
Such
are
Lenghts
,
Heighths
,
Depths
,
and
Breadths
.
Therfore
here
7
The
measure
vsed
is
a
line
.
Here
therefore
there
is
no
further
skill
required
in
the
measurer
then
a
due
application
of
the
measure
giuen
:
And
therefore
here
in
this
case
there
is
not
any
vse
of
this
our
Instrument
.
CHAP.
II.
Of
the
measuring
of
Plaines
by
the
foot
square
.
1
A
magnitude
of
many
dimensions
,
is
of
two
or
three
:
That
is
called
a
Surface
:
This
a
Solid
.
2
If
a
dimension
giuen
,
be
eyther
greater
,
or
lesser
,
then
any
of
the
numbers
vpon
the
Rular
,
you
must
take
some
lesser
,
or
greater
,
which
is
proportionall
vnto
it
.
3
A
surface
is
a
magnitude
long
and
broad
.
That
is
,
a
surface
is
a
magnitude
which
hath
two
dimensions
,
to
weet
Length
and
Breadth
.
Such
magnitudes
,
sayth
Apollonius
,
are
the
shadowes
vpon
the
ground
,
which
ouerspread
the
fields
farre
and
wide
,
but
do
not
enter
into
,
or
pierce
the
earth
:
Neither
haue
they
any
thicknes
at
all
.
The
Greek
woord
Epiphania
,
is
here
more
significant
.
For
this
worde
intimateth
no
more
but
,
The
outward
appearance
of
any
thing
.
For
of
a
magnitude
nothing
is
to
be
seene
but
the
surface
.
Such
are
bourds
esteemed
to
be
by
the
Carpentars
:
Wainscotte
,
by
the
Ioyners
:
Glasse
,
by
the
Glasiers
:
Cloth
,
both
linnen
&
Woollen
,
by
the
Drapers
:
Land
,
Medowe
,
&
Wood
,
by
the
Surueighers
:
For
in
the
measuring
of
these
,
there
is
only
Breadth
&
Length
considered
,
with
out
any
respect
at
all
had
to
the
Thicknesse
.
Therfore
2
Here
the
measure
is
a
Surface
.
Surfaces
,
according
to
their
diuerse
natures
,
are
measured
with
diuerse
and
sundry
kindes
of
measures
:
Wood
,
Land
,
&
Medowe
,
are
measured
by
the
Rod
or
Perch
:
Cloth
,
Painting
,
Pauing
,
&
Wainscotte
,
by
the
Yard
:
Bourd
and
Stone
,
by
the
Foote
.
Although
this
our
Instrument
may
be
fitted
to
all
these
,
or
any
other
like
measure
,
Yet
wee
at
this
time
intend
to
meddle
with
no
other
but
the
last
,
to
weet
With
the
Footesquare
.
4
A
surface
is
either
Plaine
or
Vneu'n
.
5
A
Plaine
surface
is
a
surface
,
which
lyeth
equally
between
his
bounds
.
A
surface
,
the
learned
knowe
is
geometrically
made
of
Lines
:
Therfore
as
lines
are
either
straight
or
Crooked
:
So
from
hence
are
all
surfaces
Straight
or
Crooked
:
Or
,
to
speak
more
properly
,
Eu'n
or
vneu'n
,
Plaine
or
Rugged
:
Yea
&
by
a
straight
line
are
surfaces
tried
,
whether
they
be
Eu'n
,
or
vneu'n
.
For
if
a
rightline
applyed
to
a
surface
euery
way
,
do
touch
it
in
all
places
,
it
is
Eu'n
:
Otherwise
,
it
is
vneu'n
.
9
Plaines
,
as
wee
sayd
,
are
measured
by
the
Foote
square
,
That
is
the
quadrate
of
12
ynches
.
A
〈◊〉
of
plaine
or
flatte
measure
is
the
quadrate
of
12
ynches
,
or
that
which
is
equall
vnto
it
.
That
is
,
it
〈◊〉
:
44
〈◊〉
Ynches
:
For
12
times
12
,
are
144.
Hauing
therefore
a
plaine
giuen
of
12
ynches
broad
,
there
is
no
question
but
12
ynches
of
that
breadth
shall
make
a
Foote
.
But
if
the
breadth
giuen
be
greater
or
lesse
then
12
,
there
is
a
question
,
What
length
,
with
the
breadth
giuen
,
shall
make
a
plaine
〈◊〉
to
the
square
144.
Here
7
Of
the
two
lines
'
giuen
,
the
one
is
the
breadth
assigned
,
the
other
is
alwayes
the
beuelling
line
12.
Here
againe
it
must
bee
remembred
,
That
onely
those
plaines
are
to
be
measured
which
are
Rightangled
parallelogramms
,
Or
to
speake
in
their
owne
Language
,
which
are
comprehended
of
a
,
Base
,
and
Heigh
which
are
rationall
betweene
〈◊〉
:
Ramus
9
e
IIII.
Those
plains
therfore
which
are
not
such
,
must
bee
reduced
vnto
these
kinde
of
figures
.
1
An
example
or
two
shall
make
all
plaine
.
A
bourd
of
16
ynches
broad
and
18
ynches
long
,
(
And
so
a
stocke
of
13
bourds
)
is
to
be
measured
.
Here
I
finde
16
,
the
line
answering
to
the
Bredth
,
to
〈◊〉
the
beueller
12
,
at
9
ynches
from
the
fore-end
of
the
Rular
.
Therefore
I
say
euery
9
ynches
of
that
length
shall
make
a
Foot
of
bourd
:
Or
which
is
all
one
,
shall
be
〈◊〉
to
144
,
the
square
of
12
ynches
.
Now
9
ynches
I
finde
to
bee
contained
in
18
foote
,
the
Length
,
24
times
:
Therefore
I
say
,
The
bourd
assigned
doth
containe
14
foote
of
bourd
.
Lastly
,
there
being
in
the
stocke
13
such
bourds
,
I
say
the
whole
stocke
doth
contayne
312
foot
of
bourd
.
II
A
Table
of
36
ynches
broad
,
and
28
foote
long
,
is
to
be
measured
.
Here
36
is
greater
then
any
of
the
parallels
found
vpon
the
Rular
:
Therefore
by
the
2
e
of
this
,
I
take
●8
the
halfe
of
it
,
which
I
finde
to
meete
with
22
,
the
beuelling
line
,
〈◊〉
8
ynches
from
the
for'end
of
the
Rular
:
Therefore
euery
8
ynches
of
length
,
of
the
bredth
18
,
shall
contayne
a
roote
of
bourd
:
But
the
breadth
giuen
is
36
ynches
:
That
is
twice
18
:
Therefore
euery
8
ynches
in
length
,
of
that
Table
shall
be
2
foote
of
bourd
.
Now
againe
I
finde
8
ynches
,
in
28
foote
42
times
:
Therefore
the
Table
containeth
twice
so
many
foot
:
That
is
84
foote
of
bourd
.
III
A
pane
of
Glasse
,
7
ynches
broad
,
is
to
bee
measured
.
Here
7
is
lesser
then
any
of
the
parallels
:
Therefore
by
the
2
e
of
this
,
I
take
14
,
the
double
thereof
:
Which
I
obserue
to
meete
with
12
,
at
10
ynches
and
2
seauenth
parts
of
an
ynch
from
the
fore-end
:
Therefore
euery
10
ynches
and
2
seauenth
partes
of
an
ynch
,
of
14
ynches
breadth
,
shall
bee
a
foote
of
Glasse
:
But
the
breadth
giuen
is
but
7
ynches
:
Therefore
euery
10
ynches
,
and
2
seauenth
partes
of
an
ynch
shall
be
but
halfe
a
foote
of
glasse
.
Of
the
measuring
of
Triangles
,
and
all
other
Rightlined
plaines
.
8
A
triangle
is
nothing
else
but
the
halfe
of
a
quadrangle
,
or
parallelogramme
:
And
if
it
haue
one
right
angle
,
it
is
the
halfe
of
a
rightangled
parallelgramme
.
Therefore
9
It
is
to
bee
measured
as
the
Rightangled-parallelogramme
,
onely
conceiue
that
the
number
found
,
shall
bee
the
double
of
that
which
is
sought
.
Here
therefore
it
must
bee
conceiued
,
That
of
the
two
sides
encluding
the
Rightangle
,
the
one
is
to
be
vnderstood
to
,
be
the
Breadth
,
the
other
the
Length
.
I
Suppose
a
Rightangled-triangle
,
whose
sides
including
the
Right-angle
,
are
18
,
and
24
,
are
to
bee
measured
.
Here
I
take
18
for
the
Heighth
,
or
Breadth
of
the
parallelogramme
which
also
I
finde
to
meete
with
the
beuelling
line
12
,
precisly
at
6
ynches
from
the
fore
end
of
the
Ruler
:
Againe
6
,
the
sayd
line
found
,
I
finde
iust
4
times
in
24
the
Lenghth
giuen
:
Therfore
I
auerre
the
Triangle
giuen
to
conteine
the
halfe
of
4
foote
,
that
is
2
foote
of
bourd
.
20
If
the
triangle
giuen
bee
not
right-angled
,
then
is
it
by
a
perpendicular
,
let
fall
within
the
triangle
,
from
one
of
the
corners
vnto
the
base
,
to
bee
reduced
vnto
two
rightangled
triangles
.
How
this
is
to
be
done
,
Euclide
teacheth
at
the
11
&
12
propositions
of
his
I
,
booke
;
And
P.
Ramus
,
at
the
9
&
10
elements
of
his
V.
booke
of
Geometry
.
It
is
also
to
bee
done
by
the
squire
.
Or
by
a
triangled
leuell
,
and
otherwise
.
II
An
Obtuseangled
triangle
,
whose
three
sides
are
25
,
40
,
and
42
,
is
to
bee
measured
.
Heere
by
one
of
those
aboue
named
wayes
,
I
finde
the
perpendicular
or
plumbline
,
falling
from
the
greater
corner
,
vnto
the
opposite
line
,
to
be
●8
.
And
24
I
finde
vpon
the
Ruler
to
meete
with
the
line
of
12
,
at
6
ynches
from
the
fore-end
of
the
same
:
Againe
6
I
find
in
42
seauen
times
:
Therefore
the
Triangle
giuen
doth
conteine
halfe
so
many
foote
,
That
is
3
foote
and
an
halfe
of
bourd
.
11
From
hence
it
is
manifest
how
any
Rhombus
,
Rhomboides
,
Trapezium
,
or
irregular
rightlined
multangles
are
to
bee
measured
.
To
weet
,
that
they
are
to
be
measured
by
parts
,
or
by
the
particular
triangles
,
which
euery
such
figure
doth
contayne
.
Examples
you
may
haue
in
the
XIIII
booke
of
Ramus's
Geometry
,
or
in
any
others
,
which
haue
written
of
Geometry
.
Of
the
measuring
of
any
ordinate
multangle
figured
.
12
Ordinate
multangled
plaines
are
measured
by
their
halfe
Perimeter
,
and
the
plumbline
from
the
center
,
vnto
the
middest
of
any
one
side
.
These
sortes
of
plaines
may
bee
measured
,
as
the
former
were
,
by
diuiding
them
into
their
seuerall
Triangles
.
But
this
last
is
farre
shorter
:
And
therefore
to
bee
embraced
&
rather
to
be
vsed
in
practise
.
Here
the
halfe
of
the
perimeter
,
or
bout-line
,
〈◊〉
to
the
Length
in
a
〈◊〉
〈◊〉
:
And
the
plumbline
here
,
is
in
stead
of
the
Heighth
or
Breadth
there
.
1
An
〈◊〉
Pentangle
,
whose
sides
are
24
ynches
a
piece
;
And
the
〈◊〉
from
the
center
,
to
the
middest
of
any
one
of
the
sides
16
,
is
to
be
measured
.
Here
10
the
Plumbline
or
Heighth
,
doth
,
vpon
the
Rular
,
meet
with
the
〈◊〉
line
12
,
at
9
ynches
from
the
oft
named
end
:
And
9
is
contayned
in
60
,
the
halfe
of
the
perimeter
,
6
times
and
two
thirds
:
Therefore
the
Pentangle
giuen
conteineth
6
foot
,
and
two
third
partes
of
a
foot
of
Bourd
.
II
A
Sexangled
ordinate
figure
,
whose
sides
are
12
ynches
broad
a
piece
,
is
to
bee
measured
.
Here
the
Plumbline
from
the
center
to
the
middest
of
any
one
side
,
is
10
ynches
,
and
8
one
and
twentyths
of
an
ynch
:
The
double
of
10
(
that
is
20.
)
and
〈◊〉
one
&
twenty
parts
of
one
ynche
,
I
obserue
to
meete
with
the
beueller
12
,
about
7
ynches
,
&
one
quarter
of
an
ynch
,
from
the
fore
end
of
the
Rular
.
Which
7
and
a
quarter
,
is
contained
in
44
six
time
,
and
two
twenty
nineth
partes
.
Therfore
I
say
the
Sexangled
figure
giuen
doth
containe
6
foote
of
bourd
,
and
some
small
quantity
more
.
The
Circle
,
or
Circular
forme
is
in
like
manner
measured
:
For
13
The
Circle
is
measured
by
the
Ray
,
and
the
halfe
of
the
perimeter
.
For
,
savth
the
Geometrician
;
Planus
è
radio
&
peripheriae
〈◊〉
est
area
〈◊〉
.
The
plaine
of
the
ray
,
and
halfe
of
the
circumference
is
the
content
of
the
circle
.
A
Round
table
,
whose
diameter
is
4
foote
,
and
8
ynches
,
(
or
56
ynches
)
is
to
be
measured
.
The
halfe
of
59
is
28
:
And
the
halfe
of
the
circumference
is
88.
Now
●8
being
geater
then
any
of
the
〈◊〉
,
I
take
14
the
half
therof
Wh
ch
I
find
to
meet
with
the
beuelling
line
12
,
at
10
ynches
,
and
a
quarter
,
from
the
for'end
of
the
Rular
:
Therfore
I
say
euery
10
ynches
,
and
a
quarter
of
an
ynche
of
that
Table
shall
be
2
foot
of
bourd
.
And
because
88
doth
containe
10
and
1
quarter
,
8
times
,
and
20
fourty
ones
;
Therefore
I
say
,
the
whole
doth
containe
16
foot
of
of
bourd
,
and
144
ynches
.
CHAP.
III.
Of
the
measuring
of
Bodies
or
Solids
by
the
Foot
.
1
A
Body
is
a
magnitude
of
three
dimensions
.
A
Body
or
Solid
is
a
magnitude
which
hath
Length
,
Breadth
,
and
Thicknes
.
2
Here
the
measure
is
also
a
body
,
to
weet
the
Cube
of
12
that
is
1728.
This
is
our
opinion
:
Yet
if
any
shall
thinke
it
a
paradox
,
or
shall
gaine
say
it
,
or
mainetaine
the
contrary
,
wee
will
not
contend
.
And
3
Of
the
three
dimensions
,
two
are
giuen
,
the
third
is
sought
.
4
Bodies
are
of
diverse
sorts
:
But
we
will
at
this
time
meddle
only
with
such
as
are
comprehended
of
parallelogrammes
,
or
with
Cylinders
.
True
it
is
,
that
this
our
instrument
may
bee
fitted
,
and
applyed
to
the
measuring
of
many
other
sorts
of
Solid
bodies
:
But
because
we
see
no
great
vse
of
it
in
the
measuring
of
any
other
then
of
these
two
sorts
:
Therefore
wee
will
declare
the
vse
of
it
,
in
the
measuring
of
these
two
onely
.
Of
these
the
first
is
the
Parallelepipedum
,
which
is
a
plaine
Solid
,
whose
opposite
sides
are
parallelogramme
.
I
A
rightangled
parallelepipedum
(
or
a
squared
〈◊〉
logge
)
of
12
Ynches
〈◊〉
,
18
broad
,
and
16
foote
long
,
is
to
be
measured
.
Here
the
〈◊〉
and
Breadth
are
giuen
:
The
Length
is
sought
.
These
I
finde
vpon
the
Rular
to
meet
at
8
ynches
from
the
〈◊〉
named
fore-end
:
Therfore
I
Say
,
Euery
8
ynches
of
that
Logge
in
〈◊〉
shall
make
a
〈◊〉
〈◊〉
of
tymber
.
And
because
I
finde
8
Ynches
,
in
16
foote
,
24
times
:
Therfore
I
say
in
the
Tymbersticke
giuen
,
there
is
24
foote
of
solid
measure
.
II
A
squared
stone
of
14
Ynches
thicke
,
fiue
〈◊〉
(
or
60
ynches
)
broad
,
and
10
foote
long
,
is
to
be
measured
.
Here
60
is
greater
then
any
of
the
parallels
vpon
the
Rular
:
Therfore
I
take
12
the
5th
part
of
it
:
And
I
obserue
12
and
14
,
to
meete
at
10
ynches
,
and
2
seaunth
partes
of
an
ynche
,
from
the
Fore-end
of
the
Rular
.
Therfore
I
say
,
That
euery
10
ynches
,
and
2
seaunth
partes
of
an
ynch
in
length
of
that
stone
shall
be
5
foote
of
solid
measure
.
And
because
that
10
foote
conteineth
10
ynches
,
and
2
seaunth
parts
of
an
〈◊〉
,
11
times
and
5
seau'nty
twoos
:
Therfore
I
say
the
whole
stone
conteineth
58
foote
,
and
one
third
parte
of
a
Foote
of
solid
measure
.
III
A
rightangled
Prisma
,
both
whose
sids
,
〈◊〉
I
meane
,
conteyning
the
rightangle
,
are
18
ynches
〈◊〉
the
whole
being
in
length
16
foot
,
is
to
be
〈◊〉
〈◊〉
vnderstand
that
,
as
before
was
shewed
,
as
a
Triangle
was
but
the
halfe
of
a
quadrangle
:
So
a
Prisma
is
nought
but
the
halfe
of
a
Parallelepipedum
,
sawne
longways
from
〈◊〉
to
corner
though
the
midd'st
:
And
hence
in
〈◊〉
it
hath
the
name
:
This
knowne
I
enter
with
the
numbers
giu'n
,
and
I
finde
18
to
meet
with
18
,
at
5
ynches
and
one
third
parte
of
an
ynche
from
the
oft
named
end
of
the
Rular
:
Therefore
I
say
,
That
〈◊〉
5
ynches
,
and
〈◊〉
third
parte
of
an
ynche
in
length
of
that
sticke
shall
be
but
halfe
a
foote
of
solid
measure
.
Nowe
because
5
ynches
,
and
1
third
of
an
ynche
is
conteined
in
16
foote
,
67
tymes
and
14
sixteen
partes
,
that
is
almost
68
times
:
The
fore
I
say
,
The
〈◊〉
giu'n
doth
conteine
〈◊〉
68
halfe
foot
's
,
or
34
foote
of
〈◊〉
measure
.
IIII
A
sispaned
solid
,
all
whole
sides
are
6
ynches
broad
a
〈◊〉
and
16
foote
long
,
is
to
bee
measured
.
Here
the
two
lines
giuen
are
,
as
aboue
was
taught
,
the
Plumblin
:
from
the
center
,
vnto
the
middest
of
any
one
of
the
sides
:
And
the
halfe
of
the
compasse
;
That
,
as
before
was
taught
,
is
5
ynches
,
and
2
〈◊〉
partes
of
an
ynche
:
This
is
,
as
you
see
18.
Now
5
and
2
eleu'nths
doth
meet
with
18
,
at
19
ynches
and
〈◊〉
fifth
parte
of
an
inche
from
the
fore-end
:
Therfore
I
say
,
That
euery
19
〈◊〉
,
and
one
〈◊〉
parte
of
an
ynch
,
shall
be
a
〈◊〉
of
solid
measure
.
Lastly
,
because
16
ynches
,
and
1
fifth
parte
is
〈◊〉
in
16
foot
,
10
times
,
and
2
fifteene
pates
,
I
say
that
the
tymber
sticke
giu'n
doth
containe
10
foot
of
solid
measure
,
and
some
small
quantity
more
.
Lastly
a
Round
columne
,
or
Cylinder
,
of
44
ynches
about
,
&
12
foote
long
,
is
to
be
measured
.
Here
according
to
that
aboue
taught
,
the
two
lines
giu'n
are
,
The
half
diameter
,
&
the
halfe
circumference
:
This
is
22
:
That
7.
Now
these
two
do
meete
vpon
the
Rular
at
11
ynches
,
and
17
seauenty
two
partes
,
of
an
ynch
,
from
the
fore-end
there
of
;
Therefore
the
sticke
containeth
about
13
foot
of
tymber
or
solid
measure
.
FINIS
.
AN
APPENDIX
TO
THE
MESOLABIVM
.
TWo
things
,
for
the
further
illustration
of
the
Instrument
,
wee
haue
thought
good
here
to
annex
vnto
the
former
.
The
one
is
a
collation
of
this
manner
of
measuring
,
with
〈◊〉
commonly
taught
and
practised
.
〈◊〉
is
of
the
Measuring
of
Land
by
the
〈◊〉
A
foot
of
solid
measure
,
as
all
doe
generally
〈◊〉
Cube
of
12
ynches
;
that
is
,
a
square
,
or
〈◊〉
〈◊〉
,
all
whose
dimensions
,
to
wit
,
Thicknesse
,
Breadth
,
and
Length
are
equall
:
And
the
content
,
in
numbers
,
is
found
by
a
continuall
〈◊〉
of
12
,
12
and
12
,
thus
:
12
〈◊〉
12
,
are
〈◊〉
:
and
12
times
〈◊〉
,
are
〈◊〉
.
Therefore
a
foot
of
solid
measure
doth
conteine
1728
seuerall
cubes
of
an
〈◊〉
,
thicknesse
,
breadth
,
and
height
.
This
is
〈◊〉
it
were
the
standard
,
whereby
this
kinde
of
measuring
is
to
be
examined
.
All
artificers
generally
do
measure
by
a
Table
of
square
numbers
.
And
therefore
if
the
body
giuen
to
be
measured
,
be
square
,
that
is
,
if
the
thicknesse
and
breadth
bee
〈◊〉
,
they
〈◊〉
giue
the
iust
content
.
But
where
these
two
dimensions
doe
differ
,
there
by
their
rules
they
doe
it
not
without
some
error
.
For
to
bring
it
to
the
vse
of
their
Table
,
they
must
first
make
the
thicknesse
and
breadth
equall
;
which
they
doe
,
by
taking
the
excesse
from
the
greater
,
and
adding
of
it
to
the
lesser
,
or
,
which
is
all
one
,
by
girding
of
the
body
about
,
and
by
taking
of
the
quarter
of
the
compasse
.
An
example
or
two
will
make
all
plaine
.
Suppose
a
body
,
giuen
to
bee
measured
,
were
10
ynches
,
thick
,
and
14
broad
.
Here
they
take
2
from
14
,
and
adde
it
vnto
10
,
and
so
doe
make
all
the
sides
equall
:
Or
by
girding
of
it
they
finde
the
compasse
to
be
48.
And
the
quarter
of
48
to
be
12.
And
their
table
for
the
square
of
12
ynches
,
doth
giue
12
ynches
for
the
length
required
to
make
a
foot
of
solid
measure
.
If
this
be
〈◊〉
,
then
10
,
14
,
and
12
,
continually
multiplyed
between
themselues
,
shall
be
equal
to
12
,
12
,
&
12
,
continually
multiplyed
between
themselues
.
But
10
,
14
,
and
12
,
doe
giue
for
the
product
1680.
And
12
,
12
,
and
12
,
as
in
former
wee
saw
,
gaue
1728.
〈◊〉
the
difference
betweene
〈◊〉
,
and
1680
,
is
48.
Therefore
the
losse
in
euery
foot
of
〈◊〉
body
,
by
their
measure
,
is
48
inches
.
〈◊〉
vpon
our
Rular
,
you
see
14
the
parallell
,
to
meet
〈◊〉
〈◊〉
beuelling
line
at
12
inches
,
and
12/35
of
an
inch
〈◊〉
end
of
the
Rular
:
And
by
multiplication
you
〈◊〉
10
,
14
,
and
12
12/35
doe
make
1728
:
Therefore
our
〈◊〉
is
true
.
If
the
body
giuen
be
8
inches
thicke
,
and
16
broad
,
they
likewise
take
4
from
16
,
and
doe
adde
it
vnto
8
,
and
so
,
as
afore
,
doe
〈◊〉
it
to
be
equall
to
12
inches
square
.
Which
,
if
it
be
true
,
then
the
product
of
8
,
16
,
and
12
,
shall
bee
equall
to
the
product
of
12
,
12
,
and
12.
But
the
product
by
the
continuall
multiplication
of
8
,
16
,
and
12
,
is
but
1536
:
And
the
product
of
12
,
12
,
and
12
,
is
1728
:
and
the
difference
betweene
1728
,
and
1536
,
is
192
:
Therefore
by
that
their
measure
there
is
lost
in
euery
foot
192
inches
.
Now
vpon
the
〈◊〉
we
finde
8
and
16
to
crosse
one
another
at
13
½
:
And
8
,
16
,
and
13
½
continually
multiplied
one
by
another
,
doe
giue
the
product
1728
:
Therefore
this
kinde
of
measuring
by
the
Rular
is
exact
.
If
it
were
6
inches
thick
,
&
18
inches
broad
,
by
the
same
reason
,
euery
12
inches
in
length
should
make
a
foot
of
solid
measure
.
For
the
summe
of
all
the
sides
added
together
,
is
48
:
And
the
quarter
of
48
is
12
:
And
their
Table
for
the
square
of
12
,
doth
assigne
12
inches
for
the
length
.
〈◊〉
6
,
18
,
and
12
,
multiplied
continually
doe
make
but
1296
,
which
differeth
from
1728
by
432.
Therefore
by
this
their
measure
,
in
euery
foot
of
solid
measure
,
there
is
lost
432
solid
inches
.
That
is
iust
one
quarter
of
a
foot
.
Vpon
this
our
instrument
6
,
and
18
,
are
obserued
to
meet
at
16
inches
from
the
said
fore
end
:
and
therefore
it
alloweth
for
a
foot
16
in
length
.
And
6
,
18
,
and
16
continually
multiplied
doe
make
〈◊〉
:
therefore
our
rule
is
true
.
Of
the
measuring
of
Plaines
,
or
Land
,
Meddows
,
and
Woods
by
the
Aker
.
ALthough
this
instrument
,
as
the
title
specifieth
,
bee
fitted
only
for
the
measuring
of
Plains
and
〈◊〉
by
the
foot
:
Yet
,
as
before
is
mentioned
,
it
may
easily
bee
applyed
to
other
like
sort
of
measuring
.
Now
,
among
others
,
there
being
none
of
more
frequent
〈◊〉
amongst
vs
,
then
the
Rod
or
Perch
for
the
measuring
of
Land
,
Meddow
,
and
Wood
by
the
Aker
:
And
this
being
either
not
easie
to
be
done
by
the
vnlearned
;
or
not
speedily
to
be
performed
by
any
,
I
haue
thought
it
not
amisse
,
for
the
further
declaration
of
the
vse
&
excellency
of
this
inuention
,
and
for
the
benefit
of
others
,
to
adde
vnto
the
former
,
something
of
this
also
.
An
Aker
of
Land
is
,
as
before
we
heard
,
an
oblong
parallelogramme
,
whose
breadth
is
4
Poles
,
and
length
40.
Therefore
an
Aker
conteineth
160
square
Rods
,
of
what
figure
or
forme
soeuer
it
be
.
For
4
times
40
,
are
160.
This
here
in
this
case
is
,
as
144
was
in
Board
,
and
1728
was
in
solid
measure
,
as
it
were
the
Standard
,
whereby
this
kinde
of
measure
is
to
be
examined
.
Land
,
Meddow
,
or
Wood
,
is
to
bee
measured
by
this
instrument
,
in
all
respects
,
as
Board
or
Glasse
was
measured
.
Only
two
things
are
here
first
to
be
knowne
:
The
first
is
,
That
as
there
the
beuelling
line
of
12
,
was
alwayes
giuen
for
the
breadth
,
as
appropriate
to
that
kinde
of
measure
:
So
here
another
,
peculiar
to
this
manner
of
measure
,
is
in
like
sort
to
bee
drawne
ouerth
wart
the
parallell
lines
from
6
〈◊〉
of
an
inch
from
the
fore-end
of
the
Rular
,
in
the
parallell
24
,
vnto
13
and
⅓
in
the
parallell
12.
The
second
is
,
That
〈◊〉
there
lines
and
spaces
did
answer
,
and
were
denominated
of
inches
and
parts
of
inches
;
so
here
the
same
lines
and
spaces
must
bee
supposed
to
signifie
Rods
and
Perches
,
and
parts
of
the
same
.
An
example
or
two
will
make
all
plaine
and
manifest
.
1
Suppose
a
right
angled
square
field
of
16
Pole
broad
,
and
30
in
length
were
to
be
measured
:
Here
I
finde
16
,
to
meet
with
the
line
of
Land
measure
at
10
inches
from
the
fore
end
of
the
Rular
:
Therefore
I
say
,
that
euery
10
rod
in
length
,
of
the
breadth
of
16
rods
shall
make
an
〈◊〉
of
Land
.
And
againe
,
because
the
said
10
is
found
in
30
the
length
giuen
3
times
.
Therefore
I
say
,
the
field
assigned
doth
conteine
3
Akers
.
2
A
square
Meddow
right-angled
of
40
Poses
in
breadth
,
and
60
in
length
,
〈◊〉
to
be
measured
.
Here
40
,
the
lesser
number
of
the
two
giuen
,
is
greater
then
any
vpon
the
Rular
:
Theresore
I
take
20
,
the
halfe
of
40
:
And
I
finde
20
to
meet
with
the
line
of
land
measure
,
at
8
inches
from
the
said
fore-end
of
the
Rular
.
Therefore
〈◊〉
I
say
.
That
euery
7
Pole
in
length
of
the
breadth
of
40
Pole
,
shall
〈◊〉
2
Akers
of
Meddow
.
Againe
because
8
is
conteined
in
60
seuen
times
and
½
:
therefore
I
auerre
,
that
the
Meddow
assigned
to
bee
measured
,
doth
conteine
15
akers
.
3
Admit
that
a
Wood
to
bee
measured
were
160
rods
square
:
that
is
,
that
euery
side
of
the
same
were
160
Poles
in
length
.
Here
160
is
farre
greater
then
any
number
vpon
the
Rular
:
Therefore
I
take
16
,
the
tenth
part
of
160
:
which
I
〈◊〉
to
crosse
the
line
of
land
measure
at
10
inches
from
the
fore
end
of
the
Rular
:
wherefore
first
I
affirme
,
that
euery
10
pole
of
the
breadth
of
160
pole
,
shall
conteine
10
Akers
of
Wood
:
Againe
,
because
160
doth
10
sixteene
times
,
I
say
,
that
the
Wood
doth
conteine
160
Akers
.
Or
,
which
is
all
one
,
that
euery
Pole
in
breadth
of
that
length
,
doth
conteine
an
Aker
.
But
some
man
may
obiect
and
say
,
this
is
not
a
matter
worth
the
learning
,
or
of
so
many
words
,
seeing
that
it
is
well
knowne
,
that
there
are
many
men
wholly
vnlearned
,
yea
,
and
some
of
no
extraordinary
parts
of
capacity
or
vnderstanding
,
which
can
measure
Land
,
Meddow
,
or
Woods
,
so
that
they
be
square
,
or
of
any
ordinary
forme
.
I
confesse
I
haue
knowne
diuers
such
.
And
yet
is
not
this
our
labour
in
〈◊〉
or
altogether
vnprofitable
:
For
first
,
what
they
doe
with
much
study
or
contention
of
minde
,
and
are
long
in
doing
of
it
,
we
teach
to
doe
with
great
facility
and
speed
.
For
they
,
although
the
field
be
a
rightangled
parallelogramme
,
must
first
measure
at
the
least
two
sides
comprehending
one
or
other
of
the
rightangles
.
And
then
multiply
these
two
sides
the
one
by
the
other
:
And
lastly
,
the
product
found
,
they
must
diuide
by
160
:
And
so
by
the
quotient
now
found
,
answer
the
question
propounded
.
We
saue
a
great
deale
of
this
labour
.
For
we
only
measure
the
breadth
of
the
field
;
then
wee
seeke
vpon
the
Rular
,
how
much
in
length
of
this
breadth
doth
make
an
Aker
.
Lastly
,
we
apply
this
last
number
found
vnto
the
whole
length
,
and
so
witho
ut
either
multiplication
or
diuision
,
doe
speedily
answere
the
demand
.
Another
thing
there
is
wherein
this
inuention
doth
goe
farre
beyond
the
reach
of
the
vnlearned
,
which
is
thus
.
These
〈◊〉
can
sooner
measure
and
cast
vp
the
whole
content
of
the
field
,
then
they
can
set
you
〈◊〉
one
,
two
,
or
three
Akers
of
the
same
.
Here
it
is
all
one
to
giue
the
content
of
the
whole
,
or
any
part
,
or
parts
of
the
same
.
And
that
which
to
those
is
most
hard
,
here
to
vs
is
most
easie
:
I
meane
to
set
out
any
one
or
more
Akers
of
the
〈◊〉
;
and
that
on
which
side
or
end
of
the
field
you
shall
thinke
good
.