MATHEMATICALL
Recreations
Or
a
Collection
of
sundrie
excellent
Problemes
out
of
ancient
&
moderne
Phylosophers
Both
vsefull
and
Recreatiue
Printed
for
William
Leake
and
are
to
be
solde
at
the
Crowne
in
fleet
streete
betweene
the
two
Temple
gates
.
TO
The
thrice
Noble
and
most
generous
Lo.
the
Lo.
Lambert
Verreyken
,
Lo.
of
Hinden
,
Wolverthem
,
&c.
My
honourable
Lo.
AMongst
the
rare
and
curious
Propositions
which
I
have
learned
out
of
the
studies
of
the
Mathematicks
in
the
famous
University
of
Pont
a
Mousson
,
I
have
taken
singular
pleasure
in
certaine
Problemes
no
lesse
ingenious
than
recreative
,
which
drew
me
unto
the
search
of
demonstrations
more
difficult
and
serious
;
some
of
which
I
have
amassed
and
caused
to
passe
the
Presse
,
and
here
dedicate
them
now
unto
your
Honour
;
not
that
I
account
them
worthy
of
your
view
,
but
in
part
to
testifie
my
affectionate
desires
to
serve
you
,
and
to
satisfie
the
curious
,
who
delight
themselves
in
these
pleasant
studies
,
knowing
well
that
the
Nobilitie
,
and
Gentrie
rather
studie
the
Mathematicall
Arts
,
to
content
and
satisfie
their
affections
,
in
the
speculation
of
such
admirable
experiments
as
are
extracted
from
them
,
than
in
hope
of
gaine
to
fill
their
Purses
.
All
which
studies
,
and
others
,
with
my
whole
indevours
,
I
shall
alwayes
dedicate
unto
your
Honour
,
with
an
ardent
desire
to
be
accounted
ever
,
Your
most
humble
and
obedient
Nephew
and
Servant
,
H.
VAN
ETTEN
.
By
way
of
advertisement
.
Five
or
six
things
I
have
thought
worthy
to
declare
before
I
passe
further
.
FIrst
,
that
I
place
not
the
speculative
demonstrations
with
all
these
Problems
,
but
content
my self
to
shew
them
as
at
the
fingers
end
:
which
was
my
plot
and
intention
,
because
those
which
understand
the
Mathematicks
can
conceive
them
easily
;
others
for
the
most
part
will
content
themselves
onely
with
the
knowledge
of
them
,
without
seeking
the
reason
.
Secondly
,
to
give
a
greater
grace
to
the
practice
of
these
things
,
they
ought
to
be
concealed
as
much
as
they
may
,
in
the
subtiltie
of
the
way
;
for
that
which
doth
ravish
the
spirits
is
,
an
admirable
effect
,
whose
cause
is
unknowne
:
which
if
it
were
discovered
,
halfe
the
pleasure
is
l●st
;
therefore
all
the
finenesse
consists
in
the
dexterity
of
the
Act
,
concealing
the
meanes
,
and
changing
often
the
streame
.
Thirdly
,
great
care
ought
to
be
had
that
one
deceive
not
himselfe
,
that
would
declare
by
way
of
Art
to
deceive
another
:
this
will
make
the
matter
contemptible
to
ignorant
Persons
,
which
will
rather
cast
the
fault
upon
the
Science
,
than
upon
him
that
shewes
it
:
when
the
cause
is
not
in
the
Mathematicall
principles
,
but
in
him
that
failes
in
the
acting
of
it
.
Fourthly
,
in
certaine
Arithmeticall
propositions
they
have
onely
their
answers
as
I
found
them
in
sundry
Authors
,
which
any
one
being
studious
of
Mathematicall
learning
,
may
finde
their
originall
,
and
also
the
way
of
their
operation
.
Fifthly
,
because
the
number
of
these
Problemes
,
and
their
dependances
are
many
,
and
intermixed
,
I
thought
it
convenient
to
gather
them
into
a
Table
:
that
so
each
one
according
to
his
fancie
,
might
make
best
choise
of
that
which
might
best
please
his
palate
,
the
matter
being
not
of
one
nature
,
nor
of
like
subtiltie
:
But
whosoever
will
have
patience
to
read
on
,
shall
finde
the
end
better
than
the
beginning
.
To
the
Reader
.
IT
hath
been
observed
by
many
,
that
sundry
fine
wits
as
well
amongst
the
Ancient
as
Moderne
,
have
sported
and
delighted
themselves
upon
severall
things
of
small
consequence
,
as
upon
the
foot
of
a
fly
,
upon
a
straw
,
upon
a
point
,
nay
upon
nothing
;
striving
as
it
were
to
shew
the
greatnesse
of
their
glory
in
the
smalnesse
of
the
subject
:
And
have
amongst
most
solid
and
artificiall
conclusions
,
composed
and
produced
sundry
Inventions
both
Philosophicall
and
Mathematicall
,
to
solace
the
minde
,
and
recreate
the
spirits
,
which
the
succeeding
ages
have
imbraced
,
and
from
them
gleaned
and
extracted
many
admirable
,
and
rare
conclusions
;
judging
that
borrowed
matter
often-times
yeelds
praise
to
the
industry
of
its
author
.
Hence
for
thy
use
(
Courteous
Reader
)
I
have
with
great
search
and
labour
collected
also
,
and
heaped
up
together
in
a
body
of
these
pleasant
and
fine
experiments
to
stirre
up
and
delight
the
affectionate
,
(
out
of
the
writings
of
Socrates
,
Plato
,
Aristotle
,
Demosthenes
,
Pythagoras
,
Democrates
,
Plinie
,
Hyparchus
,
Euclides
,
Vitruvius
,
Diaphantus
,
Pergaeus
,
Archimedes
,
Papus
Alexandrinus
,
Vitellius
,
Ptolomaeus
,
Copernicus
,
Proclus
,
Mauralicus
,
Cardanus
,
Valalpandus
,
Kepleirus
,
Gilbertus
,
Tychonius
,
Dureirus
,
Josephus
,
Clavius
,
Gallileus
Maginus
,
Euphanus
Tyberill
,
and
others
)
knowing
Art
imitating
Nature
that
glories
alwayes
in
the
variety
of
things
,
which
she
produceth
to
satisfie
the
minde
of
curious
inquisitors
.
And
though
perhaps
these
labours
to
some
humourous
persons
may
seeme
vaine
,
and
ridiculous
,
for
such
it
was
not
undertaken
:
But
for
those
which
intentively
have
desired
and
●ought
after
the
knowledge
of
those
things
,
it
being
an
invitation
and
motive
to
the
search
of
greater
matters
,
and
to
imploy
the
minde
in
usefull
knowledge
,
rather
than
to
be
busied
in
vaine
Pamphlets
,
Play-books
,
fruitlesse
Legends
,
and
prodigious
Histories
that
are
invented
out
of
fancie
,
which
abuse
many
Noble
spirits
,
dull
their
wits
,
&
alienate
their
thoughts
from
laudable
and
honourable
Studies
.
In
this
Tractate
thou
maist
therefore
make
choise
of
such
Mathematicall
Problemes
and
Conclusions
as
may
delight
thee
,
which
kinde
of
learning
doth
excellently
adorne
a
man
;
seeing
the
usefulnesse
thereof
,
and
the
manly
accomplishments
it
doth
produce
,
is
profitable
and
delightfull
for
all
sorts
of
people
,
who
may
furnish
and
adorne
themselves
with
abundance
of
matter
in
that
kinde
,
to
help
them
by
way
of
use
,
and
discourse
.
And
to
this
we
have
also
added
our
Pyrotechnie
,
knowing
that
Beasts
have
for
their
object
only
the
surface
of
the
earth
;
but
hoping
that
thy
spirit
which
followeth
the
motion
of
fire
,
will
abandon
the
lower
Elements
,
and
cause
thee
to
lift
up
thine
eyes
to
soare
in
an
higher
Contemplation
,
having
so
glittering
a
Canopie
to
behold
,
and
these
pleasant
and
recreative
fires
ascending
may
cause
thy
affections
also
to
ascend
.
The
Whole
whereof
we
send
forth
to
thee
,
that
desirest
the
scrutability
of
things
;
Nature
having
furnished
us
with
matter
,
thy
spirit
may
easily
digest
them
,
and
put
them
finely
in
order
,
though
now
in
disorder
.
A
Table
of
the
particular
heads
of
this
Book
,
contracted
according
to
the
severall
Arts
specified
in
the
Title-page
.
Experiments
of
Arithmetick
.
-
PAge
1
,
2
,
3
,
16
,
19
,
22
,
28
,
33
,
39
,
40
,
44
,
45
,
51
,
52
,
53
,
59
,
60
,
69
,
71
,
77
,
83
,
85
,
86
,
89
,
90
,
91
,
124
,
134
,
135
,
136
▪
137
,
138
,
139
,
140
,
178
179
,
181
,
182
,
183
,
184
,
185
,
188
,
208
,
210
,
213.
Experiments
●n
Geometrie
.
-
Pag.
12
,
15
,
24
,
26
,
27
,
30
,
35
,
37
,
41
,
42
,
47
,
48
,
49
,
62
,
65
,
72
,
79
,
82
,
113
,
117
,
118
,
119
,
214
,
215
,
217
,
218
,
234
,
235
,
236
,
239
,
240.
Experiments
in
Cosmographie
.
-
Pag.
14
,
43
,
75
,
106
,
107
,
219
,
220
,
225
,
227
,
228
,
229
,
230
,
232.
Experiments
in
Horologiographie
.
-
Pag.
137
,
166
,
167
,
168
,
169
,
171
,
234.
Experiments
in
Astronomie
.
-
Pag.
220
,
221
,
222
,
223
,
224.
Experiments
in
Navigation
.
-
Pag.
105
,
233
,
234
,
237
,
238.
Experiments
in
Musick
.
-
Pag.
78
,
87
,
126.
Experiments
in
Opticks
.
-
Pag.
6
,
66
,
98
,
99
,
100
,
102
,
129
,
131
,
141
,
142
,
143
,
144
,
146
,
149
,
151
,
152
,
153
,
155
,
156
,
157
,
158
,
160
,
161
,
162
,
163
,
164
,
165.
Experiments
in
Architecture
.
-
Pag.
16
,
242
,
243.
Experiments
in
Staticke
.
-
Pag.
27
,
30
,
32
,
71
,
199
,
200
,
201
,
283
,
204
,
205
,
207.
Experiments
in
Machanicks
.
-
Pag.
56
,
58
,
68
,
88
,
95
,
108
,
110
,
128
,
173
,
174
,
176
,
246
,
248
,
258
,
259.
Experiments
in
Chymistrie
.
-
Pag.
198
,
255
,
256
,
257
,
260
,
262
,
263
,
264.
Experiments
in
Water-workes
.
-
Pag.
190
,
191
,
192
,
193
,
194
,
196
,
247
,
249
,
250
,
252
,
253.
Experiments
in
Fireworkes
.
-
From
page
,
265.
to
the
end
.
FINIS
.
A
Table
of
the
Contents
,
and
chiefe
points
conteined
in
this
Book
.
PROBLEM
.
II.
-
HOw
visible
objects
that
are
without
,
and
things
that
passe
by
,
are
most
lively
represented
to
those
that
are
within
.
Page
6
-
Prob.
1
Of
finding
of
numbers
conceived
in
the
minde
.
1
,
2
,
3
-
Prob.
5
Of
a
Geographicall
Garden-plot
fit
for
a
Prince
or
some
great
personage
.
14
-
Prob.
37
Any
liquid
substance
,
as
water
or
wine
,
placed
in
a
Glasse
,
may
be
made
to
boile
by
the
motion
of
the
finger
,
and
yet
not
touching
it
.
54
-
Prob.
3
How
to
weigh
the
blow
of
ones
fist
,
of
a
Mallet
,
a
Hatchet
or
such
like
.
9.
-
Prob.
30
Two
severall
numbers
being
taken
by
two
sundry
persons
,
how
subtilly
to
discover
which
of
those
numbers
each
of
them
took
.
46
-
Prob.
4
That
a
staffe
may
be
broken
▪
placed
upon
two
Glasses
,
without
hurting
of
the
Glasses
.
12
-
Prob.
7
How
to
dispose
Lots
that
the
5
,
6
,
9
,
&c.
of
any
number
of
persons
may
escape
.
16
-
Prob.
13
How
the
weight
of
smoke
of
a
combustible
body
,
which
is
exhaled
,
may
be
weighed
.
27
-
Prob.
12
Of
three
knives
which
may
be
so
disposed
to
hang
in
the
aire
,
and
move
upon
the
Point
of
a
needle
.
27
-
Prob.
17
Of
a
deceitfull
bowle
,
to
bowle
withall
.
32
-
Prob.
16
A
ponderous
or
heavy
body
may
be
supported
in
the
aire
without
any
one
touching
it
.
30
-
Prob.
18
How
a
Peare
,
or
Apple
,
may
be
parted
into
any
parts
,
without
breaking
the
rinde
thereof
.
33
-
Prob.
15
Of
a
fine
kinde
of
dore
which
opens
and
shuts
on
both
sides
.
30
-
Prob.
9
How
the
halfe
of
a
Vessell
which
containes
8
measures
may
be
taken
,
being
but
onely
two
other
measures
,
the
one
being
3
,
and
the
other
8
measures
.
22
-
Prob.
8
Three
persons
having
taken
each
of
them
severall
things
,
to
finde
which
each
of
them
hath
taken
.
19
-
Prob.
6
How
to
dispose
three
staves
which
may
support
each
other
in
the
aire
.
15
-
Prob.
14
Many
things
being
disposed
Circular
(
or
otherwise
)
to
finde
which
of
them
any
one
thinks
upon
.
28
-
Prob.
19
To
finde
a
number
thought
upon
without
asking
questions
.
33
-
Prob.
11
How
a
Milstone
or
other
ponderosity
may
hang
upon
the
point
of
a
Needle
without
bowing
,
or
any
wise
breaking
of
it
.
26
-
Prob.
20
and
21
How
a
body
that
is
uniforme
and
inflexible
may
passe
through
a
hole
which
is
round
,
square
and
Triangular
;
or
round
,
square
and
ovall-wise
,
and
exactly
fill
those
severall
holes
.
35
,
37
-
Prob.
10
How
a
stick
may
stand
upon
ones
finger
,
or
a
Pike
in
the
middle
of
a
Court
without
falling
.
24
-
Prob.
22
To
finde
a
number
thought
upon
after
another
manner
than
those
which
are
formerly
delivered
.
39
-
Prob.
23
To
finde
out
many
numbers
that
sundry
persons
or
any
one
hath
thought
upon
.
40
-
Prob.
24
How
is
it
that
a
man
in
one
&
the
same
time
may
have
his
head
upward
,
and
his
feet
upward
,
being
in
one
and
the
same
place
?
4●
-
Prob.
25
Of
a
Ladder
by
which
two
men
ascending
at
one
time
,
the
more
they
ascend
,
the
more
they
shal
be
asunder
,
notwith
standing
the
one
be
as
high
as
the
other
.
42
-
Prob.
26
How
is
it
that
a
man
having
but
a
Rod
or
Pole
of
land
,
doth
brag
that
he
may
in
a
right
line
passe
from
place
to
place
3000
miles
.
42
-
Prob.
27
How
is
it
that
a
man
standing
upright
,
and
looking
which
way
he
will
,
he
looketh
true
North
or
South
.
43
-
Prob.
28
To
tell
any
one
what
number
remaines
after
certaine
operations
being
ended
,
without
asking
any
question
.
44
-
Prob.
29
Of
the
play
with
two
severall
things
.
45
-
Prob.
31
How
to
describe
a
circle
that
shall
touch
3
points
placed
howsoever
upon
a
plaine
,
if
they
be
not
in
a
right
line
.
47
-
Prob.
32
How
to
change
a
circle
into
a
square
forme
.
48
-
Prob.
33
With
one
and
the
same
compasses
,
and
at
one
and
the
same
extent
or
opening
,
how
to
describe
many
circles
concentricall
,
that
is
,
greater
or
lesser
one
than
another
.
49
-
Prob.
34
Any
number
under
10.
being
thought
upon
,
to
finde
what
numbers
they
were
.
51
-
Prob
,
35
Of
the
play
with
the
Ring
.
52
-
Prob.
36
The
play
of
3
,
4
,
or
more
Dice
.
53
-
Prob.
38
Of
a
fine
Vessell
which
holds
Wine
or
Water
being
cast
into
it
at
a
certain
height
,
but
being
filled
higher
it
will
runne
all
out
of
its
owne
accord
.
56
-
Prob.
39
Of
a
Glasse
very
pleasant
.
58
-
Prob.
40.
If
any
one
should
hold
in
each
hand
as
many
pieces
of
money
as
in
the
other
,
how
to
finde
how
much
there
is
.
59
-
Prob.
41
Many
Dice
being
cast
,
how
artificially
to
discover
the
number
of
the
points
that
may
arise
.
60
-
Prob.
42
Two
metals
as
Gold
and
Silver
or
of
other
kinde
,
weighing
alike
,
being
privately
placed
into
two
like
boxes
,
to
finde
in
which
of
them
the
Gold
or
Silver
is
.
62
-
Prob.
43
Two
Globes
of
divers
metals
(
as
one
Gold
the
other
Copper
)
yet
of
equall
weight
,
being
put
in
a
Box
as
B.G.
to
finde
in
which
end
the
Gold
or
Copper
is
.
65
-
Prob.
44
How
to
represent
divers
sorts
of
Rainbowes
here
below
.
66
-
Prob.
45
How
that
if
all
the
powder
in
the
World
were
inclosed
in
a
bowle
of
paper
or
glasse
,
and
being
fired
on
all
parts
,
it
could
not
break
that
bowle
.
68
-
Prob.
46
To
finde
a
number
which
being
divided
by
2.
there
will
remaine
1.
being
divided
by
3.
there
will
remaine
1.
and
so
likewise
being
divided
by
4
,
5
,
or
6.
there
will
still
remaine
one
,
but
being
divided
by
7
will
remaine
nothing
.
69
-
Prob.
47
One
had
a
certaine
number
of
Crownes
,
and
counting
them
by
2
and
2
,
there
rested
1.
counting
them
by
3
,
and
3
,
there
rested
2.
counting
them
by
4
,
and
4
,
there
rested
3.
counting
them
by
5
,
and
5
,
there
rested
4.
counting
them
by
6
,
and
6
,
there
rested
5.
but
counting
them
by
7
and
7
,
there
rested
nothing
,
how
many
Crownes
might
he
have
?
71
-
Prob.
48
How
many
sorts
of
weights
in
the
least
manner
must
there
be
to
weigh
all
sorts
of
things
betweene
one
pound
and
121
pound
,
and
so
unto
364
pound
?
71
-
Prob.
49
Of
a
deceitfull
balance
which
being
empty
seems
to
be
just
,
because
it
hangs
in
Aequilibrio
,
notwithstanding
putting
12
pound
in
one
ballance
,
and
11
in
the
other
,
it
will
remaine
in
Aequilibrio
.
72
-
Prob.
50
To
heave
or
lift
up
a
bottle
with
a
straw
.
74
-
Prob.
51
How
in
the
middle
of
a
wood
or
desert
,
without
the
sight
of
the
Sun
,
starres
,
shadow
,
or
compasse
,
to
finde
out
the
North
,
or
South
,
or
the
4
Cardinal
points
of
the
World
,
East
,
West
,
&c.
75
-
Prob.
52
Three
persons
having
taken
Counters
,
Cards
,
or
other
things
,
to
finde
how
much
each
one
hath
taken
.
7●
-
Prob.
53
How
to
make
a
consort
of
Musick
of
many
parts
with
one
voice
or
one
instrument
onely
.
78
-
Prob.
54
To
make
or
describe
an
oval
form
,
or
that
which
is
neare
resembled
unto
it
at
one
turning
,
with
a
paire
of
common
Compasses
.
79
-
Prob.
55
Of
a
purse
difficult
to
be
opened
.
80
-
Prob.
56
Whether
is
it
more
hard
and
admirable
without
Compasses
to
make
a
perfect
circle
,
or
being
made
to
finde
out
the
Centre
of
it
?
82
-
Prob.
56
Any
one
having
taken
3
Cards
,
to
finde
how
many
points
they
containe
.
83
-
Prob.
57
Many
Cards
placed
in
divers
ranks
,
to
finde
which
of
those
Cards
any
one
hath
thought
.
85
-
Prob.
58
Many
Cards
being
offered
to
sundry
persons
to
finde
which
of
those
Cards
any
one
thinketh
upon
.
86
-
Prob.
59
How
to
make
an
instrument
that
helps
to
heare
,
as
Gallileus
made
to
help
to
see
.
87
-
Prob.
60
Of
a
fine
Lamp
which
goeth
not
out
,
though
one
carries
it
in
ones
pocket
,
or
being
rolled
on
the
ground
will
still
burne
.
88
-
Prob.
61
Any
one
having
thought
a
Card
amongst
many
Cards
,
how
artificially
to
discover
it
out
.
89
-
Prob.
62
Three
Women
A
,
B
,
C.
carried
Apples
to
a
Market
to
sell
:
A
had
20.
B
had
30.
C
40.
they
sold
as
many
for
a
penny
one
as
the
other
,
and
brought
home
one
as
much
money
as
another
,
how
could
this
be
?
90
-
Prob.
63
Of
the
properties
of
some
numbers
.
91
-
Prob.
64
Of
an
excellent
Lamp
which
serves
or
furnisheth
it selfe
with
Oile
,
and
burnes
a
long
time
.
95
-
Prob.
65
Of
the
play
at
Keyles
or
Nine-pins
.
97
-
Prob.
66
Of
Spectacles
of
pleasure
-
Of
Spectacles
which
give
severall
colours
to
the
visage
.
98
-
Of
Spectacles
which
make
a
Towne
seeme
to
be
a
City
,
one
armed
man
as
a
Company
,
and
a
piece
of
Gold
as
many
pieces
.
99
-
How
out
of
a
Chamber
to
see
the
objects
which
passe
by
according
to
the
lively
perspective
.
100
-
Of
Gallileus
admirable
Optick-Glasse
,
which
helps
one
to
see
the
beginning
and
ending
of
Eclipses
,
the
spots
in
the
Sunne
,
the
Starres
which
move
about
the
Planets
,
and
perspicuously
things
far
remote
.
-
Of
the
parts
of
Gallileus
his
Glasse
.
102
-
Prob.
67
Of
the
Magnes
and
Needles
touched
therewith
.
-
How
Rings
of
Iron
may
hang
one
by
another
in
the
aire
.
103
-
Of
Mahomets
Tombe
which
hangs
in
the
aire
by
the
touch
of
the
Magnes
.
104
-
How
by
the
Magnes
only
to
finde
out
North
and
South
105
-
Of
a
secrecie
in
the
Magnes
,
for
discovering
things
farre
remote
.
106
-
Of
finding
the
Poles
by
the
Magnes
107
-
Prob.
68
Of
the
properties
of
Aeolipiles
or
Bowles
to
blow
the
fire
.
108
-
Prob.
69
Of
the
Thermometer
,
or
that
which
measures
the
degrees
of
heat
and
cold
by
the
aire
.
110
-
Of
the
proportion
of
humane
bodies
,
of
statues
,
of
Colosses
,
or
huge
Jmages
and
monstrous
Giants
.
113
-
Of
the
commensuration
of
the
parts
of
the
bodie
the
one
to
the
other
in
particular
,
by
which
the
Lion
was
measured
by
his
claw
,
the
Giant
by
his
thumbe
,
and
Hercules
by
his
foot
.
115
,
116
-
Of
Statues
or
Colosses
,
or
huge
Images
;
that
mount
Athos
metamorphosed
by
Dynocrites
into
a
statue
,
in
whose
hand
was
a
Towne
able
to
receive
ten
thousand
men
.
117
-
Of
the
famous
Colossus
at
Rhodes
which
bad
70
cubits
in
height
,
and
loaded
900.
Camels
,
which
weighed
1080000
l.
118
-
Of
Nero
his
great
Colossus
which
had
a
face
of
12
foot
large
.
119
-
Of
monstrous
Giants
-
Of
the
Giant
Og
and
Goliah
.
119
,
120
-
Of
the
Carkasse
of
a
man
found
which
was
in
length
49
foot
;
and
of
that
monster
found
in
Creet
,
which
had
46.
Cubits
of
height
.
120
-
Of
Campesius
his
relation
of
a
monster
of
300
foot
found
in
Sicile
,
whose
face
according
to
the
former
proportion
should
be
30
foot
in
length
.
121
-
Prob.
71
Of
the
game
at
the
Palme
,
at
Trap
,
at
Bowles
,
Paile-maile
,
and
others
.
122
-
Prob.
72
Of
the
game
of
square
formes
.
124
-
Prob.
73
How
to
make
the
string
of
a
Viol
sensibly
shake
without
any
one
touching
it
.
126
-
Prob.
74
Of
a
Vessell
which
containes
3
severall
kindes
of
liquor
,
all
put
in
at
one
bung-hole
,
and
drawne
out
at
one
Tap
severally
without
mixture
.
128
-
Prob.
75
Of
burning-Glasses
.
-
Archimedes
his
way
of
burning
the
ships
of
Syracuse
.
129
-
Of
Proclus
his
way
,
and
of
concave
and
sphericall
Glasses
which
burne
,
the
cause
and
demonstration
of
burning
with
Glasses
.
131
-
Of
Maginus
his
way
of
setting
fire
to
Powder
in
a
Mine
by
Glasses
.
131
-
Of
the
examination
of
burning
by
Glasses
.
133
-
Prob.
76
Of
pleasant
questions
by
way
of
Arithmetick
.
-
Of
the
Asse
and
the
Mule
.
134
-
Of
the
number
of
Souldiers
that
fought
before
old
Troy
.
135
-
Of
the
number
of
Crownes
that
two
men
had
.
136
-
About
the
houre
of
the
day
.
137
-
Of
Pythagoras
Schollers
.
137
-
Of
the
number
of
Apples
given
amongst
the
Graces
and
the
Muses
.
138
-
Of
the
testament
or
last
will
of
a
dying
Father
.
138
-
Of
the
cups
of
Croesus
.
139
-
Of
Cupids
Apples
.
139
-
Of
a
Mans
Age.
140
-
Of
the
Lion
of
Bronze
placed
upon
a
fountaine
with
his
Epigram
.
ibid.
-
Prob.
77
In
Opticks
,
excellent
experiments
.
-
Principles
touching
reflections
.
141
-
Experiments
upon
flat
and
plaine
Glasses
.
142
-
How
the
Images
seeme
to
sink
into
a
plaine
Glasse
,
and
alwayes
are
seene
perpendicular
to
the
Glasse
,
an●
also
inversed
.
143
-
The
things
which
passe
by
in
a
street
may
by
help
of
a
plaine
glasse
be
seen
in
a
Chamber
,
and
the
height
of
a
tower
or
tree
observed
.
143
-
How
severall
Candles
from
one
Candle
are
represented
in
a
plaine
Glasse
,
and
Glasses
alternately
may
be
seene
one
within
another
,
as
also
the
back-parts
of
the
body
as
well
as
the
fore-parts
are
evidently
represented
.
144
-
How
an
Image
may
be
seene
to
hang
in
the
aire
by
help
of
a
Glasse
:
and
writing
read
or
easily
understood
.
146
-
Experiments
upon
Gibbous
,
or
convex
Sphericall
Glasses
.
-
How
lively
to
represent
a
whole
City
,
fortification
,
or
Army
,
by
a
Gibbous
Glasse
.
147
-
How
the
Images
are
seen
in
Concave
Glasses
.
149
-
How
the
Images
are
transformed
by
approaching
to
the
centre
of
the
Glasse
,
or
point
of
concourse
;
and
of
an
exceeding
light
that
a
Concave
Glasse
gives
by
help
of
a
Candle
.
151
-
How
the
Images
,
as
a
man
,
a
sword
,
or
hand
,
doth
come
forth
out
of
the
Glasse
.
152
,
153
-
Of
strange
apparitions
of
Images
in
the
aire
,
by
help
of
sundry
Glasses
.
152
,
154
-
Of
the
wonderfull
augmentation
of
the
parts
of
mans
body
comming
neare
the
point
of
inflammation
,
or
centre
of
the
Glasse
.
155
-
How
writing
may
be
reverberated
from
a
Glasse
upon
a
Wall
,
and
Read
.
156
-
How
by
help
of
a
Concave
Glasse
to
cast
light
into
a
Campe
,
or
to
give
a
perspective
light
to
Pyoneers
in
a
Mine
,
by
one
Candle
only
.
156
-
How
excellently
by
help
of
a
Concave
Glasse
and
a
Candle
placed
in
the
centre
,
to
give
light
to
read
by
.
157
-
Of
other
Glasses
of
pleasure
.
158
-
Of
strange
deformed
representations
by
Glasses
;
causing
a
man
to
have
foure
eyes
,
two
Mouthes
,
two
Noses
,
two
heads
.
-
Of
Glasses
which
give
a
colour
to
the
visage
,
and
make
the
face
seeme
faire
and
foule
.
160
-
Prob.
78
How
to
shew
one
that
is
suspicious
,
what
is
in
another
Chamber
or
Roome
,
notwithstanding
the
interposition
of
that
wall
.
160
-
Corolary
,
1.
To
see
the
Besiegers
of
a
place
,
upon
the
Rampa●●t
of
a
fortification
161
-
Corolary
2.
and
3.
Notwithstanding
the
interposition
of
Walls
and
Chambers
,
by
help
of
a
Glasse
things
may
be
seen
,
which
passe
by
.
162
-
Prob.
79
How
with
a
Musket
to
strike
a
marke
not
looking
towards
it
,
as
exactly
as
one
aimed
at
it
.
162
-
How
exactly
to
shoot
out
of
a
Mu●ket
to
a
place
which
is
not
seene
,
being
hindred
by
some
obstacle
or
other
interposition
.
163
-
Prob.
80
How
to
make
an
Image
to
be
seen
hanging
in
the
aire
,
having
his
head
downward
.
164
-
Prob.
81.
How
to
make
a
company
of
representative
souldiers
seeme
to
be
as
a
regiment
,
or
how
few
in
number
may
be
multiplyed
to
seem
to
be
many
in
number
.
165
-
COROLARIE
.
Of
an
excellent
delightfull
Cabinet
made
of
plaine
Glasses
.
165
-
Prob.
82
Of
fine
and
pleasant
Dyalls
in
Horologiographie
.
-
Of
a
Dyall
of
herbs
for
a
Garden
.
166
-
Of
the
Dyall
upon
the
finger
and
hand
,
to
finde
what
of
the
Clock
it
is
.
167
-
Of
a
Dyall
which
was
about
an
Obelisk
at
Rome
.
168
-
Of
Dyals
with
Glasses
.
168
-
Of
a
Dyall
which
hath
a
Glasse
in
the
place
of
the
stile
.
169
-
Of
Dyals
with
water
,
which
the
Ancients
use●
171
-
Prob.
83
Of
shooting
out
of
Cannons
or
great
Artillery
.
-
How
to
charge
a
Cannon
without
powder
.
173
-
To
finde
how
much
time
the
Bullet
of
a
Cannon
spends
in
the
Aire
before
it
falls
to
the
ground
.
174
-
How
it
is
that
a
Cannon
shooting
upward
,
the
Bullet
flies
with
more
violence
,
than
being
shot
point
blanke
,
or
shooting
downeward
.
174
-
Whether
is
the
discharge
of
a
Cannon
so
much
the
more
violent
,
by
how
much
it
hath
the
more
length
?
176
-
Prob.
84
Of
prodigious
progressions
,
and
multiplications
of
creatures
,
plants
,
fruits
,
numbers
,
gold
,
silver
,
&c.
-
Of
graines
of
Mustardseed
,
and
that
one
graine
being
sowne
,
with
the
increase
thereof
for
20
yeares
will
produce
a
heap
greater
than
all
the
earth
a
hundred
thousand
times
.
178
-
Of
Pigges
,
and
that
the
great
Turke
with
all
his
Revenne
,
is
not
able
to
maintaine
for
one
yeare
,
a
Sow
with
all
her
increase
for
12
yeares
.
179
-
Of
graines
of
Corne
,
and
that
1
graine
with
all
its
increase
for
12
yeares
,
will
amount
to
244140625000000000000
graines
,
which
exceeds
in
value
all
the
treasures
in
the
World
.
183
-
Of
the
wonderfull
increase
af
Sheepe
.
182
-
Of
the
increase
of
Cod-fish
.
182
-
Of
the
Progressive
Multiplication
of
soules
;
that
from
one
of
Noahs
Sonnes
,
from
the
flood
unto
Nimrods
Monarchie
,
should
be
produced
111350
soules
.
183
-
Of
the
increase
of
Numbers
in
double
proportion
,
and
that
a
pin
being
doubled
as
often
as
there
are
weekes
in
the
yeare
,
the
number
of
pinnes
that
should
arise
is
able
to
load
45930
ships
of
a
thousand
Tunne
apiece
,
which
are
worth
more
than
tenne
hundred
thousand
pounds
a
day
.
183
,
184
-
Of
a
man
that
gathered
Apples
,
stones
,
or
such
like
upon
a
condition
.
185
-
Of
the
changes
in
Bells
,
in
musicall
instruments
,
transmutation
of
Places
,
in
Numbers
,
Letters
,
Men
and
such
like
▪
185
-
Of
the
wonderfull
interchange
of
the
Letters
in
the
Alphabet
:
the
exceeding
number
of
men
,
and
time
to
expresse
the
words
that
may
be
made
with
these
letters
,
and
the
number
of
Books
to
comprehend
them
.
187
,
188
-
Of
a
servant
hired
upon
certaine
condition
,
that
he
might
have
land
lent
him
to
sowe
one
graine
of
Corne
with
its
increase
for
8
yeares
time
,
which
amounted
to
more
than
four
hundred
thousand
Acres
of
Land
.
188
-
Prob.
85
Of
Fountaines
,
Hydriatiques
;
Stepticks
,
Machinecks
,
and
other
experiments
upon
water
,
or
other
liquor
.
-
First
,
how
water
at
the
foot
of
a
Mountaine
may
be
made
to
ascend
to
the
top
of
it
,
and
so
to
descend
on
the
other
side
of
it
190
-
Secondly
,
to
finde
how
much
Liquor
is
in
a
Vessell
,
onely
by
using
the
tap-hole
.
191
-
Thirdly
,
how
is
it
,
that
a
Vessell
is
said
to
hold
more
water
at
the
foot
of
a
Mountaine
,
then
at
the
top
of
it
191
-
4
How
to
conduct
water
from
the
top
of
one
Mountaine
to
the
top
of
another
192
-
5
Of
a
fine
Fountaine
which
spouts
water
very
high
and
with
great
violence
,
by
turning
of
a
Cock
193
-
6
Of
Archimedes
screw
which
makes
water
ascend
by
descending
.
194
-
7
Of
a
fine
Fountaine
of
pleasure
.
196
-
8
Of
a
fine
watering
pot
for
Gardens
.
197
-
9
How
easily
to
take
Wine
out
of
a
Vessell
at
the
bung
hole
without
piercing
a
hole
in
the
Vessell
.
198
-
10
How
to
measure
irregular
bodies
by
help
of
water
.
198
-
11
To
finde
the
weight
of
water
.
199
-
12
To
finde
the
charge
that
a
vessell
may
carry
,
as
Ships
,
Boats
or
such
like
.
200
-
13
How
comes
it
that
a
ship
having
safely
sailed
in
the
vast
Ocean
,
and
being
come
into
the
port
or
harbour
,
will
sinke
down
right
.
200
-
14
How
a
grosse
body
of
metall
may
swim
upon
the
water
.
201
-
15
How
to
weigh
the
lightnesse
of
the
aire
.
203
-
16
Being
given
a
body
,
to
mark
it
about
,
and
shew
how
much
of
it
will
sink
in
the
water
,
or
swim
above
the
water
.
204
-
17
To
finde
how
much
severall
metalls
or
other
bodies
do
weigh
lesse
in
the
water
than
in
the
aire
.
204
-
18
How
is
it
that
a
ballance
having
like
weight
in
each
scale
,
and
hanging
in
Aequilibrio
in
the
aire
,
being
removed
from
that
place
(
without
diminishing
the
weights
in
each
balance
,
or
adding
to
it
)
it
shall
cease
to
hang
in
Aequilibrio
sensibly
,
yea
by
a
great
difference
of
weight
.
205
-
19
To
shew
what
waters
are
heavier
one
than
another
,
and
how
much
.
206
-
20
How
to
make
a
pound
of
water
weigh
as
much
as
10
,
20
,
30
,
or
a
hundred
pound
of
Lead
,
nay
as
much
as
a
thousand
or
ten
thousand
pound
weight
.
207
-
Prob.
86.
Of
sundry
questions
of
Arithmetick
,
and
first
of
the
number
of
sands
calculated
by
Archimedes
and
Clavius
.
208
-
2
Divers
metalls
being
melted
together
in
one
body
,
to
finde
the
mixture
of
them
.
210
-
3
A
subtile
question
of
three
partners
about
equality
of
Wine
and
Vessels
.
213
-
4
Of
a
Ladder
which
standing
upright
against
a
wall
of
10
foot
high
,
the
foot
of
it
is
pulled
out
6
foot
from
the
wall
upon
the
pavement
,
how
much
hath
the
top
of
the
Ladder
descended
.
214
-
Prob.
87
Witty
suits
or
debates
between
Caius
and
Sempronius
,
upon
the
forme
of
figures
,
which
Geometricians
call
Isoperimeter
,
or
equall
in
circuit
,
or
Compasse
.
214
-
1
Incident
:
of
changing
a
field
of
6
measures
square
,
for
a
long
rectrangled
fiel
of
9
measures
in
length
and
3
in
breadth
:
both
equall
in
circuit
but
not
in
quantity
.
215
-
2
Incident
:
about
two
sacks
each
of
them
ho●ding
but
a
bushell
,
and
yet
were
able
to
hold
4
bushels
.
217
-
3
Incident
:
sheweth
the
deceit
of
pipes
which
conveygh
water
,
that
a
pipe
of
two
inches
diameter
,
doth
cast
out
foure
times
as
much
water
as
a
pipe
of
one
such
diameter
.
218
-
7
Heapes
of
Corne
of
10
foot
every
way
,
is
not
as
much
as
one
heap
of
Corne
of
20
foot
every
way
.
218
-
Prob.
88
Of
sundry
questions
in
matter
of
Cosmographie
,
and
Astronomy
.
-
In
what
place
the
middle
of
the
earth
is
supposed
to
be
.
219
-
Of
the
depth
of
the
earth
,
and
height
of
the
Heavens
,
and
the
compasse
of
the
World
,
how
much
.
219
-
How
much
the
starry
Firmament
,
the
Sun
,
and
the
Moone
are
distant
from
the
centre
of
the
earth
.
220
-
How
long
a
Mill-stone
would
be
in
falling
to
the
centre
of
the
earth
from
the
superficies
,
if
it
might
have
passage
thither
.
220
-
How
long
time
a
man
or
a
bird
may
be
in
compassing
the
whole
earth
.
220
-
If
a
man
should
ascend
by
supposition
20
miles
every
day
:
how
long
it
would
be
before
he
approach
to
the
Moone
.
221
-
The
Sunne
moves
more
in
one
day
than
the
Moone
in
20
dayes
.
221
-
If
a
milstone
from
the
orbe
of
the
Sun
should
descend
a
thousand
miles
in
an
houre
how
long
it
would
be
before
it
come
to
the
earth
.
221
-
Of
the
Sunnes
quick
motion
,
of
more
than
7500
miles
in
one
minute
.
221
-
Of
the
rapt
and
violent
motion
of
the
starry
Firmament
,
which
if
a
Horseman
should
ride
every
day
40
miles
,
he
could
not
in
a
thousand
yeares
make
such
a
distance
as
it
moves
every
houre
.
221
-
To
finde
the
Circle
of
the
Sunne
by
the
fingers
.
223
-
Prob.
93
Of
finding
the
new
and
full
Moone
in
each
moneth
.
224
-
Prob.
94
To
finde
the
latitude
of
Countreys
.
225
-
Prob.
95
Of
the
Climates
of
Countreys
,
and
how
to
finde
them
.
225
-
Prob.
96
Of
longitude
and
latitude
of
the
places
of
the
earth
,
and
of
the
Starres
of
the
Heavens
.
227
-
To
finde
the
Longitude
of
a
Countrey
.
228
-
Of
the
Latitude
of
a
Countrey
.
229
-
To
finde
the
Latitude
of
a
Countrey
.
230
-
To
finde
the
distance
of
places
.
230
-
Of
the
Longitude
,
Latitude
,
Declination
,
and
distance
of
the
starres
.
231
-
How
is
it
that
two
Horses
or
other
creatures
comming
into
the
World
at
one
time
,
and
dying
at
one
and
the
same
instant
,
yet
the
one
of
them
to
be
a
day
older
than
the
other
?
232
-
Certaine
fine
Observations
.
-
In
what
places
of
the
World
is
it
that
the
needle
hangs
in
Aequilibrio
,
and
verticall
?
233
-
In
what
place
of
the
world
is
it
the
sun
is
East
or
West
but
twice
in
the
yeare
?
233
-
In
what
place
of
the
World
is
it
that
the
Sunnes
Longitude
from
the
Equinoctiall
paints
and
Altitude
,
being
equall
,
the
Sunne
is
due
East
or
West
?
-
That
the
sunne
comes
twice
to
one
point
of
the
Compasse
in
the
forenoone
or
afternoone
.
233
-
That
in
some
place
of
the
World
there
are
but
two
kindes
of
winde
all
the
yeare
.
233
-
Two
ships
may
be
two
leagues
asunder
under
the
equinoctiall
,
and
sayling
North
at
a
certaine
parallell
they
will
be
but
just
halfe
so
much
.
233
-
To
what
inhabitants
,
and
at
what
time
the
sunne
will
touch
the
north-part
of
the
Horizon
at
midnight
.
234
-
How
a
man
may
know
in
his
Navigation
when
he
is
under
the
Equinoctiall
.
234
-
At
what
day
in
the
yeare
the
extremitie
of
the
styles
shadow
in
a
Dyall
makes
a
right
line
.
234
-
What
height
the
Sunne
is
of
,
and
how
far
from
the
Zenith
,
or
Horizon
,
when
a
mans
shadow
is
as
long
as
his
height
.
234
-
Prob.
97
To
make
a
Triangle
that
shall
have
three
right
Angles
.
234
-
Prob.
98
To
divide
a
line
in
as
many
parts
as
one
will
,
without
compasses
or
without
seeing
of
it
.
235
-
Prob.
99
To
draw
a
line
which
shall
incline
to
another
line
,
yet
never
meet
against
the
Axiome
of
Parallells
.
236
-
Prob.
100
To
finde
the
variation
of
the
Compasse
by
the
Sunne
shining
.
237
-
Prob.
101
To
know
which
way
the
winde
is
in
ones
Chamber
without
going
abroad
.
238
-
Prob.
102
How
to
draw
a
parallel
sphaericall
line
with
great
ease
.
239
-
Prob.
103
To
measure
an
height
onely
by
help
of
ones
Hat
.
240
-
Prob.
104
To
take
an
height
with
two
strawes
.
240
-
In
Architecture
how
statues
or
other
things
in
high
buildings
shall
beare
a
proportion
to
the
eye
below
either
equall
,
double
,
&c.
242
-
Prob.
106
Of
deformed
figures
which
have
no
exact
proportion
,
where
to
place
the
eye
to
see
them
direct
.
243
-
Prob.
107
How
a
Cannon
that
hath
shot
may
be
covered
from
the
battery
of
the
Enemy
.
244
-
Prob.
108
Of
a
fine
Lever
,
by
which
one
man
alone
may
place
a
Cannon
upon
his
Carriage
.
245
-
Prob.
109
How
to
make
a
Clock
with
one
wheele
246
-
Of
Water-workes
.
-
Prob.
110
How
a
childe
may
draw
up
a
Hogshead
of
water
with
ease
.
247
-
Prob.
111
Of
a
Ladder
of
Cords
to
cary
in
ones
pocket
,
by
which
he
may
mount
a
wall
or
Tower
alone
.
248
-
Prob.
112
Of
a
marvelous
Pump
which
drawes
up
great
quantity
of
water
.
249
-
Prob.
113
How
naturally
to
cause
water
to
ascend
out
of
a
Pit
.
250
-
Prob.
114
How
to
cast
water
out
of
a
fountaine
very
high
.
252
-
Prob.
115
How
to
empty
the
water
of
a
Pit
by
help
of
a
Cisterne
.
253
-
Prob.
116
How
to
spout
out
water
very
high
.
253
-
Prob.
117
How
to
re-animate
simples
though
brought
a
thousand
miles
.
255
-
Prob.
118
How
to
make
a
perpetuall
motion
.
255
-
Prob.
119
Of
the
admirable
invention
of
making
the
Philosophers
Tree
,
which
one
may
see
to
grow
by
little
and
little
.
256
-
Prob.
120
How
to
make
the
representation
of
the
great
world
257
-
Prob.
121
Of
a
Cone
,
or
Pyramidall
figure
that
moves
upon
a
Table
258
-
Prob.
122
How
an
Anvill
may
be
cleaved
by
the
blow
of
a
Pistoll
.
258
-
Prob.
123
How
a
Capon
may
be
rosted
in
a
mans
travells
at
his
sa●●le-bowe
.
259
-
Prob.
124
How
a
Candle
may
be
made
to
burne
three
times
longer
than
usually
it
doth
259
-
Prob.
125
How
to
draw
Wine
out
of
water
260
-
Prob.
126
Of
two
Marmouzets
,
the
one
of
which
lights
a
Candle
,
and
the
other
blowes
it
out
.
261
-
Prob.
127
How
to
make
Wine
fresh
without
Ice
or
Snow
in
the
height
of
Summer
.
262
-
Prob.
128
To
make
a
Cement
which
lastes
as
marble
,
resisting
aire
and
water
.
262
-
Prob.
129
How
to
melt
metall
upon
a
shell
with
little
fire
.
263
-
Prob.
130
Of
the
hardning
of
Iron
and
steele
.
263
-
Prob.
131
To
preserve
fire
as
long
as
you
will
,
imitating
the
inextinguible
fire
of
the
Vestales
.
264
FINIS
.
Ad
Authorem
D.D.
Henricum
Van
Etenium
,
Alumnum
Academiae
Ponta
Mousson
.
ARdua
Walkeri
sileant
secreta
profundi
,
Desinat
occultam
carpere
Porta
viam
.
Itala
Cardani
mirata
est
Lampada
docti
Terra
,
Syracusium
Graecia
tota
senem
:
Orbi
terrarum
,
Ptolemaei
Clepsydra
toti
,
Rara
dioptra
Procli
,
mira
fuêre
duo
,
Anglia
te
foveat
doctus
Pont-Mousson
alumnum
:
Quidquid
naturae
,
qui
legis
,
hortus
habet
.
Docta
,
coronet
opus
doctum
,
te
sit
tua
docto
Digna
,
Syracusii
,
arca
,
corona
,
viri
.
Arca
Syracusiis
utinam
sit
plumbea
servis
,
Aurea
sed
dominis
,
aurea
tota
suis.
MATHEMATICAL
RECREATION
.
PROBLEM
I.
To
finde
a
number
thought
upon
.
BId
him
that
he
Quadruple
the
Number
thought
upon
,
that
is
,
multiply
it
by
4
,
and
unto
it
bid
him
to
adde
6
,
8
,
10
,
or
any
Number
at
pleasure
:
and
let
him
take
the
halfe
of
the
sum
,
then
ask
how
much
it
coms
to
,
for
then
if
you
take
away
half
the
number
from
it
which
you
willed
him
at
first
to
add
to
it
,
there
shall
remain
the
double
of
the
number
thought
upon
.
Example
The
Number
thought
upon
-
5
The
Quadruple
of
it
-
20
Put
8
unto
it
,
makes
-
28
The
halfe
of
it
is
-
14
Take
away
halfe
the
number
added
from
it
,
viz
4
,
the
rest
is
-
10
The
double
of
the
number
thought
upon
,
viz.
-
10
Another
way
to
finde
what
Number
was
thought
upon
.
BId
him
which
thinketh
double
his
Number
,
and
unto
that
double
adde
4
,
and
bid
him
multiply
that
same
product
by
5
,
and
unto
that
product
bid
him
adde
12
,
and
multiply
that
last
number
by
10
(
which
is
done
easily
by
setting
a
Cypher
at
the
end
of
the
number
)
then
ask
him
the
last
number
or
product
,
and
from
it
secretly
subtract
320
,
the
remainder
in
the
hundreth
place
,
is
the
number
thought
upon
.
Example
.
The
number
thought
upon
-
7
For
which
700
account
onely
but
the
number
of
the
hundreds
viz.
7.
so
have
you
the
number
thought
upon
.
His
double
-
14
For
which
700
account
onely
but
the
number
of
the
hundreds
viz.
7.
so
have
you
the
number
thought
upon
.
To
it
add
4
,
makes
-
18
For
which
700
account
onely
but
the
number
of
the
hundreds
viz.
7.
so
have
you
the
number
thought
upon
.
Which
multiplyed
by
5
makes
-
90
For
which
700
account
onely
but
the
number
of
the
hundreds
viz.
7.
so
have
you
the
number
thought
upon
.
To
which
add
12
makes
-
102
For
which
700
account
onely
but
the
number
of
the
hundreds
viz.
7.
so
have
you
the
number
thought
upon
.
This
multiplyed
by
10
which
is
only
by
adding
a
Cypher
to
it
,
makes
-
1020
For
which
700
account
onely
but
the
number
of
the
hundreds
viz.
7.
so
have
you
the
number
thought
upon
.
From
this
subtract
-
320
For
which
700
account
onely
but
the
number
of
the
hundreds
viz.
7.
so
have
you
the
number
thought
upon
.
Rest
-
700
For
which
700
account
onely
but
the
number
of
the
hundreds
viz.
7.
so
have
you
the
number
thought
upon
.
To
finde
numbers
conceived
upon
,
otherwise
than
the
former
.
BId
the
party
which
thinks
the
number
,
that
he
triple
his
thought
,
and
cause
him
to
take
the
half
of
it
:
(
if
it
be
odde
take
the
least
half
,
and
put
one
unto
it
:
)
then
will
him
to
triple
the
half
,
and
take
half
of
it
as
before
:
lastly
,
ask
him
how
many
nines
there
is
in
the
last
half
,
and
for
every
nine
,
account
four
in
your
memory
,
for
that
shall
shew
the
number
thought
upon
,
if
both
the
triples
were
even
:
but
if
it
be
odde
at
the
first
triple
,
and
ev●n
at
the
second
,
for
the
one
added
unto
the
least
halfe
keep
one
in
memory
:
if
the
first
triple
be
even
,
and
the
second
odde
,
for
the
one
added
unto
the
least
halfe
keepe
two
in
memory
;
lastly
,
if
at
both
times
in
tripling
,
the
numbers
be
odde
,
for
the
two
added
unto
the
least
halfes
,
keep
three
in
memory
,
these
cautions
observed
,
and
added
unto
as
many
fours
as
the
party
sayes
there
is
nines
contained
in
the
last
halfe
,
shall
never
fail
you
to
declare
or
discern
truly
what
number
was
thought
upon
.
Example
.
The
number
thought
upon
-
4
or
7
The
triple
-
12
or
21
The
half
thereof
6
or
10
,
one
put
to
it
makes
-
11
The
triple
of
the
halfe
-
18
or
33
The
halfe
9
or
1●
,
one
put
to
it
makes
-
17
The
number
of
nines
in
the
last
halfe
-
1
or
1
The
first
1.
representeth
the
4.
number
thought
upon
,
and
the
last
1.
with
the
caution
makes
7.
the
other
number
thought
upon
.
Note
.
Order
your
method
so
that
you
be
not
discovered
,
which
to
help
,
you
may
with
dexterity
and
industry
make
Additions
▪
Substractions
,
Multiplications
,
Divisions
,
&c.
and
instead
of
asking
how
many
nines
there
is
,
you
may
ask
how
many
eights
tens
,
&c.
there
is
,
or
subtract
8.10
.
&c.
from
the
Number
which
remains
,
for
to
finde
out
the
number
thought
upon
.
Now
touching
the
Demonstrations
of
the
former
directions
,
and
others
which
follow
,
they
depend
upon
the
2
,
7
,
8
,
and
9
,
Books
of
the
Elements
of
Euclide
:
upon
which
2.
Book
&
4.
proposition
this
may
bee
extracted
,
for
these
which
are
more
learned
for
the
finding
of
any
number
that
any
one
thinketh
on
.
Bid
the
party
that
thinks
,
that
he
break
the
number
thought
upon
into
any
two
parts
,
and
unto
the
Squares
of
the
parts
,
let
him
adde
the
double
product
of
the
parts
,
then
ask
what
it
amounteth
unto
,
so
the
root
Quadrat
shall
be
the
number
thought
upon
.
The
number
thought
upon
5
,
the
parts
suppose
3
and
2.
The
square
of
3
makes
-
9
the
sum
of
these
three
numbers
25
,
the
squa●e
Root
of
which
is
5
,
the
number
thought
upon
The
square
of
2
makes
-
4
the
sum
of
these
three
numbers
25
,
the
squa●e
Root
of
which
is
5
,
the
number
thought
upon
The
product
of
the
parts
.
viz.
3
by
2
makes
6
,
which
6
doubled
makes
-
12
the
sum
of
these
three
numbers
25
,
the
squa●e
Root
of
which
is
5
,
the
number
thought
upon
Or
more
compendiously
it
may
be
delivered
thus
.
Break
the
number
into
two
parts
,
and
to
the
product
of
the
parts
,
adde
the
square
of
half
the
difference
of
the
parts
,
then
the
Root
Quadrat
of
the
aggregate
is
halfe
the
number
conceived
.
EXAMINATION
.
THe
Problems
which
concern
Arithmetick
,
we
examine
not
,
for
these
are
easie
to
any
one
which
hath
read
the
grounds
and
principles
of
Arithmetick
,
but
we
especially
touch
upon
that
,
which
tends
to
the
speculations
of
Physick
,
Geometry
,
and
Optickes
,
and
such
others
which
are
of
more
difficulty
,
and
more
principally
to
be
examined
and
considered
.
PROBLEM
II.
How
to
represent
to
those
which
are
in
a
Cham●er
that
which
is
without
,
or
all
that
which
passeth
by
,
It
is
pleasant
to
see
the
beautifull
and
goodly
representation
of
the
heavens
intermixed
with
clouds
in
the
Horizon
,
upon
a
woody
scituation
,
the
motion
of
Birds
in
the
Aire
,
of
men
and
other
creatures
upon
the
ground
,
with
the
trembling
of
plants
,
tops
of
trees
,
and
such
like
:
for
every
thing
will
be
seen
within
even
to
the
life
,
but
inversed
:
notwithstanding
,
this
beautifull
paint
will
so
naturally
represent
it self
in
such
a
lively
Perspective
,
that
hardly
the
most
accurate
Painter
can
represent
the
like
.
But
here
note
,
that
they
may
be
represented
right
two
manner
of
wayes
;
first
,
with
a
concave
glasse
:
secondly
,
by
help
of
another
convex
glasse
,
disposed
or
placed
between
the
paper
and
the
other
Glasse
:
as
may
be
seen
here
by
the
figure
.
Now
I
will
add
here
only
by
passing
by
,
for
such
which
affect
Painting
and
portraiture
,
that
this
experiment
may
excellently
help
them
in
the
lively
painting
of
things
perspectivewise
,
as
Topographicall
cards
,
&c.
and
for
Philosophers
,
it
is
a
fine
secret
to
explain
the
Organ
of
the
sight
,
for
the
hollow
of
the
eye
is
taken
as
the
close
Chamber
,
the
Ball
of
the
Apple
of
the
eye
,
for
the
hole
of
the
Chamber
,
the
Crystaline
humor
at
the
small
of
the
Glasse
,
and
the
bottome
of
the
eye
,
for
the
Wall
or
leafe
of
paper
.
EXAMINATION
.
THe
species
being
pressed
together
or
contracted
doth
not
perform
it
upon
a
wall
,
for
the
species
of
any
thing
doth
represent
it selfe
not
only
in
one
hole
of
a
window
,
but
in
infinite
holes
,
even
unto
the
whole
Sphere
,
or
at
least
unto
a
Hemisphere
(
intellectuall
in
a
free
medium
)
if
the
beams
or
reflections
be
not
interposed
,
and
by
how
much
the
hole
is
made
less
to
give
passage
to
the
species
,
by
so
much
the
more
lively
are
the
Images
formed
.
In
convexe
,
or
concave
Glasses
the
Images
will
be
disproportionable
to
the
eye
,
by
how
much
they
are
more
concave
,
or
convexe
,
&
by
how
much
the
parts
of
the
image
comes
neer
to
the
Axis
,
for
these
that
are
neer
are
better
proportioned
then
these
which
are
farther
off
.
But
to
have
them
more
lively
and
true
,
according
to
the
imaginary
conicall
section
,
let
the
hole
be
no
greater
than
a
pins
head
made
upon
a
piece
of
thin
brasse
,
or
such
like
,
which
hole
represents
the
top
of
the
Cone
,
and
the
Base
thereof
the
term
of
the
species
:
this
practice
is
best
when
the
sun
shines
upon
the
hole
,
for
then
the
objects
which
are
opposite
to
that
plaine
will
make
two
like
Cones
,
and
will
lively
represent
the
things
without
in
a
perfect
inversed
perspective
,
which
drawn
by
the
Pensill
of
some
artificiall
Painter
,
turn
the
paper
upside
down
,
and
it
will
be
direct
and
to
the
life
.
But
the
apparences
may
be
direct
,
if
you
place
another
hole
opposite
unto
the
former
,
so
that
the
spectator
be
under
it
;
or
let
the
species
reflect
upon
a
concave
Glass
,
and
let
that
glas
reflect
upon
a
paper
or
some
white
thing
.
PROBLEM
III.
To
tell
how
much
waighs
the
blow
of
ones
fist
,
of
a
Mallet
,
Hatchet
,
or
such
like
,
or
resting
without
giving
the
blow
SCaliger
in
his
331
exercise
against
Cardan
,
relates
that
the
Mathematicians
of
Maximillian
the
Emperour
did
propose
upon
a
day
this
Question
,
and
promised
to
give
the
resolution
;
notwithstanding
●caliger
delivered
it
not
,
and
I
conceive
it
to
be
thus
.
Take
a
Balance
,
and
let
the
Fist
,
the
Mallet
,
or
Hatchet
rest
upon
the
scale
,
or
upon
the
beam
of
the
Balance
,
and
put
into
the
other
Scale
as
much
weight
as
may
counterpoyse
it
;
then
charging
or
laying
more
waight
into
the
Scale
,
and
striking
upon
the
other
end
,
you
may
see
how
much
one
blow
is
heavier
than
another
,
and
so
consequently
how
much
it
may
waigh
for
as
Aristotle
saith
,
The
motion
that
is
made
in
striking
adds
great
waight
unto
it
,
and
so
much
the
more
,
by
how
much
it
is
quicker
:
therefore
in
effect
,
if
there
were
placed
a
thousand
mallets
,
or
a
Thousand
pounde
waight
upon
a
stone
,
nay
,
though
it
were
exceedingly
pressed
down
by
way
of
a
Vice
,
by
Levers
,
or
other
Mechanick
Engine
,
it
would
be
nothing
to
the
rigor
and
violence
of
a
blow
.
Is
it
not
evident
that
the
edge
of
a
knife
laid
upon
butter
,
and
a
hatchet
upon
a
leafe
of
paper
,
without
striking
makes
no
impression
,
or
at
least
enters
not
;
but
striking
upon
the
wood
a
little
,
you
may
presently
see
what
effect
it
hath
,
which
is
from
the
quicknesse
of
the
motion
,
which
breaks
and
enters
without
resistance
,
if
it
be
extream
quick
,
as
experience
shews
us
in
the
blows
of
Arrows
,
of
Cannons
,
Thunder-boults
,
and
such
like
.
EXAMINATION
.
THis
Problem
was
extracted
from
Scaliger
,
who
had
it
from
Aristotle
,
but
somwhat
refractory
compiled
,
&
the
strength
of
the
effect
he
says
depends
only
in
the
violence
of
the
motion
;
then
would
it
follow
that
a
little
light
hammer
upon
a
piece
of
wood
being
quickly
caused
to
smite
,
would
give
a
greater
blow
,
and
do
more
hurt
than
a
great
sledge
striking
soft
;
this
is
absurd
,
and
contrary
to
experience
:
therefore
it
consists
not
totally
in
the
motion
,
for
if
two
severall
hammers
,
the
one
being
20
times
heavier
than
the
other
,
should
move
with
like
quickness
,
the
effect
would
be
much
different
,
there
is
then
some
thing
else
to
be
considered
besides
the
Motion
which
Scaliger
understood
not
,
for
if
one
should
have
asked
him
,
what
is
the
reason
that
a
stone
falling
from
a
window
to
a
place
neer
at
hand
,
is
not
so
forceable
as
if
it
fell
farther
〈◊〉
when
a
bullet
flying
out
of
a
peece
and
striking
the
mark
neer
at
hand
〈◊〉
not
make
such
an
effect
as
striking
〈…〉
that
Scaliger
and
〈…〉
this
subiect
▪
would
not
be
less
troubled
to
resolve
this
,
than
they
have
been
in
that
.
PROBLEM
IV.
How
to
break
a
staffe
which
is
laid
upon
two
Glasses
full
of
water
,
without
breaking
the
Glasses
,
spilling
the
water
,
or
upon
two
reeds
or
straws
without
breaking
of
them
.
In
like
manner
may
you
doe
upon
two
Reeds
,
held
with
your
hands
in
the
aire
without
breaking
them
▪
thence
Kitchin
boyes
often
break
bones
of
mutton
upon
their
hand
,
or
with
a
napkin
without
any
hurt
,
in
only
striking
upon
the
middle
of
the
bone
with
a
knife
.
Now
in
this
act
,
the
two
ends
of
the
staffe
in
breaking
slides
away
from
the
Glasses
,
upon
which
they
were
placed
;
hence
it
commeth
that
the
Glasses
are
no
wise
indangered
,
no
more
than
the
knee
upon
which
a
staffe
is
broken
,
forasmuch
as
in
breaking
it
presseth
not
:
as
Aristotle
in
his
Mechanick
Questions
observeth
.
EXAMINATION
.
IT
were
necessary
here
to
note
,
that
this
thing
may
be
experimented
,
first
,
without
Glasses
,
in
placing
a
small
slender
staffe
upon
two
props
,
and
then
making
tryall
upon
it
,
by
which
you
may
see
how
the
Staffe
will
either
break
,
bow
,
or
depart
from
his
props
,
and
that
either
directly
or
obliquely
:
But
why
by
this
violence
,
that
one
Staffe
striking
another
,
(
which
is
supported
by
two
Glasses
)
will
be
broken
without
offending
the
Glasses
,
is
as
great
a
difficulty
to
be
resolved
as
the
former
.
PROBLEM
V.
How
to
make
a
faire
Ge●graphic●ll
Card
in
a
Garden
Plot
,
fit
for
a
Prince
,
or
great
personage
.
IT
is
usuall
amongst
great
men
to
have
faire
Geographicall
Maps
▪
large
Cards
,
and
great
Globes
,
that
by
them
they
may
as
at
once
have
a
view
of
any
place
of
the
World
,
and
so
furnish
themselves
with
a
generall
knowledge
,
not
only
of
their
own
Kingdoms
form
,
scituation
,
longitude
,
latitude
,
&c.
but
of
all
other
places
in
the
whole
Universe
,
with
their
magnitudes
,
positions
,
Climats
,
and
distances
.
Now
I
esteem
that
it
is
not
unworthy
for
the
meditations
of
a
Prince
,
seeing
it
carries
with
it
many
profitable
and
pleasant
contentmen●s
:
if
such
a
Card
or
Map
by
the
advice
and
direction
of
an
able
Mathematician
were
Geographically
described
in
a
Garden
plot
form
,
or
in
some
other
convenient
place
,
and
instead
of
which
generall
description
might
particularly
and
artificially
be
prefigured
his
whole
Kingdoms
and
Dominions
,
the
Mountains
and
hils
being
raised
like
small
hillocks
with
turfs
of
earth
,
the
valleys
somwhat
concave
,
which
will
be
more
agreeable
and
pleasing
to
the
eye
,
than
the
description
in
plain
Maps
and
Cards
,
within
which
may
be
presented
the
Towns
,
Villages
,
Castles
,
or
other
remarkable
edifices
in
small
green
mo●●e
banks
,
or
spring-work
proportionall
to
the
pl●tform
,
the
Forrests
and
Woods
represented
according
to
their
form
and
capacity
,
with
herbs
and
stoubs
,
the
great
Rivers
,
Lakes
,
and
Ponds
to
dilate
themselves
according
to
their
course
from
some
artificiall
Fountain
made
in
the
Garden
to
passe
through
chanels
;
then
may
there
be
composed
walks
of
pleasure
,
ascents
,
places
of
repose
,
adorned
with
all
variety
of
delightfull
herbs
and
flowers
,
both
to
please
the
eye
or
other
senses
.
A
Garden
thus
accommodated
shall
farre
exceed
that
of
my
Lord
of
Verulams
specified
in
his
●ssayes
;
that
being
only
for
delight
and
pleasure
,
this
may
have
all
the
properties
of
that
,
and
also
for
singular
use
,
by
which
a
Prince
may
in
little
time
personally
visit
his
whole
Kingdom
,
and
in
short
time
know
them
distinctly
:
and
so
in
like
manner
may
any
particular
man
Geographically
prefigure
his
own
possession
or
heritage
.
PROBLEM
VI.
How
three
staves
,
knives
,
or
like
bodies
,
may
be
conceived
to
hang
in
the
aire
,
without
being
supported
by
any
thing
but
by
themselves
.
TAke
the
first
staffe
AB
,
raise
up
in
the
aire
the
end
B
,
and
upon
him
cros-wise
place
the
staffe
CB
,
then
lastly
,
in
Triangle
wise
place
the
third
staffe
EF
▪
in
such
manner
that
it
may
be
under
AB
,
and
yet
upon
CD
.
I
say
that
these
staves
so
disposed
cannot
fall
,
and
the
space
CBE
is
made
the
stronger
,
by
how
much
the
more
it
is
pressed
downe
,
if
the
staves
break
not
,
or
sever
themselves
from
the
triangular
forme
:
so
that
alwayes
the
Center
of
gravitie
be
in
the
Center
of
the
Triangle
:
for
AB
is
supported
by
EF
,
and
EF
is
held
up
by
CD
,
and
CD
is
kept
up
from
falling
by
AB
,
therefore
one
of
these
staves
cannot
fall
,
and
so
by
consequence
none
.
PROBLEM
VII
.
How
to
dispose
as
many
men
,
or
other
things
in
such
sort
,
that
rejecting
,
or
casting
away
the
6
,
9
,
10
part
,
unto
a
certain
number
,
there
shall
remaine
these
which
you
would
have
.
ORdinarily
the
proposition
is
delivered
in
this
wise
:
15
Christians
and
15
Turkes
being
at
Sea
in
one
Shippe
,
an
extreame
tempest
being
risen
,
the
Pilot
of
the
Shippe
saith
,
it
is
necessary
to
cast
over
board
halfe
of
the
number
of
Persons
to
disburthen
the
Shippe
,
and
to
save
the
rest
:
now
it
was
agreed
to
be
done
by
lot
,
and
therefore
they
consent
to
put
themselves
in
rank
,
counting
by
nine
and
nine
,
the
ninth
Person
should
alwayes
be
cast
into
the
Sea
,
untill
there
were
halfe
throwne
over
board
;
Now
the
Pilote
being
a
Christian
indeavoured
to
save
the
Christians
,
how
ought
he
therefore
to
dispose
the
Christians
,
that
the
lot
might
fall
alwayes
upon
the
Turkes
,
and
that
none
of
the
Christians
be
in
the
ninth
place
?
The
resolution
is
ordinarily
comprehended
in
this
verse
.
Populeam
virgam
mater
regina
ferebat
.
For
having
respect
unto
the
vowels
,
making
a
one
,
e
two
,
i
three
,
o
foure
,
and
u
five
:
o
the
first
vowell
in
the
first
word
sheweth
that
there
must
be
placed
4.
Christians
;
the
next
vowel
u
,
signifieth
that
next
unto
the
4.
Christians
must
be
placed
5
Turkes
,
and
so
to
place
both
Christians
and
Turkes
according
to
the
quantity
and
value
of
the
vowels
in
the
words
of
the
verse
,
untill
they
be
all
placed
:
for
then
counting
from
the
first
Christian
that
was
placed
,
unto
the
ninth
,
the
lot
will
fall
upon
a
Turk
,
and
so
proceed
.
And
here
may
be
further
noted
that
this
Probleme
is
not
to
be
limited
,
seeing
it
extends
to
any
number
and
order
whatsoever
,
and
may
many
wayes
be
usefull
for
Captaines
,
Magistrates
,
or
others
which
have
divers
persons
to
punish
,
and
would
chastise
chiefely
the
unruliest
of
them
,
in
taking
the
10
,
20
,
or
100.
person
,
&c.
as
we
reade
was
commonly
practised
amongst
the
ancient
Romans
:
herefore
to
apply
a
generall
rule
in
counting
the
third
,
4
,
9
,
10
,
&c.
amongst
30
,
40
,
50
,
persons
,
and
more
or
lesse
;
this
is
to
be
observed
,
take
as
many
units
as
there
are
persons
,
and
dispose
them
in
order
privately
:
as
for
example
,
let
24
men
be
proposed
to
have
committed
some
outrage
,
6
of
them
especially
are
found
accessary
:
and
let
it
be
agreed
that
counting
by
8
and
8
the
eight
man
should
be
alwayes
punished
.
Take
therefore
first
24
units
,
or
upon
a
piece
of
paper
write
down
24
cyphers
,
and
account
from
the
beginning
to
the
eighth
,
which
eighth
mark
,
and
so
continue
counting
alwayes
marking
the
eighth
,
untill
you
have
markt
6
,
by
which
you
may
easily
perceive
how
to
place
those
6
men
that
are
to
be
punished
,
and
so
of
others
.
It
is
supposed
that
Josephus
the
Author
of
the
Jewish
History
escaped
the
danger
of
death
by
help
of
this
Problem
;
for
a
worthy
Author
of
beliefe
reports
in
his
eighth
chapter
of
the
third
Book
of
the
destruction
of
Jerusalem
,
that
the
Town
of
Jotapata
being
taken
by
main
force
by
Vespatian
,
Josephus
being
Governour
of
that
Town
,
accompanyed
with
a
Troop
of
forty
Souldiers
,
hid
themselves
in
a
Cave
,
in
which
they
resolved
rather
to
famish
than
to
fall
into
the
hands
of
Vespatian
:
and
with
a
bloudy
resolution
in
that
great
distresse
would
have
butchered
one
another
for
sustenance
,
had
not
Josephus
perswaded
them
to
die
by
lot
and
order
,
upon
which
it
should
fall
:
Now
seeing
that
Josephus
did
save
himselfe
by
this
Art
,
it
is
thought
that
his
industry
was
exercised
by
the
helpe
of
this
Problem
,
so
that
of
the
40
persons
which
he
had
,
the
third
was
alwayes
killed
.
Now
by
putting
himselfe
in
the
16
or
31
place
he
was
saved
,
and
one
with
him
which
he
might
kill
,
or
easily
perswade
to
yeild
unto
the
Romans
.
PROBLEM
.
VIII
.
Three
things
,
and
three
persons
proposed
,
to
finde
which
of
them
hath
either
of
these
three
things
.
LEt
the
three
things
be
a
Ring
,
a
piece
of
Gold
,
and
a
piece
of
Silver
,
or
any
other
such
like
,
and
let
them
be
known
privately
to
your self
by
these
three
Vowels
a
,
e
,
i
,
or
let
there
be
three
persons
that
have
different
names
,
as
Ambrose
,
Edmond
,
and
John
,
which
privately
you
may
note
or
account
to
your selfe
once
known
by
the
aforesaid
Vowels
,
which
signifie
for
the
first
vowel
1
,
for
the
second
vowell
2
,
for
the
third
vowell
3.
Now
if
the
said
three
persons
should
by
the
mutuall
consent
of
each
other
privately
change
their
names
,
it
is
most
facill
by
the
course
and
excellencie
of
numbers
,
distinctly
to
declare
each
ones
name
so
interchanged
,
or
if
three
persons
in
private
,
the
one
should
take
a
Ring
,
the
other
a
piece
of
Gold
,
and
the
third
should
take
a
piece
of
Silver
;
it
is
easie
to
finde
which
hath
the
Gold
,
the
Silver
,
or
the
Ring
,
and
it
is
thus
done
.
Take
30
or
40
Counters
(
of
which
there
is
but
24
necessary
)
that
so
you
may
conceale
the
way
the
better
,
and
lay
them
down
before
the
parties
,
and
as
they
sit
or
stand
,
give
to
the
first
1.
Counter
,
which
signifieth
a
,
the
first
vowell
;
to
the
second
2.
Counters
,
which
represent
e
,
the
second
vowel
;
and
to
the
third
3.
Counters
,
which
stand
for
i
,
the
third
vowell
:
then
leaving
the
other
Counters
upon
the
Table
,
retire
apart
,
and
bid
him
which
hath
the
Ring
,
take
as
many
Counters
as
you
gave
him
,
and
he
that
hath
the
Gold
,
for
every
one
that
you
gave
him
,
let
him
take
2
,
and
he
that
hath
the
Silver
for
every
one
that
you
gave
him
,
let
him
take
4.
this
being
done
,
consider
to
whom
you
gave
one
Counter
,
to
whom
two
,
and
to
whom
three
;
and
mark
what
number
of
Counters
you
had
at
the
first
,
for
there
are
necessarily
but
24.
as
was
said
before
,
the
surpluse
you
may
privately
reject
.
And
then
there
will
be
left
either
1.2.3.5.6
or
7.
and
no
other
number
can
remaine
,
which
if
there
be
,
then
they
have
failed
in
taking
according
to
the
directions
delivered
:
but
if
either
of
these
numbers
do
remaine
,
the
resolution
will
be
discovered
by
one
of
these
6
words
following
,
which
ought
to
be
had
in
memory
,
viz.
Salve
,
|
certa
,
|
anima
,
|
semita
,
|
vita
,
|
quies·
|
1.
|
2.
|
3.
|
5.
|
6.
|
7.
|
As
suppose
5.
did
remaine
,
the
word
belonging
unto
it
is
semita
,
the
vowels
in
the
first
two
syllables
are
e
and
i
,
which
sheweth
according
to
the
former
directions
,
that
to
whom
you
gave
2
Counters
,
he
hath
the
Ring
(
seeing
it
is
the
second
vowell
represented
by
two
as
before
)
and
to
whom
you
gave
the
3.
Counters
,
he
hath
the
Gold
,
for
that
i
represents
the
third
vowel
,
or
3.
in
the
former
direction
,
and
to
whom
you
gave
one
Counter
,
he
hath
the
Silver
,
and
so
of
the
rest
:
the
variety
of
changes
,
in
which
exercise
,
is
laid
open
in
the
Table
following
.
rest
|
men
|
hid
|
rest
|
men
|
hid
|
1
|
1
|
a
|
5
|
1
|
|
2
|
e
|
2
|
|
3
|
i
|
3
|
|
2
|
1
|
e
|
6
|
1
|
|
2
|
a
|
2
|
|
3
|
i
|
3
|
|
3
|
1
|
a
|
7
|
1
|
|
2
|
i
|
2
|
|
3
|
e
|
3
|
|
This
feat
may
be
done
also
without
the
former
words
by
help
of
the
Circle
A.
for
having
divided
the
Circle
into
6
parts
,
write
1.
within
and
1.
without
,
2.
within
and
5.
without
,
&c.
the
first
1.2.3
.
which
are
within
with
the
numbers
over
them
,
belongs
to
the
upper
semicircle
;
the
other
numbers
both
within
and
without
,
to
the
under
semicircle
;
now
if
in
the
action
there
remaineth
such
a
number
which
may
be
found
in
the
upper
semicircle
without
,
then
that
which
is
opposite
within
shews
the
first
,
the
next
is
the
second
,
&c.
as
if
5
remains
,
it
shews
to
whom
he
gave
2
,
he
hath
the
Ring
;
to
whom
you
gave
●
,
he
hath
the
Gold
,
&c.
But
if
the
remainder
be
in
the
under
semicircle
,
that
which
is
opposite
to
it
is
the
first
;
the
next
backwards
towards
the
right
hand
is
the
second
;
as
if
3
remains
,
to
whom
you
gave
1
he
hath
the
Ring
,
he
that
had
3
he
had
the
Gold
,
&c.
PROBLEM
IX
.
How
to
part
a
Vessel
which
is
full
of
wine
conteining
eight
pints
into
two
equall
parts
,
by
two
other
vessels
which
conteine
as
much
as
the
greater
vessell
;
as
the
one
being
5
pints
,
and
the
other
3
pints
.
LEt
the
three
vessels
be
represented
by
ABC
,
A
being
full
,
the
other
two
being
empty
;
first
,
poure
out
A
into
B
until
it
be
full
,
so
there
will
be
in
B
5
pints
,
and
in
A
but
3
pints
:
then
poure
out
of
B
into
C
untill
it
be
full
:
so
in
C
shall
be
3
pints
,
in
B
2
pints
,
and
in
A
3
pints
,
then
poure
the
wine
which
is
in
C
into
A
,
so
in
A
will
be
6
pints
,
in
B
2
pints
,
and
in
C
nothing
:
then
poure
out
the
wine
which
is
in
B
into
the
pot
C
,
so
in
C
there
is
now
2
pints
,
in
B
nothing
,
and
in
A
6
pints
,
.
Lastly
,
poure
out
of
A
into
B
untill
it
be
full
,
so
there
will
be
now
in
A
only
1
pint
in
B
5
pints
,
and
in
C
2
pints
.
But
it
is
now
evident
,
that
if
from
B
you
poure
in
unto
the
pot
C
untill
it
be
full
,
there
wil
remain
in
B
4
pints
,
and
if
that
which
is
in
C
,
viz.
3
pints
be
poured
into
the
vessell
A
,
which
before
had
1
pint
,
there
shall
be
in
the
vessel
A
,
but
halfe
of
its
liquor
that
was
in
it
at
the
first
,
viz.
4
pints
as
was
required
.
Otherwise
poure
out
of
A
into
C
untill
it
be
full
,
which
pour
into
B
,
then
poure
out
of
A
into
C
again
untill
it
be
full
,
so
there
is
now
in
A
onely
2
pints
,
in
B
3
,
and
in
C
3
,
then
pour
from
C
into
B
untill
it
be
full
,
so
in
C
there
is
now
but
1
pint
,
5
in
B
,
and
2
in
A
:
poure
all
that
is
in
B
into
A
,
then
poure
the
wine
which
is
in
C
into
B
,
so
there
is
in
C
nothing
,
in
B
onely
1
pint
,
and
in
7
A
7
pints
:
Lastly
,
out
of
A
fill
the
pot
C
,
so
there
will
remain
in
A
4
pints
,
or
be
but
halfe
full
:
then
if
the
liquor
in
C
be
poured
into
B
,
it
will
be
the
other
half
.
In
like
manner
might
be
taken
the
half
of
a
vessell
which
conteins
12
pints
,
by
having
but
the
measures
5
and
7
,
or
5
and
8.
Now
such
others
might
be
proposed
,
but
we
omit
many
,
in
one
and
the
same
nature
.
PROBLEM
.
X.
To
make
a
stick
stand
upon
the
tip
of
ones
finger
,
without
falling
.
FAsten
the
edges
of
two
knives
or
such
like
of
equall
poise
,
at
the
end
of
the
stick
,
leaning
out
somewhat
from
the
stick
,
so
that
they
may
counterpoise
one
another
;
the
stick
being
sharp
at
the
end
,
and
held
upon
the
top
of
the
finger
,
will
there
rest
without
supporting
:
if
it
fall
,
it
must
fall
together
,
and
that
perpendicular
or
plumb-wise
,
or
it
must
fall
side-wise
or
before
one
another
;
in
the
first
manner
it
cannot
:
for
the
Centre
of
gravitie
is
supported
by
the
top
of
the
finger
:
and
seeing
that
each
part
by
the
knives
is
counterpoised
,
it
cannot
fall
sidewise
,
therefore
it
can
fall
no
wise
.
In
like
manner
may
great
pieces
of
Timber
,
as
Joists
,
&c
be
supported
,
if
unto
one
of
the
ends
be
applied
convenient
proportionall
counterpoises
,
yea
a
Lance
or
Pike
,
may
stand
perpendicular
in
the
Aire
upon
the
top
of
ones
finger
:
or
placed
in
the
midst
of
a
Court
by
help
of
his
Centre
of
gravitie
.
EXAMINATION
.
THis
Proposition
seems
doubtfull
;
for
to
imagine
absolutely
,
that
a
Pike
,
or
such
like
,
armed
with
two
Knives
,
or
other
things
,
shall
stand
upright
in
the
Aire
,
and
so
remain
without
any
other
support
,
seeing
that
all
the
parts
have
an
infinite
difference
of
propensity
to
fall
;
and
it
is
without
question
that
a
staff
so
accommodated
upon
his
Centre
of
gravity
,
but
that
it
may
incline
to
some
one
part
without
some
remedy
be
applied
,
and
such
as
is
here
specified
in
the
Probleme
will
not
warrant
the
thing
,
nor
keep
it
from
falling
;
and
if
more
Knives
should
be
placed
about
it
,
it
should
cause
it
to
fall
more
swiftly
,
forasmuch
as
the
superiour
parts
(
by
reason
of
the
Centricall
motion
)
is
made
more
ponderous
,
and
therefore
lesse
in
rest
.
To
place
therefore
this
prop
really
,
let
the
two
Knives
,
or
that
which
is
for
counterpoise
,
be
longer
always
then
the
staffe
,
and
so
it
will
hang
together
as
one
body
:
and
it
will
appear
admirable
if
you
place
the
Centre
of
gravity
,
neer
the
side
of
the
top
of
the
finger
or
point
;
for
it
will
then
hang
Horizontall
,
and
seem
to
hang
onely
by
a
touch
,
yet
more
strange
,
if
you
turn
the
point
or
top
of
the
finger
upside
down
.
PROBLEM
XI
.
How
a
milstone
or
other
Ponderosity
,
may
be
supported
by
a
small
needle
,
without
breaking
o●
any
wise
bowing
the
same
.
LEt
a
needle
be
set
perpendicular
to
the
Horizon
,
and
the
center
of
gravitie
of
the
stone
be
placed
on
the
top
of
the
needle
:
it
is
evident
that
the
stone
cannot
fall
,
forasmuch
as
it
hangs
in
aequilibra
,
or
is
counterpoysed
in
all
parts
alike
;
and
moreover
it
cannot
bow
the
needle
more
on
the
one
side
then
on
the
other
,
the
needle
will
not
therefore
be
either
broken
or
bowed
;
if
otherwise
then
the
parts
of
the
needle
must
penetrate
and
sinke
one
with
another
:
that
which
is
absurd
and
impossible
to
nature
;
therefore
it
shall
be
supported
.
The
experiments
which
are
made
upon
trencher
plates
,
or
such
like
lesser
thing
doth
make
it
most
credible
in
greater
bodies
.
But
here
especially
is
to
be
noted
,
that
the
needle
ought
to
be
uniforme
in
matter
and
figure
,
and
that
it
be
erected
perpendicular
to
the
Horizon
,
and
lastly
,
that
the
Center
of
gravity
be
exactly
found
.
PROBLEM
XII
.
To
make
three
Knives
hang
and
move
upon
the
point
of
a
Needle
.
FIt
the
three
Knives
in
form
of
a
Ballance
,
and
holding
a
Needle
in
your
hand
,
and
place
the
back
of
that
,
Knife
which
lyes
cross-wise
to
the
other
two
,
upon
the
point
of
the
Needle
:
as
the
figure
here
sheweth
you
;
for
then
in
blowing
softly
upon
them
,
they
will
easily
turne
and
move
upon
the
point
of
the
Needle
with
●ou
falling
.
PROBLEM
XIII
.
To
finde
the
weight
of
Smoak
,
which
is
exhaled
of
any
combustible
body
whatsoever
.
LEt
it
be
supposed
that
a
great
heape
of
Fagots
,
or
a
load
of
straw
weighing
500
pound
should
be
fired
,
it
is
evident
that
this
grosse
substance
will
be
all
inverted
into
smoak
and
ashes
:
now
it
seems
that
the
smoak
weighs
nothing
;
seeing
it
is
of
a
thin
substance
now
dilated
in
the
Aire
,
notwithstanding
if
it
were
gathered
and
reduced
into
the
thickest
that
it
was
at
first
,
it
would
be
sensibly
weighty
:
weigh
therefore
the
ashes
which
admit
50
pound
,
now
seeing
that
the
rest
of
the
matter
is
not
lost
,
but
is
exhaled
into
smoake
,
it
must
necessarily
be
,
that
the
rest
of
the
weight
(
to
wit
)
450
pound
,
must
be
the
weight
of
the
smoak
required
.
EXAMINATION
.
NOw
although
it
be
thus
delivered
,
yet
here
may
be
noted
,
that
a
ponderosity
in
his
own
medium
is
not
weighty
:
for
things
are
said
to
be
weighty
,
when
they
are
out
of
their
place
,
or
medium
,
and
the
difference
of
such
gravity
,
is
according
to
the
motion
:
the
smoak
therefore
certainly
is
light
being
in
its
true
medium
(
the
aire
,
)
if
it
should
change
his
medium
,
then
would
we
change
our
discourse
.
PROBLEM
XVI
.
Many
things
being
disposed
circular
,
(
or
otherwise
)
to
finde
which
of
them
,
any
one
thinks
upon
▪
SUppose
that
having
ranked
10
things
,
as
ABCDEFGHIK
,
Circular
(
as
the
figure
sheweth
)
and
that
one
had
touched
or
thought
upon
G
,
which
is
the
7
:
ask
the
party
at
what
letter
he
would
begin
to
account
(
for
account
he
must
,
otherwise
it
cannot
be
done
)
which
suppose
,
at
E
which
is
the
5
place
,
then
add
secretly
to
this
5
,
10
(
which
is
the
number
of
the
Circle
)
and
it
makes
15
,
bid
him
account
15
backward
from
E
,
beginning
his
account
with
that
number
hee
thought
upon
,
so
at
E
he
shal
account
to
himself
7
,
at
D
account
8
,
at
C
account
9
,
&c.
So
the
account
of
15
wil
exactly
fall
upon
G
,
the
thing
or
number
thought
upon
:
and
so
of
others
:
but
to
conceal
it
the
more
,
you
may
will
the
party
from
E
to
account
25
,
35
,
&c.
and
it
will
be
the
same
.
There
are
some
that
use
this
play
at
Cards
,
turned
upside
downe
,
as
the
ten
simple
Cards
,
with
the
King
and
Queen
,
the
King
standing
for
12
,
and
the
Queene
for
11
,
and
so
knowing
the
situation
of
the
Cards
:
and
thinking
a
certain
houre
of
the
day
:
cause
the
party
to
account
from
what
Card
he
pleaseth
:
with
this
Proviso
,
that
when
you
see
where
he
intends
to
account
,
set
12
to
that
number
,
so
in
counting
as
before
,
the
end
of
the
account
shall
fall
upon
the
Card
:
which
shall
denote
or
shew
the
houre
thought
upon
,
which
being
turned
up
will
give
grace
to
the
action
,
and
wonder
to
those
that
are
ignorant
in
the
cause
.
PROBLEM
XV.
How
to
make
a
Door
or
Gate
,
which
shall
open
on
both
sides
.
ALL
the
skill
and
subtilty
of
this
,
rests
in
the
artificiall
disposer
of
foure
plates
of
Iron
,
two
at
the
higher
end
,
and
two
at
the
lower
end
of
the
Gate
:
so
that
one
side
may
move
upon
the
hooks
or
hinges
of
the
Posts
,
and
by
the
other
end
may
be
made
fast
to
the
Gate
,
and
so
moving
upon
these
hinges
,
the
Gate
will
open
upon
one
side
with
the
aforesaid
plates
,
or
hooks
of
Iron
:
and
by
help
of
the
other
two
plates
,
will
open
upon
the
other
side
.
PROBLEM
XVI
.
To
shew
how
a
Ponderosity
,
or
heavy
thing
,
may
be
supported
upon
the
end
of
a
staffe
(
or
such
like
)
upon
a
Table
,
and
nothing
holding
or
touching
it
.
TAke
a
pale
which
hath
a
handle
,
and
fill
it
full
of
water
(
or
at
pleasure
:
)
then
take
a
staffe
or
stick
which
may
not
rowle
upon
the
Table
as
EC
,
and
place
the
handle
of
the
Pale
upon
the
staffe
;
then
place
another
staffe
,
or
stick
,
under
the
staffe
CE
,
which
may
reach
from
the
bottom
of
the
Pale
unto
the
former
staffe
CE
,
perpendicular
wise
:
which
suppose
FG
,
then
shall
the
Pale
of
water
hang
without
falling
,
for
if
it
fall
it
must
fall
perpendicularly
,
or
plumbe
wise
:
and
that
cannot
be
seeing
the
staffe
CE
supports
it
,
it
being
parallel
to
the
Horizon
and
susteined
by
the
Table
,
and
it
is
a
thing
admirable
that
if
the
staffe
CE
were
alone
from
the
table
,
and
that
end
of
the
staffe
which
is
upon
the
Table
were
greater
and
heavier
than
the
other
:
it
would
be
constrained
to
hang
in
that
nature
.
EXAMINATION
.
NOw
without
some
experience
of
this
Probleme
,
a
man
would
acknowledge
either
a
possibility
or
impossibity
;
therefore
it
is
that
very
touchstone
of
knowledge
in
any
thing
,
to
discourse
first
if
a
thing
be
possible
in
nature
,
and
then
if
it
can
be
brought
to
experience
and
under
sence
without
seeing
it
done
.
At
the
first
,
this
proposition
seems
to
be
absurd
,
and
impossible
.
Notwithstanding
,
being
supported
with
two
sticks
,
as
the
figure
declareth
,
it
is
made
facile
:
for
the
Horizontall
line
to
the
edge
of
the
Table
,
is
the
Centre
of
motion
;
and
passeth
by
the
Centre
of
gravity
,
which
necessarily
supporteth
it
.
PROBLEM
XVII
.
Of
a
deceitfull
Bowle
to
play
withall
.
MAke
a
hole
in
one
side
of
the
Bowle
,
and
cast
molten
Lead
therein
,
and
then
make
up
the
hole
close
,
that
the
knavery
or
deceit
be
not
perceived
:
you
will
have
pleasure
to
see
,
that
notwithstanding
the
Bowle
is
cast
directly
to
the
play
,
how
it
wil
turn
away
side-wise
:
for
that
on
that
part
of
the
Bowle
which
is
heavier
upon
the
one
side
then
on
the
other
,
it
never
will
go
truly
right
,
if
artificially
it
be
not
corrected
;
which
will
hazard
the
game
to
those
which
know
it
not
:
but
if
it
be
known
that
the
leady
side
in
rolling
be
always
under
or
above
,
it
may
go
indifferently
right
;
if
otherwise
,
the
weight
will
carry
it
always
side-wise
.
PROBLEM
.
XVIII
.
To
part
an
Apple
into
2.4
.
or
8.
like
parts
,
without
breaking
the
Rinde
.
PAsse
a
needle
and
threed
under
the
kinde
of
the
Apple
,
and
then
round
it
with
divers
turnings
,
untill
you
come
to
the
place
where
you
began
:
then
draw
out
the
threed
gently
,
and
part
the
Apple
into
as
many
parts
as
you
think
convenient
:
and
so
the
parts
may
be
taken
out
between
the
parting
of
the
Rind
,
and
the
rind
remaining
alwayes
whole
.
PROBLEM
XIX
.
To
finde
a
number
thought
upon
without
asking
of
any
question
,
certaine
operations
being
done
.
BId
him
adde
to
the
number
thought
(
as
admit
15
)
halfe
of
it
,
if
it
may
be
,
if
not
the
greatest
halfe
that
exceeds
the
other
but
by
an
unite
,
which
is
8
;
and
it
makes
23.
Secondly
,
unto
this
23.
adde
the
halfe
of
it
if
it
may
be
,
if
not
,
the
greatest
halfe
,
viz.
12.
makes
35.
in
the
meane
time
,
note
that
if
the
number
thought
upon
cannot
be
halfed
at
the
first
time
,
as
here
it
cannot
,
then
for
it
keep
3
in
the
memory
,
if
at
the
second
time
it
will
not
be
equally
halfed
,
reserve
2
in
memory
,
but
if
at
both
times
it
could
not
be
equally
halved
,
then
may
you
together
reserve
five
in
memory
:
this
done
,
cause
him
from
the
last
summe
,
viz.
35.
to
subtract
the
double
of
the
number
thought
,
viz.
30.
rest
5.
will
him
to
take
the
halfe
of
that
if
he
can
,
if
not
,
reject
1.
and
then
take
the
halfe
of
the
rest
,
which
keep
in
your
memory
:
then
will
him
to
take
the
halfe
againe
if
he
can
,
if
not
,
take
one
from
it
,
which
reserve
in
your
memory
,
and
so
perpetually
halveing
untill
1.
remaine
:
for
then
mark
how
many
halfes
there
were
taken
,
for
the
first
halfe
account
2
,
for
the
second
4
,
for
the
third
8
,
&c.
and
adde
unto
those
numbers
the
ones
which
you
reserved
in
memory
,
so
there
being
5
remaining
in
this
proposition
,
there
were
2
halfings
:
for
which
last
!
account
4
,
but
because
it
could
not
exactly
be
halved
without
rejecting
of
1.
I
adde
the
1
therefore
to
this
4
,
makes
5
,
which
halfe
or
summe
alwayes
multiplied
by
4
,
makes
20.
from
which
subtract
the
first
3
and
2
,
because
the
halfe
could
not
be
formerly
added
,
leaves
15
,
the
number
thought
upon
.
Other
Examples
.
The
number
thought
upon
.
The
number
thought
-
12
The
halfe
of
it
-
6
The
summe
-
18
The
halfe
of
it
-
9
The
summe
of
it
-
27
The
double
of
the
number
,
-
24
Which
taken
away
,
rests
-
3
The
halfe
of
it
-
1
For
which
account
-
2
and
1
put
to
it
because
the
-
3
could
not
be
halfed
,
makes
-
3
this
multiplied
by
4
makes
-
12
The
number
thought
-
79
The
greatest
halfe
-
40
3
The
summe
-
119
The
greatest
halfe
of
which
is
-
60
2
The
summe
of
it
is
-
179
The
double
of
79
is
-
158
Which
taken
from
it
,
rests
-
21
The
lesser
half
10.
which
halve
:
-
The
halfe
of
this
is
5
which
makes
-
The
half
of
this
is
2
which
is
10
-
The
half
of
this
is
1
,
with
10
and
11
is
21.
-
this
21
which
is
the
double
of
the
last
halfe
with
the
remainder
being
multiplied
by
4.
makes
84
,
from
which
take
the
aforesaid
3
and
2
,
●●st
79
,
the
number
thought
upon
.
-
PROBLEM
.
XX.
How
to
make
an
uniforme
,
&
an
inflexible
body
,
to
passe
through
two
small
holes
of
divers
formes
,
as
one
being
circular
,
and
the
other
square
,
Quadrangular
,
and
Triangular-wise
,
yet
so
that
the
holes
shall
be
exactly
filled
.
THis
Probleme
is
extracted
from
Geometricall
observations
,
and
seemes
at
the
first
somewhat
obscure
,
yet
that
which
may
be
extracted
in
this
nature
,
will
appeare
more
difficult
and
admirable
.
Now
in
all
Geometricall
practises
,
the
lesser
or
easier
Problemes
do
alwayes
make
way
to
facilitate
the
greater
:
and
the
aforesaid
Probleme
is
thus
resolved
.
Take
a
Cone
or
round
Pyramide
,
and
make
a
Circular
hole
in
some
board
,
or
other
hard
material
,
which
may
be
equall
to
the
bases
of
the
Cone
,
and
also
a
Triangular
hole
,
one
of
whose
sides
may
be
equall
to
the
Diameter
of
the
circle
,
and
the
other
two
sides
equall
to
the
length
of
the
Cone
:
Now
it
is
most
evident
,
that
this
Conicall
or
Pyramidall
body
,
will
fill
up
the
Circular
hole
,
and
being
placed
side-wise
will
fill
up
the
Triangular
hole
.
Moreover
,
if
you
cause
a
body
to
be
turned
,
which
may
be
like
to
two
Pyramides
conjoyned
,
then
if
a
Circular
hole
be
made
,
whose
Diameter
is
equal
to
the
Diameter
of
the
Cones
conjoyned
,
and
a
Quadrangular
hole
,
whose
sloping
sides
be
equall
to
the
length
of
each
side
of
the
Pyramide
,
and
the
breadth
of
the
hol
equal
to
the
Diameter
of
the
Circle
,
this
conjoyned
Pyramide
shall
exactly
fill
both
the
Circular
hole
,
and
also
the
Quadrangle
hole
.
PROBLEM
.
XXI
.
How
with
one
uniforme
body
or
such
like
to
fill
three
severall
holes
:
of
which
the
one
is
round
,
the
other
a
just
square
,
and
the
third
an
ovall
forme
?
THis
Proposition
seemes
more
subtill
then
the
former
,
yet
it
may
be
practised
two
wayes
:
for
the
first
,
take
a
Cylindricall
body
as
great
or
little
as
you
please
:
Now
it
is
evident
that
it
will
fill
a
Circular
hole
,
which
is
made
equall
to
the
basis
of
it
,
if
it
be
placed
downe
right
,
and
will
also
fill
a
long
square
;
whose
sides
are
equall
unto
the
Diameter
and
length
of
the
Cylinder
,
and
acording
to
Pergeus
,
Archimedes
,
&c.
in
their
Cylindricall
demonstrations
,
a
true
Ovall
is
made
when
a
Cylinder
is
cut
slopewise
,
therefore
if
the
oval
have
breadth
equall
unto
the
Diameter
of
the
Basis
of
the
Cylinder
,
&
any
length
whatsoever
:
the
Cylinder
being
put
into
his
owne
Ovall
hole
shall
also
exactly
fill
it
.
The
second
way
is
thus
,
make
a
Circular
hole
in
some
board
,
&
also
a
square
hole
,
the
side
of
which
Square
may
be
equall
to
the
Diameter
of
the
Circle
:
and
lastly
,
make
a
hole
Oval-wise
,
whose
breadth
may
be
equal
unto
the
diagonall
of
the
Square
;
then
let
a
Cylindricall
body
be
made
,
whose
Basis
may
be
equall
unto
the
Circle
,
and
the
length
equall
also
to
the
same
:
Now
being
placed
downe
right
shall
fall
in
the
Circle
,
and
flat-wise
will
fit
the
Square
hole
,
and
being
placed
sloping-wise
will
fill
the
Ovall
.
EXAMINATION
.
YOu
may
note
upon
the
last
two
Problemes
farther
,
that
if
a
Cone
be
cut
Ecliptick-wise
,
it
may
passe
through
an
Issoc●●●
Triangle
through
many
Scalen
Triangles
,
and
through
an
Ellipsis
;
and
if
there
be
a
Cone
cut
scalen-wise
,
it
will
passe
through
all
the
former
,
only
for
the
Ellipsis
place
a
Circle
:
and
further
,
if
a
solid
colume
be
cut
Ecliptick-wise
it
may
fill
a
Circle
,
a
Square
,
divers
Parallelogrammes
,
and
divers
Ellipses
,
which
have
different
Diameters
.
PROBLEM
XXII
.
To
finde
a
number
thought
upon
●fter
another
manner
,
then
what
is
formerly
delivered
BId
him
that
he
multiply
the
number
thought
upon
,
by
what
number
he
pleaseth
,
then
bid
him
divide
that
product
by
any
other
number
,
and
then
multiply
that
Quotient
by
some
other
number
;
and
that
product
againe
divide
by
some
other
,
and
so
as
often
as
he
will
:
and
here
note
,
that
he
declare
or
tell
you
by
what
number
he
did
multiply
&
divide
Now
in
the
same
time
take
a
number
at
pleasure
,
and
secretly
multiply
and
divide
as
often
as
he
did
:
then
bid
him
divide
the
last
number
by
that
which
he
thought
upon
.
In
like
manner
do
yours
privately
,
then
will
the
Quotient
of
your
divisor
be
the
same
with
his
,
a
thing
which
seemes
admirable
to
those
which
are
ignorant
of
the
cause
.
Now
to
have
the
number
thought
upon
without
seeming
to
know
the
last
Quotient
,
bid
him
adde
the
number
thought
upon
to
it
,
and
aske
him
how
much
it
makes
:
then
subtract
your
Quotient
from
it
,
there
will
remaine
the
number
thought
upon
For
example
,
suppose
the
number
thought
upon
were
5
,
multiply
it
by
4
makes
20.
this
divided
by
2
,
the
Quotient
makes
10
,
which
multiplyed
by
6
,
makes
60
,
and
divided
by
4
,
makes
1●
.
in
the
same
time
admit
you
think
upon
4
,
which
multiplied
by
4
,
makes
16
,
this
divided
by
2
,
makes
8
,
which
multiplied
by
6
makes
48
,
and
divided
by
4
makes
1●
;
then
divide
1●
by
the
number
thought
,
which
was
5
,
the
Quotient
is
●
;
divide
also
12
by
the
number
you
took
,
viz.
4
,
the
Quotient
is
also
3.
as
was
declared
;
therefore
if
the
Quo●ient
●
be
added
unto
the
number
thought
,
viz.
●
,
it
makes
8
,
which
being
known
,
the
number
thought
upon
is
also
knowne
.
PROBLEM
XXIII
.
To
finde
out
many
numbers
that
sundry
persons
,
or
one
man
hath
thought
upon
.
IF
the
multitude
of
numbers
thought
upon
be
odde
,
as
three
numbers
,
five
numbers
,
seven
,
&c.
as
for
example
,
let
5
numbers
thought
upon
be
these
●
2
,
3
,
4
,
5
,
6.
bid
him
declare
the
sum
of
the
first
and
second
,
which
will
be
5
,
the
second
and
third
,
which
makes
7
,
the
third
and
fourth
,
which
makes
9
,
the
fourth
and
fifth
,
which
makes
11
,
and
so
alwayes
adding
the
two
next
together
,
aske
him
how
much
the
first
and
last
makes
together
,
which
is
8.
then
take
these
summes
,
and
place
them
in
order
,
and
adde
all
these
together
,
which
were
in
the
odde
places
:
that
is
the
first
,
third
,
and
fifth
,
viz.
5
,
9
,
●
,
makes
22.
In
like
manner
adde
all
these
numbets
together
,
which
are
in
the
even
places
,
that
is
in
the
second
and
fourth
places
,
viz.
7
and
1●
makes
18
,
substract
this
from
the
former
22
,
then
there
will
remaine
the
double
of
the
first
number
thought
upon
,
viz.
4.
which
known
,
the
rest
is
easily
known
:
seeing
you
know
the
summe
of
the
first
and
second
;
but
if
the
multitude
of
numbers
be
even
as
these
six
numbers
,
viz.
2
,
●
,
4
,
5
,
6
,
7
,
cause
the
partie
to
declare
the
summe
of
each
two
,
by
antecedent
and
consequent
,
and
also
the
summe
of
the
second
and
last
,
which
will
be
5
,
7
,
9
,
11
,
13
,
10
,
then
adde
the
odde
places
together
,
except
the
first
,
that
is
9
,
and
13
,
makes
22
,
adde
also
the
even
places
together
,
that
is
7
,
11
,
10
,
which
makes
28
,
substract
the
one
from
the
other
,
there
shall
remaine
the
double
of
the
second
number
thought
upon
,
which
known
all
the
rest
are
knowne
.
PROBLEM
XXIV
.
How
is
it
that
a
man
in
one
and
the
same
time
,
may
have
his
head
upward
,
and
his
feet
upward
,
being
in
one
and
the
same
place
?
THe
answer
is
very
facill
,
for
to
be
so
he
must
be
supposed
to
be
in
the
centre
of
the
earth
:
for
as
the
heaven
is
above
on
every
side
,
Coelum
undique
sursum
,
all
that
which
looks
to
the
heavens
being
distant
from
the
centre
is
upward
;
and
it
is
in
this
sense
that
Ma●●olyeus
in
his
Cosmographie
,
&
first
dialogue
,
reported
of
one
that
thought
he
was
led
by
one
of
the
Muses
to
hell
,
where
he
saw
Lucifer
sitting
in
the
middle
of
the
World
,
and
in
the
Centre
of
the
earth
,
as
in
a
Throne
:
having
his
head
and
feet
upward
.
PROBLEM
.
XXV
.
Of
a
Ladder
by
which
two
men
ascending
at
one
time
;
the
more
they
ascend
,
the
more
they
shall
be
asunder
,
notwithstanding
one
being
as
high
as
another
THis
is
most
evident
,
that
if
there
were
a
Ladder
halfe
on
this
side
of
the
Centre
of
the
earth
,
and
the
other
halfe
on
the
other
side
:
and
that
two
at
the
Centre
of
the
World
at
one
instant
being
to
ascend
,
the
one
towards
us
,
and
the
other
towards
our
Antipodes
,
they
should
in
ascending
go
farther
and
farther
,
one
from
another
;
notwithstanding
both
of
them
being
of
like
height
.
PROBLEM
.
XXVI
.
How
it
is
that
a
man
having
but
a
Rod
or
Pole
of
Land
,
doth
bragge
that
he
may
in
a
right
line
passe
from
place
to
place
above
3000
miles
.
THe
opening
of
this
is
easie
,
forasmuch
as
he
that
possesseth
a
Rod
of
ground
possesseth
not
only
the
exterior
surface
of
the
earth
,
but
is
master
also
of
that
which
extends
even
to
the
Centre
of
the
earth
,
and
in
this
wise
all
heritages
&
possessions
are
as
so
many
Pyramides
,
whose
summets
or
points
meet
in
the
centre
of
the
earth
,
and
the
basis
of
them
are
nothing
else
but
each
mans
possession
,
field
,
or
visible
quantity
;
and
therefore
if
there
were
made
or
imagined
so
to
be
made
,
a
descent
to
go
to
the
bottome
of
the
heritage
,
which
would
reach
to
the
centre
of
the
earth
;
it
would
be
above
3000
miles
in
a
right
line
as
before
.
PROBLEM
.
XXVII
.
How
it
is
,
that
a
man
standing
upright
,
and
looking
which
way
he
will
,
he
looketh
either
true
North
or
true
South
.
THis
happeneth
that
if
the
partie
be
under
either
of
the
Poles
,
for
if
he
be
under
the
North-pole
,
then
looking
any
way
he
looketh
South
,
because
all
the
Meridians
concurre
in
the
Poles
of
the
world
,
and
if
he
be
under
the
South-pole
,
he
looks
directly
North
by
the
same
reason
.
PROBLEM
XXVIII
.
To
tell
any
one
what
number
remaines
after
certaine
operations
being
ended
,
without
asking
any
question
.
BId
him
to
think
upon
a
number
,
and
will
him
to
multiply
it
by
what
number
you
think
convenient
:
and
to
the
pro●●ct
bid
him
adde
what
number
you
please
,
or
〈◊〉
that
secretly
you
consider
,
that
it
ma●
be
divided
by
that
which
multiplied
,
and
〈…〉
divide
the
sum
by
the
number
which
he
〈…〉
by
,
and
substract
from
this
Quotient
the
number
thought
upon
:
In
the
same
time
divide
apart
the
number
which
was
add●d
by
that
which
multiplied
,
so
then
your
Quotient
shall
be
equall
to
his
remainder
,
wherefore
without
asking
him
any
thing
,
you
shall
tell
him
what
did
remaine
,
which
will
seem
strange
to
him
that
knoweth
not
the
cause
:
for
example
,
suppose
he
thought
7
,
which
multiplied
by
5
makes
35
,
to
which
adde
10
,
makes
45
,
which
divided
by
5
,
yields
9
,
from
which
if
you
take
away
one
the
number
thought
,
(
because
the
Multiplier
divided
by
the
Divisor
gives
the
Quotient
1
,
)
the
rest
will
be
two
,
which
will
be
also
proved
,
if
10
the
number
which
was
added
,
were
divided
by
5
,
viz.
2.
PROBLEM
XXIX
.
Of
the
play
with
two
severall
things
.
IT
is
a
pleasure
to
see
and
consider
how
the
science
of
numbers
doth
furnish
us
,
not
only
〈…〉
recreate
the
spirits
,
but
also
〈…〉
knowledge
of
admirable
things
,
〈…〉
measure
be
shewen
in
this
〈…〉
the
meane
time
to
produce
alwayes
some
of
them
:
suppose
that
a
man
hold
divers
things
in
his
hand
,
as
Gold
and
●ilver
▪
and
in
one
hand
he
held
the
Gold
,
and
in
the
other
hand
he
held
the
Silver
:
to
know
subtilly
,
and
by
way
of
divination
,
or
artificially
in
which
hand
the
Gold
or
Silver
is
;
attribu●e
t●
the
Gold
,
or
suppose
it
have
a
certaine
price
,
and
so
likewise
attribute
to
the
Silver
another
price
,
conditionally
that
the
one
be
odd
,
and
the
other
even
:
as
for
example
,
bid
h●m
that
the
Gold
be
valued
at
4
Crownes
,
or
Shillings
,
and
the
Silver
at
●
Crownes
,
or
3
Shillings
,
or
any
other
number
,
so
that
one
be
odde
▪
and
the
other
even
,
as
before
;
then
bid
him
triple
that
which
is
in
the
right
hand
,
&
double
that
which
is
in
the
left
hand
,
and
bid
him
adde
these
two
products
together
,
and
aske
him
if
it
be
even
or
odde
;
if
it
be
even
,
then
the
Gold
is
in
the
right
hand
;
if
odde
,
the
Gold
is
in
the
left
hand
.
PROBLEM
.
XXX
.
Two
numbers
being
proposed
unto
two
severall
parties
,
to
tell
which
of
these
numbers
is
taken
by
each
of
them
.
AS
for
example
:
admit
you
had
proposed
unto
two
men
whose
names
were
Peter
and
John
,
two
numbers
,
or
pieces
of
money
,
the
one
even
,
and
the
other
odde
,
as
10.
and
9.
and
let
the
one
of
them
take
one
of
the
numbers
,
and
the
other
partie
take
the
other
number
,
which
they
place
privately
to
themselves
:
how
artificially
,
according
to
the
congruity
,
and
excellency
of
numbers
,
to
finde
which
of
them
did
take
10.
and
which
9.
without
asking
any
qustion
:
and
this
seems
most
subtill
,
yet
delivered
howsoever
differing
little
from
the
former
,
and
is
thus
performed
:
Take
privately
to
your selfe
also
two
numbers
,
the
one
even
,
and
the
other
odde
,
as
4.
and
3.
then
bid
Peter
that
he
double
the
number
which
he
took
,
and
do
you
privately
double
also
your
greatest
number
;
then
bid
John
to
triple
the
number
which
he
hath
,
and
do
you
the
like
upon
your
last
number
:
adde
your
two
products
together
,
&
mark
if
it
be
even
or
odde
,
then
bid
the
two
parties
put
their
numbers
together
,
and
bid
them
take
the
halfe
of
it
,
which
if
they
cannot
do
,
then
immediately
tell
Peter
he
took
10.
and
John
9.
because
the
aggregate
of
the
double
of
4.
and
the
triple
of
3.
makes
odde
,
and
such
would
be
the
aggregate
or
summe
of
the
double
of
Peters
number
and
Johns
number
,
if
Peter
had
taken
10.
if
otherwise
,
then
they
might
have
taken
halfe
,
and
so
John
should
have
taken
10.
and
Peter
9.
as
suppose
Peter
had
taken
10.
the
double
is
20.
and
the
triple
of
9.
the
other
●umber
is
27.
which
put
together
makes
47.
odde
:
in
like
manner
the
double
of
your
number
conceived
in
minde
,
viz.
4.
makes
8.
and
the
triple
of
the
3.
the
other
number
,
makes
9.
which
set
together
makes
17.
odde
.
Now
you
cannot
take
the
halfe
of
17
,
nor
47.
which
argueth
that
Peter
had
the
greater
number
,
for
otherwise
the
double
of
9.
is
18.
&
the
triple
of
10.
is
30.
which
set
together
makes
48.
the
halfe
of
it
may
be
taken
:
therefore
in
such
case
Peter
the
took
lesse
number
:
and
John
the
greater
,
and
this
being
don
cleanly
carries
much
grace
with
it
.
PROBLEM
.
XXXI
.
How
to
describe
a
Circle
that
shall
touch
3
:
Points
placed
howsoever
upon
a
plaine
,
if
they
be
not
in
a
right
line
.
LEt
the
three
points
be
A.B.C.
put
one
foot
of
the
Compasse
upon
A.
and
describe
an
Arch
of
a
Circle
at
pleasure
:
and
placed
at
B.
crosse
that
Arch
in
the
two
points
E.
and
F.
and
placed
in
C.
crosse
the
Arch
in
G.
and
H.
then
lay
a
ruler
upon
G.H.
and
draw
a
line
,
and
place
a
Ruler
upon
E.
and
F.
cut
the
other
line
in
K
▪
so
K
is
the
Centre
of
the
Circumference
of
a
Circle
,
which
will
passe
by
the
said
three
points
A.B.C.
or
it
may
be
inverted
,
having
a
Circle
drawne
;
to
finde
the
Centre
of
that
Circle
,
make
3.
points
in
the
circumference
,
and
then
use
the
same
way
:
so
shall
you
have
the
Centre
,
a
thing
most
facill
to
every
practitioner
in
the
principles
of
Geometrie
.
PROBLEM
.
XXXII
.
How
to
change
a
Circle
into
a
square
forme
?
M●ke
a
Circle
upon
past-board
or
other
materiall
,
as
the
Circle
A.C.D.E.
of
which
A.
is
the
Centre
;
then
cut
it
into
4
quarters
,
and
dispose
them
so
,
that
A.
at
the
centre
of
the
Circle
may
alwayes
be
at
the
Angle
of
the
square
,
and
so
the
foure
quarters
of
the
Circle
being
placed
so
,
it
will
make
a
perfect
square
,
whose
side
A.A.
is
equall
to
the
Diameter
B.D.
Now
here
is
to
be
noted
that
the
square
is
greater
then
the
Circle
by
the
vacuity
in
the
middle
,
viz.
M.
PROBLEM
.
XXXIII
.
With
one
and
the
same
comp●sses
,
and
at
one
and
the
same
extent
,
or
opening
,
how
to
describe
many
Circles
concentricall
,
that
is
,
greater
or
lesser
one
then
another
?
IT
is
not
without
cause
that
many
admire
how
this
Proposition
is
to
be
resolved
;
yea
in
the
judgement
of
some
it
is
thought
impossible
:
who
consider
not
the
industrie
of
an
ingenious
Geometrician
,
who
makes
it
possible
,
and
that
most
facill
,
sundry
wayes
;
for
in
the
first
place
if
you
make
a
Circle
upon
a
fine
plaine
,
and
upon
the
Centre
of
that
Circle
,
a
small
pegge
of
wood
be
placed
,
to
be
raised
up
and
put
downe
at
pleasure
by
help
of
a
small
ho●e
made
in
the
Centre
,
then
with
the
same
opening
of
the
Compasses
,
you
may
describe
Circles
Concentricall
,
that
is
,
one
greater
or
lesser
than
another
;
for
the
higher
the
Center
is
lifted
up
,
the
lesser
the
Circle
will
be
.
Secondly
,
the
compasse
being
at
that
extent
upon
a
Gibus
body
,
a
Circle
may
be
described
,
which
will
be
lesse
than
the
former
,
upon
a
plaine
,
and
more
artificially
upon
a
Globe
,
or
round
bowle
:
and
this
againe
is
most
obvious
upon
a
round
Pyramide
,
placing
the
Compasses
upon
the
top
of
it
,
which
will
be
farre
lesse
than
any
of
the
former
;
and
this
is
demonstrated
by
the
20.
Prop.
of
the
first
of
Euclids
,
for
the
Diameter
●
.
D.
is
lesse
than
the
line
AD.A.E.
taken
together
,
and
the
lines
AD.AE.
being
equall
to
the
Diameter
BC.
because
of
the
same
distance
or
extent
of
opening
the
compasses
,
it
followes
that
the
Diameter
E.D.
and
all
his
Circles
together
is
much
lesse
than
the
Diameter
,
and
the
Circle
BC.
which
was
to
be
performed
.
PROBLEM
XXXIV
.
Any
numbers
under
10.
being
thought
upon
,
to
finde
what
numbers
they
were
.
LEt
the
first
number
be
doubled
,
and
unto
it
adde
5.
and
multiply
that
summe
by
5.
and
unto
it
adde
10.
and
unto
this
product
add
the
next
number
thought
upon
;
multiply
this
same
againe
by
10.
and
adde
unto
it
the
next
number
,
and
so
proceed
:
now
if
he
declare
the
last
summe
;
marke
if
he
thought
but
upon
one
figure
,
for
then
subtract
only
35.
from
it
,
and
the
first
figure
in
the
place
of
tennes
is
the
number
thought
upon
:
if
he
thought
upon
two
figures
,
then
subtract
also
the
said
●5
.
from
his
last
summe
,
and
the
two
figures
which
remaine
are
the
number
thought
upon
:
if
he
thought
upo●
three
figures
,
then
subtract
350.
and
then
the
first
three
figures
are
the
numbers
thought
upon
,
&c.
so
if
one
thought
upon
these
numbers
5.7.9.6
.
double
the
first
,
makes
1●
.
to
which
adde
5.
makes
15.
this
multiplied
by
5.
makes
75.
to
which
adde
1●
.
makes
85.
to
this
adde
the
next
number
,
viz.
7.
makes
92.
this
multiplied
by
10.
makes
920.
to
which
adde
the
next
number
,
viz.
9.
makes
929.
which
multiplied
by
10.
makes
9290.
to
which
adde
6.
makes
9296.
from
which
subtract
3500.
resteth
5796.
the
foure
numbers
thought
upon
.
Now
because
the
two
last
figures
are
like
the
two
numbers
thought
upon
:
to
conceale
this
,
bid
him
take
the
halfe
of
it
,
or
put
first
12.
or
any
other
number
to
it
,
and
then
it
will
not
be
so
open
.
PROBLEM
.
XXXV
.
Of
the
Play
with
the
Ring
.
AMongst
a
company
of
9.
or
10.
persons
,
one
of
them
having
a
Ring
,
or
such
like
:
to
finde
out
in
which
hand
:
upon
which
finger
,
&
joynt
it
is
;
this
will
cause
great
astonishment
to
ignorant
spirits
,
which
will
make
them
beleeve
that
he
that
doth
it
works
by
Magick
,
or
Witchcraft
:
But
in
effect
it
is
nothing
else
but
a
nimble
act
of
Arithmetick
,
founded
upon
the
precedent
Probleme
:
for
first
it
is
supposed
that
the
persons
stand
or
sit
in
order
,
that
one
is
first
,
the
next
second
,
&c.
likewise
there
must
be
imagined
that
of
these
two
hands
the
one
is
first
,
and
the
other
second
:
and
also
of
the
five
fingers
,
the
one
is
first
,
the
next
is
second
,
and
lastly
of
the
joynts
,
the
one
is
as
1.
the
other
is
as
2.
the
other
as
3.
&c.
from
whence
it
appeares
that
in
performing
this
Play
there
is
nothing
else
to
be
done
than
to
think
4.
numbers
:
for
example
,
if
the
fourth
person
had
the
Ring
in
his
left
hand
,
and
upon
the
fifth
finger
,
and
third
joynt
,
and
I
would
divine
and
finde
it
out
:
thus
I
would
proceed
,
as
in
the
34
Problem
:
in
causing
him
to
double
the
first
number
:
that
is
,
the
number
of
persons
,
which
was
4.
and
it
makes
8.
to
which
add
5.
makes
13.
this
multiplied
by
5.
makes
65.
put
10.
to
it
,
makes
75.
unto
this
put
●
.
for
the
number
belonging
to
the
left
hand
,
and
so
it
makes
77.
which
multiplied
by
10.
makes
770.
to
this
adde
the
number
of
the
fingers
upon
which
the
Ring
is
,
viz.
5.
makes
775.
this
multiplied
by
10.
makes
7750.
to
which
adde
the
number
for
the
joynt
upon
which
the
Ring
is
,
viz
the
third
joynt
,
makes
7●53
.
to
which
cause
him
to
adde
14.
or
some
other
number
,
to
conceale
it
the
better
:
and
it
makes
7767.
which
being
declared
unto
you
,
substract
3514
▪
and
there
will
remaine
4.2.5.3
.
which
figures
in
order
declares
the
whol
mystery
of
that
which
is
to
be
known
:
4.
signifieth
the
fourth
person
,
2.
the
left
hand
,
5.
the
fifth
finger
,
and
3.
the
third
joynt
of
that
finger
.
PROBLEM
.
XXXVI
.
The
Play
of
34.
or
more
Dice
.
THat
which
is
said
of
the
two
precedent
Problemes
may
be
applied
to
this
of
Dice
(
and
many
other
particular
things
)
to
finde
what
number
appeareth
upon
each
Dice
being
cast
by
some
one
,
for
the
points
that
are
upon
any
side
of
a
Dice
are
alwayes
lesse
than
10
and
the
points
of
each
side
of
a
Dice
may
be
taken
for
a
number
thought
upon
:
therefore
the
Rule
will
be
as
the
former
:
As
for
example
,
one
having
thrown
three
Dice
,
and
you
would
declare
the
numbers
of
each
one
,
or
how
much
they
make
together
,
bid
him
double
the
points
of
one
of
the
Dice
,
to
which
bid
him
adde
5
,
then
multiply
that
by
5.
and
to
it
adde
10
,
and
to
the
summe
bid
him
adde
the
number
of
the
second
Dice
:
and
multiply
that
by
10
:
lastly
,
to
this
bid
him
adde
the
number
of
the
last
Dice
,
and
then
let
him
declare
the
whole
number
:
then
if
from
it
you
subtract
●50
.
there
will
remaine
the
number
of
the
three
Dice
throwne
.
PROBLEM
.
XXXVII
.
How
to
make
water
in
a
Glasse
seeme
to
boyle
and
sparkle
?
TAke
a
Glasse
neere
full
of
water
or
other
liquor
;
and
setting
one
hand
upon
the
foot
of
it
,
to
hold
it
fast
:
turne
slightly
one
of
the
fingers
of
your
other
hand
upon
the
brimme
,
or
edge
of
the
Glasse
;
having
before
privately
wet
your
finger
:
and
so
passing
softly
on
with
your
finger
in
pressing
a
little
:
for
then
first
,
the
Glasse
will
begin
to
make
a
noyse
:
secondly
,
the
parts
of
the
Glasse
will
sensibly
appeare
to
tremble
,
with
notable
rarefaction
and
condensation
:
thirdly
,
the
water
will
shake
,
seeme
to
boyle
:
fourthly
,
it
will
cast
it selfe
out
of
the
Glasse
,
and
leap
out
by
small
drops
,
with
great
astonishment
to
the
standers
by
;
if
they
be
ignorant
of
the
cause
of
it
,
which
is
onely
in
the
Rarefaction
of
the
parts
of
the
Glasse
,
occasioned
by
the
motion
and
pressure
of
the
finger
.
EXAMINATION
.
THe
cause
of
this
,
is
not
in
the
rarefaction
of
the
parts
of
the
Glasse
,
but
it
is
rather
in
the
quick
locall
motion
of
the
finger
,
for
reason
sheweth
us
that
by
how
much
a
Body
draweth
nearer
to
a
quality
,
the
lesse
is
it
subject
or
capable
of
another
which
is
contrary
unto
it
?
now
condensation
,
and
rarefaction
are
contrary
qualities
,
and
in
this
Probleme
there
are
three
bodies
considered
,
the
Glasse
,
the
Water
,
and
the
Aire
,
now
it
is
evident
that
the
Glasse
being
the
most
solid
,
and
impenitrable
Body
,
is
lesse
subject
and
capable
of
rarefaction
than
the
water
,
the
water
is
lesse
subject
than
the
Aire
,
and
if
there
be
any
rarefaction
,
it
is
rather
considerable
in
the
Aire
then
in
the
Water
,
which
is
inscribed
by
the
Glasse
,
and
above
the
Water
,
and
rather
in
the
Water
then
in
the
Glasse
:
the
agitation
,
or
the
trembling
of
the
parts
of
the
Glasse
to
the
sense
appeares
not
:
for
it
is
a
continued
body
;
if
in
part
,
why
then
not
in
the
whole
?
and
that
the
Water
turnes
in
the
Glasse
,
this
appeares
not
,
but
only
the
upper
contiguous
parts
of
the
Water
:
that
at
the
bottome
being
lesse
subiect
to
this
agitation
,
and
it
is
most
certaine
that
by
how
much
quicker
the
Circular
motion
of
the
finger
upon
the
edge
of
the
Glasse
is
,
by
so
much
the
more
shall
the
Aire
be
agitated
,
and
so
the
water
shall
receive
some
apparant
affection
more
or
lesse
from
it
,
according
to
that
motion
:
as
we
see
from
the
quicknesse
of
winde
upon
the
Sea
,
or
c●lme
thereof
,
that
there
is
a
greater
or
lesser
agitation
in
the
water
;
and
for
further
examination
,
we
leave
it
to
the
search
of
those
which
are
curious
.
PROBLEM
.
XXXVIII
.
Of
a
fine
vessell
which
holds
wine
or
water
,
being
cast
in●o
it
at
a
certaine
height
,
but
being
filled
higher
,
it
will
runne
out
of
its
owne
accord
.
LEt
there
be
a
vessell
A.B.C.D.
in
the
middle
of
which
place
a
Pipe
;
whose
ends
both
above
at
E
,
and
below
at
the
bottom
of
the
vessell
as
at
●
▪
are
open
;
let
the
end
●
be
somewhat
lower
than
the
brimme
of
the
Glasse
:
about
this
Pipe
,
place
another
Pipe
as
H.
L
,
which
mounts
a
little
above
E
and
let
it
most
diligently
be
closed
at
H
,
that
no
Aire
enter
in
thereby
,
and
this
Pipe
at
the
bottome
may
have
a
small
hole
to
give
passage
unto
the
water
;
then
poure
in
water
or
wine
,
and
as
long
as
it
mounts
not
above
E
,
it
is
safe
;
but
if
you
poure
in
the
water
so
that
it
mount
above
it
,
farewell
all
:
for
it
will
not
cease
untill
it
be
all
gone
out
;
the
same
may
be
done
in
disposing
any
crooked
Pipe
in
a
vessell
in
the
manner
of
a
Faucet
or
funnell
,
as
in
the
figure
H
,
for
fill
it
under
H
,
at
pleasure
,
and
all
will
go
well
;
but
if
you
fill
it
unto
H.
you
will
see
fine
sport
,
for
then
all
the
vessell
will
be
empty
incontinent
,
and
the
subtiltie
of
this
will
seeme
more
admirable
,
if
you
conceale
the
Pipe
by
a
Bird
,
Serpent
,
or
such
like
,
in
the
middle
of
the
Glasse
.
Now
the
reason
of
this
is
not
difficult
to
those
which
know
the
nature
of
a
Cock
or
Faucet
;
for
it
is
a
bowed
Pipe
,
one
end
of
which
is
put
into
the
water
or
liquor
,
and
sucking
at
the
other
end
untill
the
Pipe
be
full
,
then
will
it
run
of
it selfe
,
and
it
is
a
fine
secret
in
nature
to
see
,
that
if
the
end
of
the
Pipe
which
is
out
of
the
water
,
be
lower
then
the
water
,
it
will
run
out
without
ceasing
:
but
if
the
mouth
of
the
Pipe
be
higher
then
the
water
or
levell
with
it
,
it
will
not
runne
,
although
the
Pipe
which
is
without
be
many
times
bigger
than
that
which
is
in
the
water
:
for
it
is
the
property
of
water
to
keep
alwayes
exactly
levell
▪
EXAMINATION
.
HEre
is
to
be
noted
,
that
if
the
face
of
the
water
without
be
in
one
and
the
same
plaine
,
with
that
which
is
within
,
though
the
outtermost
Pipe
be
ten
times
greater
than
that
which
is
within
;
the
water
naturally
will
not
runne
,
but
if
the
plaine
of
the
water
without
be
any
part
lower
then
that
which
is
within
,
it
will
freely
runne
:
and
here
may
be
noted
further
,
that
if
the
mouth
of
the
Pipe
which
is
full
of
water
,
doth
but
only
touch
the
superficies
of
the
water
within
,
although
the
other
end
of
the
pipe
without
be
much
lower
than
that
within
,
the
water
it
will
not
run
at
all
:
which
contradicts
the
first
ground
;
hence
we
gather
that
the
pressure
or
ponderosity
of
the
water
within
,
is
the
cause
of
running
in
some
respect
.
PROBLEM
.
XXXIX
.
Of
a
Glasse
very
pleasant
.
SOmetimes
there
are
Glasses
which
are
made
of
a
double
fashion
,
as
if
one
Glasse
were
within
another
,
so
that
they
seem
but
one
,
but
there
is
a
little
space
between
them
.
No●
poure
Wine
or
other
liquor
between
the
two
edges
by
help
of
a
Tunnell
,
into
a
little
hole
left
to
this
end
,
so
will
there
appeare
two
fine
delusions
or
fallacies
;
for
though
there
be
not
a
drop
of
Wine
within
the
hollow
of
the
Glasse
,
it
will
seem
to
those
which
behold
it
that
it
is
an
ordinary
Glasse
full
of
Wine
,
and
that
especially
to
those
which
are
side-wise
of
it
,
and
if
any
one
move
it
,
it
will
much
confirme
it
,
because
of
the
motion
of
the
Wine
;
but
that
which
will
give
most
delight
,
is
that
,
if
any
one
shall
take
the
Glasse
,
and
putting
it
to
his
mouth
shall
think
to
drink
the
Wine
,
instead
of
which
he
shall
sup
the
Aire
,
and
so
will
cause
laughter
to
those
that
stand
by
,
who
being
deceived
,
will
hold
the
Glass
to
the
light
,
&
thereby
considering
that
the
raies
or
beames
of
the
light
are
not
reflected
to
the
eye
,
as
they
would
be
if
there
were
a
liquid
substance
in
the
Glasse
,
hence
they
have
an
assured
proofe
to
conclude
,
that
the
hollow
of
the
Glasse
is
totally
empty
.
PROBLEM
.
XL.
If
any
one
should
hold
in
each
hand
,
as
many
pieces
of
money
as
in
the
other
,
how
to
finde
how
much
there
is
?
BId
him
that
holds
the
money
that
he
put
out
of
one
hand
into
the
other
what
number
you
think
convenient
:
(
provided
that
it
may
be
done
,
)
this
done
,
bid
him
that
out
of
the
hand
that
he
put
the
other
number
into
,
that
he
take
out
of
it
as
many
as
remaine
in
the
other
hand
,
and
put
it
into
that
hand
:
for
then
be
assured
that
in
the
hand
which
was
put
the
first
taking
away
:
there
will
be
found
just
the
double
of
the
number
taken
away
at
the
first
.
Example
,
admit
there
were
in
each
hand
12
Shi●lings
or
Counters
,
and
that
out
of
the
right
hand
you
bid
him
take
7.
and
put
it
into
the
left
:
and
then
put
into
the
right
hand
from
the
left
as
many
as
doth
remaine
in
the
right
,
which
is
5.
so
there
will
be
in
the
left
hand
●4
,
which
is
the
double
of
the
number
taken
out
of
the
right
hand
,
to
wit
7.
then
by
some
of
the
rules
before
delivered
,
it
is
easie
to
finde
how
much
is
in
the
right
hand
,
viz.
10.
PROBLEM
.
XLI
.
Many
Dice
being
cast
,
how
artificially
to
discover
the
number
of
the
points
that
may
arise
.
SVppose
any
one
had
cast
three
Dice
secretly
,
bid
him
that
he
adde
the
points
that
were
upmost
together
:
then
putting
one
of
the
Dice
apart
,
unto
the
former
summe
adde
the
points
which
are
under
the
other
two
,
then
bid
him
throw
these
two
Dice
,
and
mark
how
many
points
a
paire
are
upwards
,
which
adde
unto
the
former
summe
:
then
put
one
of
these
Dice
away
not
changing
the
side
,
mark
the
points
which
are
under
the
other
Dice
,
and
adde
it
to
the
former
summe
:
lastly
,
throw
that
one
Dice
,
and
whatsoever
appeares
upward
adde
it
unto
the
former
summe
;
and
let
the
Dice
remaine
thus
:
this
done
,
comming
to
the
Table
,
note
what
points
do
appeare
upward
upon
the
three
Dice
,
which
adde
privately
together
,
and
unto
it
adde
●1
or
3
times
7
:
so
this
Addition
or
summe
shall
be
equall
to
the
summe
which
the
party
privately
made
of
all
the
operations
which
he
formerly
made
.
As
if
he
should
throw
three
Dice
,
and
there
should
appeare
upward
5
,
3
,
2.
the
sum
of
them
is
10.
and
setting
one
of
them
apart
,
(
as
5.
)
unto
10
,
adde
the
points
which
are
under
3
and
2
,
which
is
4
and
5
,
and
it
makes
19.
then
casting
these
two
Dice
suppose
there
should
appeare
4
and
1
,
this
added
unto
19
makes
24.
and
setting
one
of
these
two
Dice
apart
as
the
4.
unto
the
former
24
,
I
adde
the
number
of
points
which
is
under
the
other
Dice
,
viz.
under
1
,
that
is
6
,
which
makes
30.
Last
of
all
I
throw
that
one
Dice
,
and
suppose
there
did
appeare
2
,
which
I
adde
to
the
former
30
,
and
it
makes
32
,
then
leaving
the
3
dice
thus
,
the
points
which
are
upward
will
be
these
,
5
,
4
,
2
unto
which
adde
secretly
21
,
(
as
before
was
said
)
so
have
you
32
,
the
same
number
whi●h
he
had
;
and
in
the
same
manner
you
may
practise
with
4
,
5
,
6
,
or
many
Dice
or
other
bodies
,
observing
only
that
you
must
adde
the
points
opposite
of
the
Dice
;
for
upon
which
depends
the
whole
demonstration
or
secret
of
the
play
;
for
alwayes
that
which
is
above
and
underneath
makes
7.
but
if
it
make
another
number
,
then
must
you
adde
as
often
that
number
.
PROBLEM
.
XLII
.
Two
mettals
,
as
Gold
and
Silver
,
or
of
other
kin●●
weighing
alike
,
being
privately
placed
into
two
like
Boxes
,
to
finde
which
of
them
the
Gold
or
Silver
is
in
.
But
because
that
this
experiment
in
water
hath
divers
accidents
,
and
therefore
subject
to
a
caution
;
and
namely
,
because
the
matter
of
the
chest
,
mettall
or
other
things
may
hinder
.
Behold
here
a
more
subtill
and
certaine
invention
to
finde
and
discover
it
out
without
weighing
it
in
the
water
▪
Now
experience
and
reason
sheweth
us
that
two
like
bodies
or
magnitudes
of
equall
weight
,
and
of
divers
mettalls
,
are
not
of
equal
quantity
:
and
seeing
that
Gold
is
the
heaviest
of
all
mettalls
,
it
will
occupie
less
roome
or
place
;
from
which
will
follow
that
the
like
weight
of
Lead
in
the
same
forme
,
will
occupie
or
take
up
more
roome
or
place
.
Now
let
there
be
therefore
presented
two
Globes
or
Chests
of
wood
or
other
matter
alike
,
&
equall
one
to
the
other
,
in
one
of
which
in
the
middle
there
is
another
Globe
or
body
of
lead
weighing
12.
l.
(
as
C
,
)
and
in
the
other
a
Globe
or
like
body
of
Gold
weighing
12
pound
(
as
B.
)
Now
it
is
supposed
that
the
wooden
Globes
or
Chests
are
of
equall
weight
,
forme
,
and
magnitude
:
and
to
discover
in
which
the
Gold
or
Lead
is
in
,
take
a
broad
paire
of
Compasses
,
and
clip
one
of
the
Coffers
or
Globes
somewhat
from
the
middle
,
as
at
D.
then
fix
in
the
Chest
or
Globe
a
small
piece
of
Iron
between
the
feet
of
the
Compasses
,
as
EK
,
at
the
end
of
which
hang
a
weight
G
,
so
that
the
other
end
may
be
counterpoysed
,
and
hang
in
aequilibrio
:
and
do
the
like
to
the
other
Chest
or
Globe
.
Now
if
that
the
other
Chest
or
Globe
being
clipped
in
like
distance
from
the
end
,
and
hanging
at
the
other
end
the
same
weight
G.
there
be
found
no
difference
;
then
clip
them
nearer
towards
the
middle
,
that
so
the
points
of
the
Compasse
may
be
against
some
of
the
mettall
which
is
inclosed
;
or
just
against
the
extremitie
of
the
Gold
as
in
D
,
and
suppose
it
hang
thus
in
aequilibrio
;
it
is
certaine
that
in
the
other
Coffer
is
the
Lead
;
for
the
points
of
the
Compasses
being
advanced
as
much
as
before
,
as
at
F
,
which
takes
up
a
part
of
the
Lead
,
(
because
it
occupies
a
greater
place
than
the
Gold
)
therefore
that
shall
help
the
weight
G.
to
weigh
,
and
so
will
not
hang
in
aequilibrio
,
except
G
be
placed
neare
to
F.
hence
we
may
conclude
,
that
there
is
the
Lead
;
and
in
the
other
Chest
or
Globe
,
there
is
the
Gold
.
EXAMINATION
.
IF
the
two
Boxes
being
of
equall
magnitude
weighed
in
the
aire
be
found
to
be
of
equall
weight
,
they
shall
necessarily
take
up
like
place
in
the
water
,
and
therefore
weigh
also
one
as
much
as
another
:
hence
there
is
no
possibilitie
to
finde
the
inequalitie
of
the
mettalls
which
are
inclosed
in
these
Boxes
in
the
water
:
the
intention
of
Archimedes
was
not
upon
contrary
mettalls
inclosed
in
〈…〉
Boxes
,
but
consisted
of
comparing
metta●●●
,
simple
in
the
water
one
with
another
:
therefore
the
inference
is
false
and
absurd
.
PROBLEM
.
XLIII
.
Two
Globes
of
diverse
mettalls
,
(
as
one
Gold
,
and
the
other
Copper
)
yet
of
equall
weight
being
put
into
a
box
,
as
BG
,
to
finde
in
which
end
the
Gold
or
Copper
is
.
THis
is
discovered
by
the
changing
of
the
places
of
the
two
Bowles
or
Globes
,
having
the
same
counterpoyse
H
to
be
hung
at
the
other
side
,
as
in
N.
and
if
the
Gold
which
is
the
lesser
Globe
,
were
before
the
nearest
to
the
handle
D●
,
having
now
changed
his
place
will
be
farthest
from
the
handle
DE
,
as
in
K.
therefore
the
Centre
of
gravity
of
the
two
Globes
taken
together
,
shall
be
farther
separate
from
the
middle
of
the
handle
(
under
which
is
the
Centre
of
gravity
of
the
Box
)
than
it
was
before
,
and
seeing
that
the
handle
is
alwayes
in
the
middle
of
the
Box
,
the
weight
N.
must
be
augmented
▪
to
keep
it
in
equil●●●●
and
by
this
way
one
may
know
,
that
if
at
the
second
time
,
the
counterpoise
be
too
light
,
it
is
a
signe
that
the
Gold
is
farthest
off
the
handle
,
as
at
the
first
triall
it
was
nearest
.
PROBLEM
.
XLIIII
.
How
to
represent
diverse
sorts
of
Rainebowes
here
below
?
THe
Rainbowe
is
a
thing
admirable
in
the
world
,
which
ravisheth
often
the
eyes
and
spirits
of
men
in
consideration
of
his
rich
intermingled
colours
which
are
seen
under
the
clouds
,
seeming
as
the
glistering
of
the
Starres
,
precious
stones
,
and
ornaments
of
the
most
beauteous
flowers
:
some
part
of
it
as
the
resplendent
stars
,
or
as
a
Rose
,
or
burning
Cole
of
fire
▪
in
it
one
may
see
Dyes
of
sundry
sorts
,
the
Violet
,
the
Blew
,
the
Orange
,
the
Saphir
,
the
Jacinct
,
and
the
Emerald
colours
,
as
a
lively
plant
placed
in
a
green
soile
:
and
as
a
most
rich
treasure
of
nature
,
it
is
a
high
work
of
the
Sun
who
casteth
his
raies
or
beames
as
a
curious
Painter
drawes
strokes
with
his
pensill
,
and
placeth
his
colours
in
an
exquisite
situation
;
and
Solomon
saith
,
Eccles.
43.
it
is
a
chiefe
and
principall
work
of
God
.
Notwithstanding
there
is
left
to
industrie
how
to
represent
it
from
above
,
here
below
,
though
not
in
perfection
,
yet
in
part
,
with
the
same
intermixture
of
colours
that
is
above
.
Have
you
not
seen
how
by
Oares
of
a
Boate
it
doth
exceeding
quickly
glide
upon
the
water
with
a
pleasant
grace
?
Aristotle
sayes
,
that
it
coloureth
the
water
,
and
makes
a
thousand
atomes
,
upon
which
the
beames
of
the
Sunne
reflecting
,
make
a
kinde
of
coloured
Rainbowe
:
or
may
we
not
see
in
houses
or
Gardens
of
pleasure
artificiall
fountaines
,
which
poure
forth
their
droppie
streames
of
water
,
that
being
between
the
Sunne
and
the
fountaine
,
there
will
be
presented
as
a
continuall
Rainbowe
?
But
not
to
go
farther
,
I
will
shew
you
how
you
may
do
it
at
your
doore
,
by
a
fine
and
facill
experiment
.
Take
water
in
your
mouth
,
and
turne
your
back
to
the
Sunne
,
and
your
face
against
some
obscure
place
,
then
blow
out
the
water
which
is
in
your
mouth
,
that
it
may
be
sprinkled
in
small
drops
and
vapours
:
you
shall
see
those
atomes
vapours
in
the
beames
of
the
Sunne
to
turne
into
a
faire
Rainebowe
,
but
all
the
griefe
is
,
that
it
lasteth
not
,
but
soone
is
vanished
.
But
to
have
one
more
stable
and
permanent
in
his
colours
:
Take
a
Glasse
full
of
water
,
and
expose
it
to
the
Sunne
,
so
that
the
raies
that
passe
through
strike
upon
a
shadowed
place
,
you
will
have
pleasure
to
see
the
fine
forme
of
a
Rainebowe
by
this
reflection
.
Or
take
a
Trigonall
Glasse
or
Crystall
Glasse
of
diverse
Angles
,
and
look
through
it
,
or
let
the
beames
of
the
Sunne
passe
through
it
;
or
with
a
candle
let
the
appearances
be
received
upon
a
shadowed
place
:
you
will
have
the
same
contentment
.
PROBLEM
XLV
.
How
that
if
all
the
Powder
in
the
world
were
in
closed
within
a
bowle
of
paper
or
glasse
,
and
being
fired
on
all
parts
,
it
could
not
break
that
bowle
?
IF
the
bowle
and
the
powder
be
uniforme
in
all
his
parts
,
then
by
that
means
the
powder
would
presse
and
move
equally
on
each
side
,
in
which
there
is
no
possibility
whereby
it
ought
to
begin
by
one
side
more
than
another
.
Now
it
is
impossible
that
the
bowle
should
be
broken
in
all
his
parts
:
for
they
are
infinite
.
Of
like
fineness
or
subtiltie
may
it
be
that
a
bowle
of
Iron
falling
from
a
high
place
upon
a
plaine
pavement
of
thin
Glasse
,
it
were
impossible
any
wise
to
break
it
;
if
the
bowle
were
perfectly
round
,
and
the
Glasse
flat
and
uniforme
in
all
his
parts
▪
for
the
bowle
would
touch
the
Glasse
but
in
one
point
,
which
is
in
the
middle
of
infinite
parts
which
are
about
it
:
neither
is
there
any
cause
why
it
ought
more
on
one
side
than
on
another
,
seeing
that
it
may
not
be
done
with
all
his
sides
together
;
it
may
be
concluded
as
speaking
naturally
,
that
such
a
bowle
falling
upon
such
a
glasse
will
not
break
it
.
But
this
matter
is
meere
Metaphysicall
,
and
all
the
workmen
in
the
world
cannot
ever
with
all
their
industrie
make
a
bowle
perfectly
round
,
or
a
Glasse
uniforme
.
PROBLEM
.
XLVI
.
To
finde
a
number
which
being
divided
by
2
,
there
will
remaine
1
,
being
divided
by
3
,
there
will
remaine
1
;
and
so
likewise
being
divided
by
4
,
5
,
or
6
,
there
would
still
remaine
1
;
but
being
didivided
by
7
,
there
will
remaine
nothing
.
IN
many
Authors
of
Arithmetick
this
Probleme
is
thus
proposed
:
A
woman
carrying
Egges
to
Market
in
a
basket
,
met
an
unruly
fellow
who
broke
them
:
who
was
by
order
made
to
pay
for
them
:
and
she
being
demanded
what
number
she
had
,
she
could
not
tell
:
but
she
remembred
that
counting
them
by
2
&
2
,
there
remained
1
▪
likewise
by
3
and
3
by
4
and
4
,
by
5
and
5
,
by
6
and
6
;
there
still
remained
1.
but
when
she
counted
them
by
7
and
7
,
there
remained
nothing
:
Now
how
may
the
number
of
Egges
be
discovered
?
Finde
a
number
which
may
exactly
be
measured
by
7
,
and
being
measured
by
2
,
3
,
4
,
5
,
and
6
;
there
will
still
remaine
a
unite
▪
multiply
these
numbers
together
,
makes
720
,
to
which
adde
1
;
so
have
you
the
number
,
viz.
721.
in
like
manner
301
will
be
measured
by
2
,
3
,
4
,
5
,
6
;
so
that
1
remaines
:
but
being
measured
by
7
,
nothing
will
remaine
;
to
which
continually
adde
220
,
and
you
have
other
numbers
which
will
do
the
same
:
hence
it
is
doubtfull
what
number
she
had
,
therefore
not
to
faile
,
it
must
be
known
whether
they
did
exceed
400
,
800
,
&c.
in
which
it
may
be
conjectured
that
it
could
not
exceed
4
or
5
hundred
,
seeing
a
man
or
woman
could
not
carry
7
or
8
hundred
Egges
,
therefore
the
number
was
the
former
●01
.
which
she
had
in
her
Basket
:
which
being
counted
by
2
and
2
,
there
will
remaine
1
,
by
3
and
3
,
&c.
but
counted
by
7
and
7
,
there
will
remaine
nothing
.
PROBLEM
.
XLVII
.
One
had
a
certaine
number
of
crownes
,
and
counting
them
by
2
and
2
,
there
rested
1.
counting
them
by
3
and
3
,
there
rested
2.
counting
them
by
4
and
4
,
there
rested
3.
counting
them
by
5
and
5
,
there
rested
4.
counting
them
by
6
&
6
,
there
rested
5.
but
counting
them
by
7
and
7
,
there
remained
nothing
:
how
many
crownes
might
he
have
?
THis
Question
hath
some
affinitie
to
the
precedent
,
and
the
resolution
is
almost
in
the
same
manner
:
for
here
there
must
be
found
a
number
,
which
multiplied
by
7
,
and
then
divided
by
2
,
3
,
4
,
5
,
6
;
there
may
alwayes
remaine
a
number
lesse
by
1
than
the
Divisor
:
Now
the
first
number
which
arrives
in
this
nature
is
119
,
unto
which
if
420
be
added
,
makes
539
,
which
also
will
do
the
same
:
and
so
by
adding
420
,
you
may
have
other
numbers
to
resolve
this
proposition
.
PROBLEM
.
XLVIII
.
How
many
sorts
of
weights
in
the
least
manner
must
there
be
to
weigh
all
sorts
of
things
between
1
pound
and
40
pound
,
and
so
unto
121
,
&
364
pound
.
TO
weigh
things
between
1
and
40
,
take
numbers
in
triple
proportion
,
so
that
their
summe
be
equall
,
or
somewhat
greater
than
40
,
as
are
the
numbers
1
3.9.27
.
I
say
that
with
●
such
weights
,
the
first
being
of
1
pound
,
the
second
being
3
pound
,
the
third
being
9
pound
,
and
the
fourth
being
27
:
any
weight
between
1
and
40
pound
may
be
weighed
.
As
admit
to
weigh
21
pound
,
put
unto
the
thing
that
is
to
be
weighed
the
9
pound
weight
,
then
in
the
other
ballance
put
27
pound
and
3
pound
,
which
doth
counterpoise
21
pound
and
9
pound
,
and
if
20
pound
were
to
be
weighed
,
put
to
it
in
the
ballance
9
and
1
,
and
in
the
other
ballance
put
27
and
3
,
and
so
of
others
In
the
same
manner
take
those
5
weights
,
1
,
3
,
9
,
27
▪
81
,
you
may
weigh
with
them
between
1
pound
,
and
121
pound
:
and
taking
those
6
weights
▪
as
1
,
3
,
9
,
2●
,
81
,
243
,
you
may
weigh
even
from
1
pound
unto
364
pound
:
this
depends
upon
the
property
of
continued
proportionals
,
the
latter
of
which
containing
twice
all
the
former
.
PROBLEM
.
XLIX
.
Of
a
deceitfull
ballance
which
being
c●●●ty
seemes
i●
be
just
,
because
it
hangs
in
aequilibrio
:
not●ithstanding
putting
12
pound
in
one
ballance
,
and
11
in
the
other
,
it
will
remaine
in
aequilibrio
.
ARistotle
maketh
mention
of
this
ballance
in
his
mechanick
Questions
,
and
saith
,
that
the
Merchants
of
purpose
in
his
time
used
them
to
deceive
the
world
:
the
subtiltie
or
craft
of
which
is
thus
,
that
one
arme
of
the
ballance
is
longer
then
another
,
by
the
same
proportion
,
that
one
weight
is
heavier
then
another
:
As
if
the
beame
were
23
inches
long
,
and
the
handle
placed
so
that
12
inches
should
be
on
one
side
of
it
,
and
11
inches
on
the
other
side
:
conditionally
that
the
shorter
end
should
be
as
heavy
as
the
longer
,
a
thing
easie
to
be
done
:
then
afterwards
put
into
the
ballance
two
unequal
weights
in
such
proportion
as
the
parts
of
the
beame
have
one
unto
another
,
which
is
12
to
11
,
but
so
that
the
greater
be
placed
in
the
ballance
which
hangs
upon
the
shorter
part
of
the
beame
,
and
the
lesser
weight
in
the
other
ballance
:
it
is
most
certaine
that
the
ballances
will
hang
in
aequilibrio
,
which
will
seem
most
sincere
and
just
;
though
it
be
most
deceitfull
,
abominable
,
and
false
.
The
reason
of
this
is
drawne
from
the
experiments
of
Archimedes
,
who
shewes
that
two
unequall
weights
will
counterpoyse
one
another
,
when
there
is
like
proportion
betweene
the
parts
of
the
beame
(
that
the
handle
separates
)
and
the
weights
themselves
:
for
in
one
and
the
same
counterpoise
,
by
how
much
it
is
farther
from
the
Centre
of
the
handle
,
by
so
much
it
seems
heavier
,
therefore
if
there
be
a
diversitie
of
distance
that
the
ballances
hang
from
the
handle
,
there
must
necessarily
be
an
ineqality
of
weight
in
these
ballances
to
make
them
hang
in
aequilibrio
,
and
to
discover
if
there
be
deceit
,
change
the
weight
into
the
other
ballance
,
for
as
soone
as
the
greater
weight
is
placed
in
the
ballance
that
hangs
on
the
longer
parts
of
the
beame
:
it
will
weigh
downe
the
other
instantly
.
PROBLEM
.
L.
To
heave
or
lift
up
a
bottle
with
a
straw
.
TAke
a
straw
that
is
not
bruised
,
bow
it
that
it
make
an
Angle
,
and
put
it
into
the
bottle
so
that
the
greatest
end
be
in
the
neck
,
then
the
Reed
being
put
in
the
bowed
part
wil
cast
side-wise
,
and
make
an
Angle
as
in
the
figure
may
be
seen
:
then
may
you
take
the
end
which
is
out
of
the
bottle
in
your
hand
,
and
heave
up
the
bottle
,
and
it
is
so
much
surer
,
by
how
much
the
Angle
is
acuter
or
sharper
;
and
the
end
which
is
bowed
approacheth
to
the
other
perpendicular
parts
which
come
out
of
the
bottle
.
PROBLEM
.
LI.
How
in
the
middle
of
a
wood
or
desert
,
without
the
sight
of
the
Sunne
,
Starres
,
Shadow
or
Compasse
,
to
finde
out
the
North
or
South
,
or
the
foure
Cardinall
points
of
the
world
,
East
,
West
,
&
c
?
IT
is
the
opinion
of
some
,
that
the
windes
are
to
be
observed
in
this
:
if
it
be
hot
,
the
South
is
found
by
the
windes
that
blow
that
way
,
but
this
observation
is
uncertaine
and
subject
to
much
error
:
nature
will
help
you
in
some
measure
to
make
it
more
manifest
than
any
of
the
former
,
from
a
tree
thus
:
Cut
a
small
tree
off
,
even
to
the
ground
,
and
mark
the
many
circles
that
are
about
the
sap
or
pith
of
the
tree
,
which
seem
nearer
together
in
some
part
than
in
other
,
which
is
by
reason
of
the
Suns
motion
about
the
tree
:
for
that
the
humiditie
of
the
parts
of
the
tree
towards
the
South
by
the
heat
of
the
Sun
is
rarified
,
and
caused
to
extend
:
and
the
S●n
not
giving
such
heat
towards
the
North-part
of
the
tree
,
the
sap
is
lesser
rarefied
,
but
condensed
;
by
which
the
circles
are
nearer
together
on
the
North-part
,
than
on
the
South-part
:
therefore
if
a
line
be
drawne
from
the
widest
to
the
narrowest
part
of
the
circles
,
it
shall
shew
the
North
&
South
of
the
world
.
Another
Experiment
may
be
thus
:
Take
a
small
needle
,
such
as
women
work
with
:
place
it
gently
downe
flatwise
upon
still
water
,
and
it
will
not
sink
,
(
which
is
against
the
generall
tenet
that
Iron
will
not
swimme
)
which
needle
will
by
little
and
little
turne
to
the
North
and
South-points
.
But
if
the
needle
be
great
and
will
not
swim
,
thrust
it
through
a
small
piece
of
Cork
,
or
some
such
like
thing
,
and
then
it
will
do
the
same
:
for
such
is
the
property
of
Iron
when
it
is
placed
in
aequilibrio
,
it
strives
to
finde
out
the
Poles
of
the
world
or
points
of
North
and
South
in
a
manner
as
the
magnes
doth
.
EXAMINATION
.
HEre
is
observable
,
that
the
moisture
which
aideth
to
the
growth
of
the
tree
,
is
dilated
and
rarefied
by
the
Meridionall
heat
,
and
contracted
by
the
Septentrionall
cold
:
this
rarefaction
works
upon
the
part
of
the
humour
or
moisture
that
is
more
thinne
,
which
doth
easily
dissipate
and
evaporate
:
which
evaporation
carries
a
part
of
the
salt
with
it
;
and
because
that
solidation
or
condensation
,
so
that
there
is
left
but
a
part
of
the
nourishment
which
the
heat
bakes
up
and
consumes
:
so
contrarily
on
the
other
side
the
condensation
and
restrictive
quality
of
the
moisture
causeth
lesse
evaporation
and
perdition
:
and
so
consequently
there
remaines
more
nourishment
,
which
makes
a
greater
increase
on
that
side
than
on
the
other
side
:
for
as
trees
have
their
growth
in
winter
,
because
of
their
pores
and
these
of
the
earth
are
shut
up
:
so
in
the
spring
when
their
pores
are
open
,
and
when
the
sappe
and
moisture
is
drawne
by
it
,
there
is
not
such
cold
on
the
North-side
that
it
may
be
condensed
at
once
:
But
contrarily
to
the
side
which
is
South
,
the
heat
may
be
such
,
that
in
little
time
by
continuance
,
this
moisture
is
dissipated
greatly
:
and
cold
is
nothing
but
that
which
hardneth
and
contracteth
the
moisture
of
the
tree
,
and
so
converteth
it
into
wood
.
PROBLEM
.
LII
.
Three
persons
having
taken
Counters
,
Cards
,
or
other
things
,
to
finde
how
much
each
one
hath
taken
.
CAuse
the
third
party
to
take
a
number
which
ma●
be
divided
by
4
,
and
as
often
as
he
takes
4
,
let
the
second
party
take
7
,
and
the
first
take
13
,
then
cause
them
to
put
them
all
together
,
and
declare
the
summe
of
it
;
which
secretly
divide
by
3
,
and
the
Quotient
is
the
double
of
the
number
which
the
third
person
did
take
.
Or
cause
the
third
to
give
unto
the
second
and
first
,
as
many
as
each
of
them
hath
;
then
let
the
second
give
unto
the
first
and
third
,
as
many
as
each
of
them
hath
;
lastly
,
let
the
third
give
unto
the
second
and
first
,
as
many
as
each
of
them
hath
;
and
then
aske
how
much
one
of
them
hath
;
(
for
they
will
have
then
all
alike
,
)
so
halfe
of
that
number
is
the
number
that
the
third
person
had
at
the
first
:
which
knowne
all
is
knowne
.
PROBLEM
.
LIII
.
How
to
make
a
consort
of
musick
of
many
parts
with
one
voyce
,
or
one
instrument
only
?
THis
Probleme
is
resolved
,
so
that
a
finger
or
player
upon
an
instrument
,
be
neare
an
Echo
which
answereth
his
voice
or
instrument
;
and
if
the
Echo
answereth
but
once
at
a
time
,
he
may
make
a
double
;
if
twice
,
then
a
triple
,
if
three
times
,
then
an
harmonie
of
foure
parts
,
for
it
must
be
such
a
one
that
is
able
to
exercise
both
tune
and
note
as
occasion
requires
.
As
when
he
begins
ut
,
before
the
Echo
answer
,
he
may
begin
sol
,
and
pronounce
it
in
the
same
tune
that
●he
Echo
answereth
,
by
which
meanes
you
●ave
a
fifth
,
agreeable
consort
of
musick
:
then
in
the
same
time
that
the
Echo
followeth
,
to
sound
the
second
note
sol
,
he
may
sound
forth
another
sol
higher
or
lower
to
make
an
eight
,
the
most
perfect
consort
of
musick
,
and
so
of
others
,
if
he
will
continue
his
voice
with
the
Echo
,
and
sing
alone
with
two
parts
.
Now
experience
sheweth
this
to
be
true
,
which
often
comes
to
passe
in
many
Churches
,
making
one
to
beleeve
that
there
are
many
more
parts
in
the
musick
of
a
Quire
,
then
in
effect
truly
there
are
because
of
the
resounding
and
multiplying
of
the
voic●
,
and
redoubling
of
the
Quire
.
PROBLEM
.
LIIII
.
T●
make
or
describe
an
Ovall
form
,
or
that
which
neare
resembles
unto
it
,
at
one
turning
with
a
paire
of
common
Compasses
.
THere
are
many
fine
wayes
in
Geometricall
practices
,
to
make
an
Ovall
figure
or
one
neare
unto
it
,
by
severall
centres
:
any
of
which
I
will
not
touch
upon
,
but
shew
how
it
may
be
done
promptly
upon
one
centre
only
.
In
which
I
will
say
nothing
of
the
Ovall
forme
,
which
appeares
,
when
one
describeth
circles
with
the
points
of
a
common
Compasses
,
somewhat
deep
upon
a
skinne
stretched
forth
hard
:
which
contracting
it selfe
in
some
parts
of
the
skinne
maketh
an
Ovall
forme
.
But
it
will
more
evidently
appeare
upon
a
Columne
or
Cylinder
:
if
paper
be
placed
upon
it
,
then
with
a
paire
of
Compasses
describe
as
it
were
a
circle
upon
it
,
which
paper
afterwards
being
extended
,
will
not
be
circular
but
ovall-wise
:
and
a
paire
of
Compasses
may
be
so
accommodated
,
that
it
may
be
done
also
upon
a
plaine
thus
.
As
let
the
length
of
the
Ovall
be
H.
K
,
fasten
2
pinnes
or
nailes
neare
the
end
of
that
line
as
F.
G
,
and
take
a
threed
which
is
double
to
the
length
of
G.
H
,
or
F.
K
,
then
if
you
take
a
Compasse
which
may
have
one
foot
lower
than
another
,
with
a
spring
between
his
legges
:
and
placing
one
foot
of
this
Compasse
in
the
Centre
of
the
Ovall
,
and
guiding
the
threed
by
the
other
foot
of
the
Compasses
,
and
so
carrying
it
about
:
the
spring
will
help
to
describe
and
draw
the
Ovall
forme
.
But
in
stead
of
the
Compasses
it
may
be
done
with
ones
hand
only
,
as
in
the
figure
may
appeare
.
PROBLEM
.
LV.
Of
a
pu●se
difficult
to
be
opened
.
IT
is
made
to
shut
and
open
with
Rings
:
first
at
each
side
there
is
a
strap
or
string
,
as
AB
.
and
CD
,
at
the
end
of
which
are
2
rings
,
B
&
D
,
and
the
string
CD
passeth
through
the
ring
B
,
so
that
it
may
not
come
out
againe
;
or
be
parted
one
from
another
:
and
so
that
the
ring
B
,
may
slide
up
and
downe
upon
the
string
CD
,
then
over
the
purse
,
there
is
a
piece
of
Leather
EFGH
,
which
covers
the
opening
of
the
purse
,
and
there
is
another
piece
of
Leather
AE
,
which
passeth
through
many
rings
:
which
hath
a
slit
towards
the
end
I
,
so
great
that
the
string
BC
may
slide
into
it
:
Now
all
the
cunning
or
craft
is
how
to
make
fast
or
to
open
the
purse
,
which
consists
in
making
the
string
BC
slide
through
the
side
at
I
,
therefore
bring
down
B
to
I
,
then
make
the
end
I
passe
through
the
ring
B
,
and
also
D
with
his
string
to
passe
through
the
slit
I
,
so
shall
the
purse
be
fast
,
and
then
may
the
strings
be
put
as
before
,
and
it
will
seem
difficult
to
discover
how
it
was
done
.
Now
to
open
the
purse
,
put
through
the
end
I
through
the
ring
B
,
and
then
through
the
slit
I
,
by
which
you
put
through
the
string
DC
,
by
this
way
the
purse
will
be
opened
.
PROBLEM
.
LVI
.
Whether
it
is
more
hard
and
admirable
without
Compasses
to
make
a
perfect
circle
,
or
being
made
to
finde
out
the
Centre
of
it
?
IT
is
said
that
upon
a
time
past
,
two
Mathematicians
met
,
and
they
would
make
tryall
of
their
industry
:
the
one
made
instantly
a
Perfect
circle
without
Compasses
,
and
the
other
immediately
pointed
out
the
Centre
thereof
with
the
point
of
a
needle
;
now
which
is
the
chiefest
action
?
it
seems
the
first
,
for
to
draw
the
most
noblest
figure
upon
a
plaine
Table
without
other
help
than
the
hand
,
and
the
minde
,
is
full
of
admiration
;
to
finde
the
Centre
is
but
to
finde
out
only
one
point
,
but
to
draw
a
round
,
there
must
be
almost
infinite
points
,
equidistant
from
the
Centre
or
middle
;
that
in
conclusion
it
is
both
the
Circle
and
the
Centre
together
.
But
contrarily
it
may
seem
that
to
finde
the
Centre
is
more
difficult
,
for
what
attention
,
vivacitie
,
and
subtiltie
must
there
be
in
the
spirit
,
in
the
eye
,
in
the
hand
,
which
will
chuse
the
true
point
amongst
a
thousand
other
points
?
He
that
makes
a
circle
keeps
alwayes
the
same
distance
,
and
is
guided
by
a
halfe
distance
to
finish
the
rest
;
but
he
that
must
finde
the
Centre
,
must
in
the
same
time
take
heed
to
the
parts
about
it
,
and
choose
one
only
point
which
is
equall
distant
from
an
infinite
of
other
points
which
are
in
the
circumference
;
which
is
very
difficult
.
Aristotle
confirmes
this
amongst
his
morals
,
and
seems
to
explaine
the
difficultie
which
is
to
be
found
in
the
middle
of
vertue
;
for
it
may
want
a
thousand
wayes
,
and
be
farre
separated
from
the
true
Centre
of
the
end
of
a
right
mediocritie
of
a
vertuous
action
;
for
to
do
well
it
must
touch
the
middle
point
which
is
but
one
,
and
there
must
be
a
true
point
which
respects
the
end
,
and
that
's
but
one
only
.
Now
to
judge
which
is
the
most
difficult
,
as
before
is
said
,
either
to
draw
the
round
or
to
finde
the
Centre
,
the
round
seems
to
be
harder
than
to
finde
the
Centre
,
because
that
in
finding
of
it
,
it
is
done
at
once
,
and
hath
an
equall
distance
from
the
whole
;
But
,
as
before
,
to
draw
a
round
there
is
a
visible
point
imagined
,
about
which
the
circle
is
to
be
drawne
.
I
esteeme
that
it
is
as
difficult
therefore
,
if
not
more
,
to
make
the
circle
without
a
Centre
,
as
to
finde
the
middle
or
Centre
of
that
circle
.
PROBLEM
.
LVII
.
Any
one
having
taken
3
Cards
,
to
finde
how
many
points
they
containe
THis
is
to
be
exercised
upon
a
full
Pack
of
Cards
of
52
,
then
let
one
choose
any
three
at
pleasure
secretly
from
your
sight
,
and
bid
him
secretly
account
the
points
in
each
Card
,
and
will
him
to
take
as
many
Cards
as
will
make
up
15
to
each
of
the
points
of
his
Cards
,
then
will
him
to
give
you
the
rest
of
the
Cards
,
for
4
of
them
being
rejected
,
the
rest
shew
the
number
of
points
that
his
three
Cards
which
he
took
at
the
first
did
conteine
.
As
if
the
3
Cards
were
7
,
10
,
and
4
;
now
7
wants
of
15
,
8.
take
8
Cards
therefore
for
your
first
Card
:
the
10
wants
of
15
▪
5
,
take
5
cards
for
your
second
card
:
lastly
4
wants
of
15
,
11
,
take
11
Cards
for
your
third
Card
,
&
giving
him
the
rest
of
the
Cards
,
there
will
be
25
;
from
which
take
4
,
there
remaines
21
,
the
number
of
the
three
Cards
taken
,
viz.
7
,
10
,
and
4.
Whosoever
would
practise
this
play
with
4
,
5
,
6
,
or
more
Cards
,
and
that
the
whole
number
of
Cards
be
more
or
lesse
than
52
;
and
that
the
terme
be
15
,
14
,
12
,
&c
,
this
generall
rule
ensuing
may
serve
:
multiply
the
terme
by
the
number
of
Cards
taken
at
first
:
to
the
product
adde
the
number
of
Cards
taken
,
then
subtract
this
summe
from
the
whole
number
of
Cards
;
the
remainder
is
the
number
which
must
be
subtracted
from
the
Cards
,
which
remaines
to
make
up
the
game
:
if
there
remaine
nothing
after
the
Subtraction
,
then
the
number
of
Cards
remaining
doth
justly
shew
the
number
of
points
which
were
in
the
Cards
chosen
.
If
the
Subtraction
cannot
be
made
,
then
subtract
the
number
of
Cards
from
that
number
,
and
the
remainder
added
unto
the
Cards
that
did
remaine
,
the
summe
will
be
the
number
of
points
in
the
Cards
taken
,
as
if
the
Cards
were
7
,
10
,
5
,
8
,
and
the
terme
given
were
12
;
so
the
first
wants
5
,
the
second
wants
2
,
the
third
wants
7
,
and
the
fourth
wants
4
Cards
,
which
taken
,
the
party
gives
you
the
rest
of
the
Cards
:
then
secretly
multiply
12
by
4
,
makes
48
;
to
which
adde
4
,
the
number
of
Cards
taken
makes
52
,
from
which
52
should
be
taken
,
rest
nothing
:
therefore
according
to
the
direction
of
the
remainder
of
the
Cards
which
are
30
,
is
equall
to
the
points
of
the
foure
Cards
taken
,
viz.
7
,
10
,
5
,
8.
Againe
,
let
these
five
Cards
be
supposed
to
be
taken
,
8
,
6
,
10
,
3
,
7
;
their
differences
to
15
,
the
termes
are
7
,
9
,
5
,
12
,
8
,
which
number
of
Cards
taken
,
there
will
remaine
but
6
Cards
:
then
privately
multiply
15
by
5
,
makes
75
,
to
which
adde
5
makes
80
,
from
this
take
52
the
number
of
Cards
,
rest
28
,
to
which
add
the
remainder
of
Cards
,
make
34.
the
summe
with
8
,
6
,
10
,
3
,
7.
PROBLEM
.
LVII
.
Many
Cards
placed
in
diverse
ranks
,
to
finde
which
of
these
Cards
any
one
hath
thought
.
TAke
15
Cards
,
and
place
them
in
3
heaps
in
rank-wise
,
5
in
a
heap
:
now
suppose
any
one
had
thought
one
of
these
Cards
in
any
one
of
the
heaps
,
it
is
easie
to
finde
which
of
the
Cards
it
is
,
and
it
is
done
thus
;
ask
him
in
which
of
the
heaps
it
is
,
which
place
in
the
middle
of
the
other
two
;
then
throw
downe
the
Cards
by
1
and
1
into
three
severall
heaps
in
rank-wise
,
untill
all
be
cast
downe
,
then
aske
him
in
which
of
the
rankes
his
Card
is
,
which
heap
place
in
the
middle
of
the
other
two
heaps
alwayes
,
and
this
do
foure
times
at
least
,
so
in
putting
the
Cards
altogether
,
look
upon
the
Cards
,
or
let
their
back
be
towards
you
,
and
throw
out
the
eight
Card
,
for
that
was
the
Card
thought
upon
without
faile
.
PROBLEM
.
LVIII
.
Many
Cards
being
offered
to
sundry
persons
,
to
finde
which
of
these
Cards
any
one
thinketh
upon
.
ADmit
there
were
4
persons
,
then
take
4
Cards
,
and
shew
them
to
the
first
,
bid
him
think
one
of
them
,
and
put
these
4
away
,
then
take
4
other
Cards
,
and
shew
them
in
like
manner
to
the
second
person
,
and
bid
him
think
any
one
of
these
Cards
,
and
so
do
to
the
third
person
,
and
so
the
fourth
,
&c.
Then
take
the
4
Cards
of
the
first
person
,
and
dispose
them
in
4
rankes
,
and
upon
them
the
4
Cards
of
the
second
person
,
upon
them
also
these
of
the
third
person
,
and
lastly
,
upon
them
these
of
the
fourth
person
,
then
shew
unto
eaeh
of
these
parties
each
of
these
ranks
,
and
aske
him
if
his
Card
be
in
it
which
he
thought
,
for
infallibly
that
which
the
first
partie
thought
upon
will
be
in
the
first
rank
,
and
at
the
bottome
,
the
Card
of
the
second
person
will
be
in
the
second
ranke
,
the
Card
of
the
third
thought
upon
will
be
in
the
third
rank
,
and
the
fourth
mans
Card
will
be
in
the
fourth
rank
,
and
so
of
others
,
if
there
be
more
persons
use
the
same
method
.
This
may
be
practised
by
other
things
,
ranking
them
by
certaine
numbers
:
allotted
to
pieces
of
money
,
or
such
like
things
.
PROBLEM
.
LIX
.
How
to
make
an
instrument
to
help
hearing
,
as
Galileus
made
to
help
the
sight
?
THink
not
that
the
Mathematickes
(
which
hath
furnished
us
with
such
admirable
helps
for
seeing
)
is
wanting
for
that
of
hearing
,
it
s
well
knowne
that
long
trunks
or
pipes
make
one
heare
well
farre
off
,
and
experience
shewes
us
that
in
certaine
places
of
the
Orcades
in
a
hollow
vault
,
that
a
man
speaking
but
softly
at
one
corner
thereof
,
may
be
audibly
understood
at
the
other
end
:
notwithstanding
those
which
are
between
the
parties
cannot
heare
him
speak
at
all
:
And
it
is
a
generall
principle
,
that
pipes
do
greatly
help
to
strengthen
the
activitie
of
naturall
causes
:
we
see
that
〈◊〉
contracted
in
a
pipe
,
burnes
4
or
5
foot
high
,
which
would
scarce
heat
,
being
in
the
open
aire
:
the
rupture
or
violence
of
water
issuing
out
of
a
fountaine
,
shewes
us
that
water
being
contracted
into
a
pipe
,
causeth
a
violence
in
its
passage
.
The
Glasses
of
Galeileus
makes
us
see
how
usefull
pipes
or
trunkes
are
to
make
the
light
and
species
more
visible
,
and
proportionable
to
our
eye
.
It
is
said
that
a
Prince
of
Italy
hath
a
faire
hall
,
in
which
he
can
with
facility
heare
distinctly
the
discourses
of
those
which
walk
in
the
adjacent
Gardens
,
which
is
by
certaine
vessels
and
pipes
that
answer
from
the
Garden
to
the
Hall
.
Vitruvius
makes
mention
also
of
such
vessels
and
pipes
,
to
strengthen
the
voice
and
action
of
Comedians
:
and
in
these
times
amongst
many
noble
personages
▪
the
new
kinde
of
trunkes
are
used
to
help
the
hearing
,
being
made
of
silver
,
copper
,
or
other
resounding
materiall
;
in
funnell-wise
putting
the
widest
end
to
him
which
speaketh
,
to
the
end
to
contract
the
voice
,
that
so
by
the
pipe
applied
to
the
eare
it
may
be
more
uniform
and
lesse
in
danger
to
dissipate
the
voice
,
and
so
consequently
more
fortified
.
PROBLEM
.
LX.
Of
a
fine
lamp
which
goes
not
out
,
though
one
carry
it
in
ones
pocket
:
or
being
rolled
upon
the
ground
will
still
burne
.
IT
must
be
observed
that
the
vessell
in
which
the
oile
is
put
into
,
have
two
pinnes
on
the
sides
of
it
,
one
against
another
,
being
included
within
a
circle
:
this
circle
ought
to
have
two
other
pinnes
,
to
enter
into
another
circle
of
brasse
,
or
other
solid
matter
:
lastly
,
this
second
circle
hath
two
pinnes
,
which
may
hang
within
some
box
to
containe
the
whole
lamp
,
in
such
manner
,
that
there
be
6
pinnes
in
different
position
:
Now
by
the
aid
of
these
pegges
or
pinnes
,
the
lamp
that
is
in
the
middle
will
be
alwayes
well
situated
according
to
his
Centre
of
gravity
,
though
it
be
turned
any
way
:
though
if
you
endeavour
to
turne
it
upside
downe
,
it
will
lie
levell
▪
which
is
pleasant
and
admirable
to
behold
to
those
which
know
not
the
cause
:
And
it
is
facil
from
his
to
make
a
place
to
rest
quiet
in
,
though
there
be
great
agitation
in
the
outward
parts
.
PROBLEM
.
LXI
.
Any
one
having
thought
a
Card
amongst
many
Cards
,
how
artificially
to
discover
it
out
?
TAke
any
number
of
Cards
as
10
,
12
,
&c.
and
open
some
4
or
5
to
the
parties
sight
,
and
bid
him
think
one
of
them
,
but
let
him
note
whether
it
be
the
first
,
second
,
third
,
&c.
then
with
promptness
learn
what
number
of
Cards
you
had
in
your
hands
,
and
take
the
other
part
of
the
Cards
,
and
place
them
on
the
top
of
these
you
hold
in
your
hand
;
and
having
done
so
,
aske
him
whether
his
Card
were
the
first
,
second
,
&c.
then
before
knowing
the
number
of
Cards
that
were
at
the
bottome
,
account
backwards
untill
you
come
to
it
:
so
shall
you
easily
take
out
the
card
that
he
thought
upon
.
PROBLEM
.
LXII
.
Three
Women
AB.C.
carried
apples
to
a
marke
to
sell
,
A
had
20
,
B
30
▪
and
C
40
,
they
sold
as
many
for
a
penny
,
the
one
as
the
other
:
and
brought
home
one
as
much
money
as
another
,
how
could
this
be
?
THe
answer
to
the
Probleme
is
easie
▪
as
suppose
at
the
beginning
of
the
Market
:
A
▪
sold
her
apples
at
a
penny
an
apple
:
and
sold
but
2.
which
was
2
pence
,
and
so
she
had
18
left
:
but
B.
sold
17.
which
was
17
pence
,
and
so
had
13
left
:
C.
sold
32.
which
was
32
pence
,
and
so
had
8
apples
left
▪
then
A
said
she
would
not
sell
her
apples
so
cheap
,
but
would
sell
them
for
3
pence
the
peece
,
which
she
did
:
and
so
her
apples
came
to
54
pence
,
and
B
having
left
but
13
apples
sold
them
at
the
same
rate
,
which
came
to
39
pence
:
and
lastly
▪
C.
had
but
8
apples
,
which
at
the
same
rate
came
to
24
pence
:
these
summes
of
money
which
each
others
before
received
come
to
56
pence
,
and
so
much
each
one
received
;
and
so
consequently
brought
home
one
as
much
as
another
.
PROBLEM
.
LXIII
.
Of
the
properties
of
some
numbers
.
FIrst
,
any
two
numbers
is
just
the
summe
of
a
number
,
that
have
equall
distance
from
the
halfe
of
that
number
▪
the
one
augmenting
,
and
the
other
diminishing
,
as
7
and
7
,
of
8
and
●
,
of
9
and
5
,
of
10
and
4
,
of
11
and
3
,
of
12
and
2
,
of
13
and
●
.
as
the
one
is
more
than
the
halfe
,
the
other
is
lesse
.
Secondly
,
it
is
difficult
to
finde
two
numbers
whose
summe
and
product
is
alike
,
(
that
is
)
if
the
numbers
be
multiplied
one
by
another
,
and
added
together
,
will
be
equall
,
which
two
numbers
are
2
and
2
,
for
to
multiply
2
by
2
makes
4
,
and
adding
2
unto
2
makes
the
same
:
this
property
is
in
no
other
two
whole
numbers
,
but
in
broken
numbers
there
are
infinite
,
whose
summe
and
product
will
be
equall
one
to
another
.
As
Clavius
shewes
upon
the
36
Pro.
of
the
9th
book
of
Euclide
.
Thirdly
,
the
numbers
5
and
6
are
called
circular
numbers
,
because
the
circle
turnes
to
the
point
from
whence
it
begins
:
so
these
numbers
multiplied
by
themselves
,
do
end
alwayes
in
5
and
6
,
as
5
times
5
makes
25
,
that
againe
by
5
makes
125
,
so
6
times
6
makes
36
,
and
that
by
6
makes
216
,
&c.
Fourthly
,
the
number
6
,
is
the
first
which
Arithmeticians
call
a
perfect
number
,
that
is
,
whose
parts
are
equall
unto
it
,
so
the
6
part
of
it
is
1
,
the
third
part
is
2
,
the
halfe
is
3
,
which
are
all
his
parts
:
now
1
,
2
,
and
3
,
is
equall
to
6.
It
is
wonderfull
to
conceive
that
there
is
so
few
of
them
,
and
how
rare
these
numbers
are
▪
50
of
perfect
men
:
for
betwixt
1
&
1000000000000
numbers
there
is
but
ten
,
that
is
;
6
,
28
,
486.
8128.
120816.
2096128.
33550336.
536854528.
8589869056
,
&
137438691328
▪
with
this
admirable
property
,
that
alternately
they
end
all
in
6
and
8
,
&
the
twentieth
perfect
number
is
151115727451553768931328.
Fiftly
,
the
number
9
amongst
other
priviledges
carries
with
it
an
excellent
property
:
for
take
what
number
you
will
,
either
in
grosse
or
in
part
,
the
nines
of
the
whole
or
in
its
parts
rejected
,
and
taken
simply
will
be
the
same
,
as
●7
it
makes
3
times
9
,
so
whether
the
nines
be
rejected
of
27
,
or
of
the
summe
of
2
and
7
,
it
is
all
one
,
so
if
the
nines
were
taken
away
of
240.
it
is
all
one
,
if
the
nines
were
taken
away
of
2
,
4
,
and
0
;
for
there
would
remaine
6
in
either
;
and
so
of
others
.
Sixtly
,
11
being
multiplied
by
2
,
4
,
5
,
6
,
7
,
8
,
or
9
,
will
end
and
begin
with
like
numbers
;
so
11
multiplied
by
5
makes
55
,
if
multiplied
by
8
,
it
makes
88
,
&c.
Seventhly
,
the
numbers
220
and
284
being
unequall
,
notwithstanding
the
parts
of
the
one
number
do
alwayes
equalize
the
other
number
:
so
the
aliquot
parts
of
220
are
110
,
54
,
44
,
22
,
20
,
11
,
10
,
5
,
4
,
2
,
1
,
which
together
makes
284.
the
aliquot
parts
of
284
,
are
142
,
71
,
4
,
2
,
1.
which
together
makes
220
,
a
thing
rare
and
admirable
,
and
difficult
to
finde
in
other
numbers
.
I●
one
be
taken
from
any
square
number
which
is
odde
,
the
square
o●
halfe
of
it
being
added
to
the
first
square
,
will
make
a
square
number
.
The
square
of
halfe
any
even
number
+
.
1
being
added
to
that
even
number
makes
a
square
number
,
and
the
even
number
taken
from
it
leaves
a
square
number
.
If
odde
numbers
be
continually
added
from
the
unitie
successively
,
there
will
be
made
all
square
numbers
,
and
if
cubick
numbers
be
added
successively
from
the
unitie
,
there
will
be
likewise
made
square
numbers
.
PROBLEM
.
LXIV
.
Of
an
excellent
lamp
,
which
serves
or
furnisheth
it selfe
with
oile
,
and
burnes
a
long
time
.
I
Speak
not
here
of
a
common
lamp
which
Ca●danus
writes
upon
in
his
book
de
subtilita●●
,
for
that
's
a
little
vessell
in
columne-wise
,
which
is
full
of
Oile
,
and
because
there
is
but
one
little
hole
at
the
bottome
neare
the
weeke
or
match
;
the
oile
runnes
not
,
for
feare
that
there
be
emptinesse
above
:
when
the
match
is
kindled
it
begins
to
heat
the
lamp
,
and
rarefying
the
oile
it
issueth
by
this
occasion
:
and
so
sends
his
more
airie
parts
above
to
avoid
vacuitie
.
It
is
certaine
that
such
a
lampe
the
Atheniaus
used
,
which
lasted
a
whole
yeare
without
being
touched
:
which
was
placed
before
the
statue
of
Minerva
,
for
they
might
put
a
certaine
quantitie
of
oile
in
the
lamp
CD
,
and
a
match
to
burne
without
being
consumed
:
such
as
the
naturalists
write
of
,
by
which
the
lamp
will
furnish
it selfe
,
and
so
continue
in
burning
:
and
here
may
be
noted
that
the
oile
may
be
poured
in
,
at
the
top
of
th●
vessell
at
a
little
hole
,
and
then
made
fast
againe
that
the
aire
get
not
in
.
PROBLEM
.
LXV
.
Of
the
play
at
Keyles
or
nine
Pinnes
.
YOu
will
scarce
beleeve
that
with
one
bowle
and
at
one
blow
playing
freely
,
one
may
strike
downe
all
the
Keyles
at
once
:
yet
from
Mathematicall
principles
it
is
easie
to
be
demonstrated
,
that
if
the
hand
of
him
that
playes
were
so
well
assured
by
experience
,
as
reason
induceth
one
thereto
;
one
might
at
one
blow
strike
downe
all
the
Keyles
,
of
at
least
7
or
8
,
or
such
a
number
as
one
pleaseth
.
For
they
are
but
9
in
all
disposed
or
placed
in
a
perfect
square
,
having
three
every
way
.
Let
us
suppose
then
that
a
good
player
beginning
to
play
at
1
somewhat
low
,
should
so
strike
it
,
that
it
should
strike
down
the
Keyles
2
and
5
,
and
these
might
in
their
violence
strike
downe
the
Keyles
3
,
6
,
and
9
,
and
the
bowle
being
in
motion
may
strike
down
the
Keyle
4
,
and
7
;
which
4
Keyle
may
strike
the
Keyle
8
,
&
so
all
the
9
Keyles
may
be
striken
down
at
once
.
PROBLEM
.
LXIV
.
Of
Spectacles
of
pleasure
.
SImple
Spectacles
of
blew
,
yellow
,
red
or
green
colour
,
are
proper
to
recreate
the
sight
,
and
will
present
the
objects
died
in
like
colour
that
the
Glasses
are
,
only
those
of
the
greene
do
somewhat
degenerate
;
instead
of
shewing
a
lively
colour
it
will
represent
a
pale
dead
colour
,
and
it
is
because
they
are
not
dyed
greene
enough
,
or
receive
not
light
enough
for
greene
:
and
colour
these
images
that
passe
through
these
Glasses
unto
the
bottome
of
the
eye
.
EXAMINATION
.
IT
is
certaine
,
that
not
onely
Glasses
dyed
green
,
but
all
other
Glasses
coloured
,
yield
the
app●arances
of
objects
strong
or
weak
in
colour
according
to
the
quantity
of
the
dye
,
more
or
lesse
,
as
one
being
very
yellow
,
another
a
pale
yellow
;
now
all
colours
are
not
proper
to
Glasses
to
give
colour
,
hence
the
defect
is
not
that
they
want
facultie
to
receive
light
,
or
resist
the
penetration
of
the
beams
;
for
in
the
same
Glasses
those
which
are
most
dyed
,
give
alwayes
the
objects
more
high
coloured
and
obscure
,
and
those
which
are
lesse
dyed
give
them
more
pale
and
cleare
:
and
this
is
daily
made
manifest
by
the
painting
of
Glasse
,
which
hinders
more
the
penetration
of
the
light
than
dying
doth
,
where
all
the
matter
by
fire
is
forced
into
the
Glasse
,
leaving
it
in
all
parts
transparent
.
Spectacles
of
Crystall
cut
with
divers
Angles
diamond-wise
do
make
a
marvellous
multiplication
of
the
appearances
,
for
looking
towards
a
house
it
becomes
as
a
Towne
,
a
Towne
becomes
like
a
Citie
,
an
armed
man
seems
as
a
whole
company
caused
solely
by
the
diversity
of
refractions
,
for
as
many
plaines
as
there
are
on
the
outside
of
the
spectacle
,
so
many
times
will
the
object
be
multiplied
in
the
appearance
,
because
of
diverse
Images
cast
into
the
eye
.
These
are
pleasurable
spectacles
for
avaricious
persons
that
love
Gold
and
silver
,
for
one
piece
will
seeme
many
,
or
one
heap
of
money
will
seeme
as
a
treasury
:
but
all
the
mischiefe
is
,
he
will
not
have
his
end
in
the
enjoying
of
it
,
for
indeavouring
to
take
it
,
it
will
appeare
but
a
deceitfull
Image
,
or
delusion
of
nothing
.
Here
may
you
note
that
if
the
finger
be
directed
by
one
and
the
same
ray
or
beam
,
which
pointeth
to
one
and
the
same
object
,
then
at
the
first
you
may
touch
that
visible
object
without
being
deceived
:
otherwise
you
may
faile
often
in
touching
that
which
you
see
.
Againe
,
there
are
Spectacles
made
which
do
diminish
the
thing
seen
very
much
,
and
bring
it
to
a
faire
perspective
forme
,
especially
if
one
look
upon
a
faire
Garden
plat
,
a
greater
walk
,
a
stately
building
,
or
great
Court
,
the
industry
of
an
exquisite
Painter
cannot
come
neare
to
expresse
the
lively
forme
of
it
as
this
Glasse
will
represent
it
;
you
will
have
pleasure
to
see
it
really
experimented
,
and
the
cause
of
this
is
,
that
the
Glasses
of
th●se
Spectacles
are
hollow
and
thinner
in
the
middle
,
than
at
the
edges
by
which
the
visuall
Angle
is
made
lesser
:
you
may
observe
a
further
secret
in
these
Spectacles
,
for
in
placing
them
upon
a
window
one
may
see
those
that
passe
to
and
fro
in
the
streets
,
without
being
seen
of
any
,
for
their
property
is
to
raise
up
the
objects
that
it
lookes
upon
.
Now
I
would
not
passe
this
Probleme
without
saying
something
of
Galileus
admirable
Glasse
,
for
the
common
simple
perspective
Glasses
give
to
aged
men
but
the
eyes
or
sight
of
young
men
,
but
this
of
Galileus
gives
a
man
an
Eagles
eye
,
or
an
eye
that
pierceth
the
heavens
:
first
it
discovereth
the
spottie
and
shadowed
opacous
bodies
that
are
found
about
the
Sunne
,
which
darknet
and
diminisheth
the
splendor
of
that
beautifull
and
shining
Luminary
:
secondly
,
it
shewes
the
new
Planets
that
accompany
Saturne
and
Jupiter
:
thirdly
,
in
Venus
is
seen
the
new
,
full
,
and
quartill
increase
;
as
in
the
Moon
by
her
separation
from
the
Sunne
:
fourthly
,
the
artificiall
structure
of
this
instrument
helpeth
us
to
see
an
innumerable
number
of
stars
,
which
otherwise
are
obscured
,
by
reason
of
the
naturall
weaknesse
of
our
sight
,
yea
the
starres
in
via
lactea
are
seen
most
apparantly
;
where
there
seem
no
starres
to
be
,
this
Instrument
makes
apparantly
to
be
seen
,
and
further
delivers
them
to
the
eye
in
their
true
and
lively
colour
,
as
they
are
in
the
heavens
:
in
which
the
splendor
of
some
is
as
the
Sunne
in
his
most
glorious
beauty
.
This
Glasse
hath
also
a
most
excellent
use
in
observing
the
body
of
the
Moone
in
time
of
Eclipses
,
for
it
augments
it
manifold
,
and
most
manifestly
shewes
the
true
forme
of
the
cloudy
substance
in
the
Sunne
;
and
by
it
is
seene
when
the
shadow
of
the
earth
begins
to
eclipse
the
Moon
,
&
when
totally
she
is
over
shadowed
:
besides
the
celestiall
uses
which
are
made
of
this
Glasse
,
it
hath
another
noble
property
;
it
farre
exceedeth
the
ordinary
perspective
Glasses
,
which
are
used
to
see
things
remote
upon
the
earth
,
for
as
this
Glasse
reacheth
up
to
the
heavens
and
excelleth
them
there
in
his
performance
,
so
on
the
earth
it
claimeth
preheminency
,
for
the
objects
which
are
farthest
remote
,
and
most
obscure
,
are
seen
plainer
than
those
which
are
neere
at
hand
,
scorning
as
it
were
all
small
and
triviall
services
,
as
leaving
them
to
an
inferiour
help
:
great
use
may
be
made
of
this
Glass
in
discovering
of
Ships
,
Armies
,
&c.
Now
the
apparell
or
parts
of
this
instrument
or
Glasse
,
is
very
meane
or
simple
,
which
makes
it
the
more
admirable
(
seeing
it
performes
such
great
service
)
having
but
a
convex
Glasse
thickest
in
the
middle
,
to
unite
and
amasse
the
rayes
,
and
mak
the
object
the
greater
:
to
the
augmenting
the
visuall
Angle
,
as
also
a
pipe
or
trunk
to
amasse
the
Species
,
and
hinder
the
greatness
of
the
light
which
is
about
it
:
(
to
see
well
,
the
object
must
be
well
inlightened
,
and
the
eye
in
obscurity
;
)
then
there
is
adjoyned
unto
it
a
Glasse
of
a
short
sight
to
distinguish
the
rayes
,
which
the
other
would
make
more
confused
if
alone
.
As
for
the
proportion
of
those
Glasses
to
the
Trunk
,
though
there
be
certaine
rules
to
make
them
,
yet
it
is
often
by
hazard
that
there
is
made
an
excellent
one
there
being
so
many
difficulties
in
the
action
,
therefore
many
ought
to
be
tryed
,
seeing
that
exact
proportion
,
in
Geometricall
calculation
cannot
serve
for
diversity
of
sights
in
the
observation
.
PROBLEM
.
LXVII
.
Of
the
Adamant
or
Magnes
,
and
the
needles
touched
therewith
.
WHo
would
beleeve
if
he
saw
not
with
his
eyes
,
that
a
needle
of
steel
being
once
touched
with
the
magnes
,
turnes
not
once
,
not
a
yeare
▪
but
as
long
as
the
World
lasteth
;
his
end
towards
the
North
and
South
,
yea
though
one
remove
it
,
and
turne
it
from
his
position
,
it
will
come
againe
to
his
points
of
North
and
South
.
Who
would
have
ever
thought
that
a
brute
stone
black
and
ill
formed
,
touching
a
ring
of
Iron
,
should
hang
it
in
the
aire
,
and
that
ring
support
a
second
,
that
to
support
a
third
,
and
so
unto
10
,
12
,
or
more
,
according
to
the
strength
of
the
magnes
;
making
as
it
were
a
chaine
without
a
line
,
without
souldering
together
,
or
without
any
other
thing
to
support
them
onely
;
but
a
most
occult
and
hidden
vertue
,
yet
most
evident
in
this
effect
,
which
penetrateth
insensibly
from
the
first
to
the
second
,
from
the
second
to
the
third
,
&c.
What
is
there
in
the
world
that
is
more
capable
to
cast
a
deeper
astonishment
in
our
minds
than
a
great
massie
substance
of
Iron
to
hang
in
the
aire
in
the
middest
of
a
building
without
any
thing
in
the
world
touching
it
,
only
but
the
aire
?
As
some
histories
assure
us
,
that
by
the
aid
of
a
Magnes
or
Adamant
,
placed
at
the
roof
of
one
of
the
Turkish
Synagogues
in
Meca
:
the
sepulchre
of
that
infamous
Mah●met
rests
suspended
in
the
aire
;
and
Plinie
in
his
naturall
Historie
writes
that
the
Architect
or
Democrates
did
begin
to
vault
the
Temple
of
A●sin●e
in
Alexandria
,
with
store
of
magnes
to
produce
the
like
deceit
,
to
hang
the
sepulchre
of
that
Goddesse
likewise
in
the
aire
.
I
should
passe
the
bounds
of
my
counterpoise
,
if
I
should
divulge
all
the
secrets
of
this
stone
,
and
should
expose
my selfe
to
the
laughter
of
the
world
:
if
I
should
brag
to
shew
others
the
cause
how
this
appeareth
,
than
in
its
owne
naturall
sympathy
,
for
why
is
it
that
a
magnes
with
one
end
will
cast
the
Iron
away
,
&
attract
it
with
the
other
?
from
whence
commeth
it
that
all
the
magnes
is
not
proper
to
give
a
true
touch
to
the
needle
,
but
only
in
the
two
Poles
of
the
stone
:
which
is
known
by
hanging
the
stone
by
a
threed
in
the
aire
untill
it
be
quiet
,
or
placed
upon
a
peece
of
Cork
in
a
dish
of
water
,
or
upon
some
thinne
board
,
for
the
Pole
of
the
stone
will
then
turne
towards
the
Poles
of
the
world
,
and
point
out
the
North
and
South
,
and
so
shew
by
which
of
these
ends
the
needle
is
to
be
touched
?
From
whence
comes
it
that
there
is
a
variation
in
the
needle
,
and
pointeth
not
out
truly
the
North
and
South
of
the
world
,
but
only
in
some
place
of
the
earth
?
How
is
it
that
the
needle
made
with
pegges
and
inclosed
within
two
Glasses
,
sheweth
the
height
of
the
Pole
,
being
elevated
as
many
degrees
as
the
Pole
is
above
the
Horizon
?
What
's
the
cause
that
fire
and
Garlick
takes
away
the
propertie
of
the
magnes
?
There
are
many
great
hidden
mysteries
in
this
stone
,
which
have
troubled
the
heads
of
the
most
learned
in
all
ages
;
and
to
this
time
the
world
remaines
ignorant
of
declaring
the
rrue
cause
thereof
.
Some
say
,
that
by
help
of
the
Magnes
persons
which
are
absent
may
know
each
others
minde
,
as
if
one
being
here
at
London
,
and
another
at
Prague
in
Germany
:
if
each
of
them
had
a
needle
touched
with
one
magnes
,
then
the
vertue
is
such
that
in
the
same
time
that
the
needle
which
is
at
Prague
shall
move
,
this
that
is
at
London
shall
also
;
provided
that
the
parties
have
like
secret
notes
or
alphabets
,
and
the
observation
be
at
a
set
houre
of
the
day
or
night
;
and
when
the
one
party
will
declare
unto
the
other
,
then
let
that
party
move
the
needle
to
these
letters
which
will
declare
the
matter
to
the
other
,
and
the
moving
of
the
other
parties
needle
shall
open
his
intention
.
The
invention
is
subtile
,
but
I
doubt
whether
in
the
world
there
can
be
found
so
great
a
stone
▪
or
such
a
Magnes
which
carries
with
it
such
vertue
:
neither
is
it
expedient
,
for
treasons
would
be
then
too
frequent
and
open
.
EXAMINATION
.
THe
experimentall
difference
of
rejection
,
and
attraction
proceeds
not
from
the
different
nature
of
Stones
,
but
from
the
quality
of
the
Iron
;
and
the
vertue
of
the
stone
consisteth
only
,
and
especially
in
his
poles
,
which
being
hanged
in
the
Aire
,
turnes
one
of
his
ends
alwayes
naturally
towards
the
South
,
and
the
other
towards
the
North
:
but
if
a
rod
of
Iron
be
touched
with
one
of
the
ends
thereof
,
it
hath
the
like
property
in
turning
North
and
South
,
as
the
magnes
hath
:
notwithstanding
the
end
of
the
Iron
Rod
touched
,
hath
a
contrary
position
,
to
that
end
of
the
stone
that
touched
it
;
yet
the
same
end
will
attract
it
,
and
the
other
end
reject
it
:
and
so
contrarily
this
may
easily
be
experimented
upon
two
needles
touched
with
one
or
different
stones
,
though
they
have
one
and
the
same
position
;
for
as
you
come
unto
them
apply
one
end
of
the
magnes
neare
unto
them
,
the
North
of
the
one
will
abhorre
the
North
of
the
other
,
but
the
North
of
the
one
will
alwayes
approach
to
the
South
of
the
other
:
and
the
same
affection
is
in
the
stones
themselves
.
For
the
finding
of
the
Poles
of
the
magnes
,
it
may
be
done
by
holding
a
small
needle
between
your
fingers
softly
,
and
so
moving
it
from
part
to
part
over
the
stone
untill
it
be
held
perpendicular
,
for
that
shall
be
one
of
the
Poles
of
the
stone
which
you
may
marke
out
;
in
like
manner
finde
out
the
other
Pole
:
Now
to
finde
out
which
of
those
Poles
is
North
or
South
,
place
a
needle
being
touched
with
one
of
the
Poles
upon
a
smooth
convex
body
,
(
as
the
naile
of
ones
finger
or
such
like
,
)
and
marke
which
way
the
end
of
the
needle
that
was
touched
turneth
:
if
to
the
South
,
then
the
point
that
touched
it
was
the
South-Pole
,
&c.
and
it
is
most
certain
and
according
to
reason
and
experience
:
that
if
it
be
suspended
in
aequilibrio
in
the
aire
,
or
supported
upon
the
water
,
it
will
turne
contrary
to
the
needle
that
toucheth
it
;
for
then
the
pole
that
was
marked
for
the
South
shall
turne
to
the
North
,
&c.
PROBLEM
.
LXVIII
.
Of
the
properties
of
Aeolipiles
or
bowels
to
blow
the
fire
.
THese
are
concave
vessels
of
Brass
or
Copper
or
other
material
,
which
may
indure
the
,
fire
:
having
a
small
hole
very
narrow
,
by
which
it
is
filled
with
water
,
then
placing
it
to
the
fire
,
before
it
be
hot
there
is
no
effect
seen
;
but
assoone
as
the
heat
doth
penetrate
it
,
the
water
begins
to
rarefie
,
&
issueth
forth
with
a
hidious
and
marvelous
force
;
it
is
pleasure
to
see
how
it
blowes
the
fire
with
great
noise
.
Now
touching
the
forme
of
these
vessels
,
they
are
not
made
of
one
like
fashion
:
some
makes
them
like
a
bowle
,
some
like
a
head
painted
representing
the
winde
,
some
make
them
like
a
Peare
:
as
though
one
would
put
it
to
rost
at
the
fire
,
when
one
would
have
it
to
blow
,
for
the
taile
of
it
is
hollow
,
in
forme
of
a
funnell
,
having
at
the
top
a
very
little
hole
no
greater
than
the
head
of
a
pinne
.
Some
do
accustome
to
put
within
the
Aeolipile
a
crooked
funnell
of
many
foldings
,
to
the
end
that
the
winde
that
impetuously
rolles
▪
to
and
fro
within
,
may
imitate
the
noise
of
thunder
.
Others
content
themselves
with
a
simple
funnell
placed
right
upward
,
somewhat
wider
at
the
top
than
elsewhere
like
a
Cone
,
whose
basis
is
the
mouth
of
the
funnell
:
and
there
may
be
placed
a
bowle
of
Iron
or
Brasse
,
which
by
the
vapours
that
are
cast
out
will
cause
it
to
leap
up
,
and
dance
over
the
mouth
of
the
Aeolipile
.
Lastly
,
some
apply
near
to
the
hole
smal
Windmils
,
or
such
like
,
which
easily
turne
by
reason
of
the
vapours
;
or
by
help
of
two
or
more
bowed
funnels
,
a
bowle
may
be
made
to
turne●
these
Aeolipiles
are
of
excellent
use
for
the
melting
of
mettalls
and
such
like
.
Now
it
is
cunning
and
subtiltie
to
fill
one
of
these
Aeolipiles
with
water
at
so
little
a
hole
,
and
therefore
requires
the
knowledge
of
a
Philosopher
to
finde
it
out
:
and
the
way
is
thus
.
Heat
the
Aeolipiles
being
empty
,
and
the
aire
which
is
within
it
will
become
extreamely
rarefied
;
then
being
thus
hot
throw
it
into
water
,
and
the
aire
will
begin
to
be
condensed
:
by
which
meanes
it
will
occupie
lesse
roome
,
therefore
the
water
will
immediately
enter
in
at
the
hole
to
avoide
vacuitie
:
thus
you
have
some
practicall
speculation
upon
the
Aeolipile
.
PROBLEM
.
LXIX
.
Of
the
Thermometer
:
or
an
instrument
to
measure
the
degrees
of
heat
and
cold
in
the
aire
.
THis
Instrument
is
like
a
Cylindricall
pipe
of
Glasse
,
which
hath
a
little
ball
or
bowle
at
the
toppe
▪
the
small
end
of
which
is
placed
into
a
vessell
of
water
below
,
as
by
the
figure
may
be
seene
.
Then
put
some
coloured
liquor
into
the
Cylindricall
glasse
,
as
blew
,
red
,
yellow
,
green
,
or
such
like
:
such
as
is
not
thick
.
This
being
done
the
use
may
be
thus
.
Those
that
will
determine
this
change
by
numbers
and
degrees
,
may
draw
a
line
upon
the
Cylinder
of
the
Thermometer
;
and
divide
it
into
4
degrees
,
according
to
the
ancient
Philosophers
,
or
into
4
degrees
according
to
the
Physicians
,
dividing
each
of
these
8
into
8
others
:
to
have
in
all
64
divisions
,
&
by
this
way
they
may
not
only
distinguish
upon
what
degree
the
water
ascendeth
in
the
morning
,
at
midday
,
&
at
any
other
houre
:
but
also
one
may
know
how
much
one
day
is
hotter
or
colder
than
another
:
by
marking
how
many
degrees
the
water
ascendeth
or
descendeth
,
one
may
compare
the
hottest
and
coldest
dayes
in
a
whole
year
together
with
these
of
another
year
:
againe
one
may
know
how
much
hotter
one
roome
is
than
another
,
by
which
also
one
might
keep
a
chamber
,
a
furnace
,
a
stove
,
&c.
alwayes
in
an
equalitie
of
heat
,
by
making
the
water
of
the
Thermometer
rest
alwayes
upon
one
&
the
same
degree
:
in
brief
,
one
may
judge
in
some
measure
the
burning
of
Fevers
,
and
neare
unto
what
extension
the
aire
can
be
rarefied
by
the
greatest
heat
.
Many
make
use
of
these
glasses
to
judge
of
the
weather
:
for
it
is
observed
that
if
the
water
fall
in
3
or
4
hours
a
degree
or
thereabout
,
that
raine
insueth
;
and
the
water
will
stand
at
that
stay
,
untill
the
weather
change
:
marke
the
water
at
your
going
to
bed
,
for
if
in
the
morning
it
hath
descended
raine
followeth
,
but
if
it
be
mounted
higher
,
it
argueth
faire
weather
:
so
in
very
cold
weather
,
if
it
fall
suddenly
,
it
is
snow
or
some
sleekey
weather
that
wiil
insue
,
PROBLEM
.
LXX
.
Of
the
proportion
of
humane
bodies
of
statues
,
of
Colossus
or
huge
images
,
and
of
monstrous
Giants
.
PYthagoras
had
reason
to
say
that
man
is
the
measure
of
all
things
.
First
,
because
he
is
the
most
perfect
amongst
all
bodily
creatures
,
&
according
to
the
Maxime
of
Philosophers
,
that
which
is
most
perfect
and
the
first
in
rank
,
measureth
all
the
rest
.
Secondly
,
because
in
effect
the
ordinary
measure
of
a
foot
,
the
inch
,
the
cubit
,
the
pace
,
have
taken
their
names
and
greatnesse
from
humane
bodies
.
Thirdly
,
because
the
symmetrie
and
concordancie
of
the
parts
is
so
admirable
,
that
all
workes
which
are
well
proportionable
,
as
namely
the
building
of
Temples
,
of
Shippes
,
of
Pillars
,
and
such
like
pieces
of
Architecture
,
are
in
some
measure
fashioned
and
composed
after
his
proportion
.
And
we
know
that
the
Arke
of
Noah
built
by
the
commandement
of
God
,
was
in
length
300
Cubits
,
in
breadth
50
Cubits
,
in
height
or
depth
30
cubits
,
so
that
the
length
containes
the
breadth
6
times
,
and
10
times
the
depth
:
now
a
man
being
measured
you
will
finde
him
to
have
the
same
proportion
in
length
,
breadth
,
and
depth
.
Vilalpandus
treating
of
the
Temple
of
Solomon
(
that
chieftaine
of
works
)
was
modulated
all
of
good
Architecture
,
and
curiously
to
be
observed
in
many
pieces
to
keep
the
same
proportion
as
the
body
to
his
parts
:
so
that
by
the
greatnesse
of
the
work
and
proportionable
symmetrie
,
some
dare
assure
themselves
that
by
knowledge
of
one
onely
part
of
that
building
,
one
might
know
all
the
measures
of
that
goodly
structure
.
Some
Architects
say
that
the
foundation
of
houses
,
and
basis
of
columnes
,
are
as
the
foot
;
the
top
,
and
roofe
as
the
head
;
the
rest
as
the
body
:
those
which
have
beene
somewhat
more
curious
,
have
noted
that
as
in
humane
bodies
,
the
parts
are
uniforme
,
as
the
nose
,
the
mouth
,
&c.
these
which
are
double
are
put
on
one
side
or
other
,
with
a
perfect
equality
in
the
same
Architecture
.
In
like
manner
,
some
have
been
yet
more
curious
than
solid
;
comparing
all
the
ornaments
of
a
Corinth
to
the
parts
of
the
face
,
as
the
brow
,
the
eyes
,
the
nose
,
the
mouth
;
the
rounding
of
Pillars
,
to
the
writhing
of
haire
,
the
channells
of
columnes
,
to
the
fouldings
of
womens
Robes
,
&c.
Now
building
being
a
work
of
the
best
Artist
,
there
is
much
reason
why
man
ought
to
make
his
imitation
from
the
chiefe
work
of
nature
;
which
is
man
.
Hence
it
is
that
Vitru●ius
in
his
third
book
,
and
all
the
best
Architectes
,
treate
of
the
proportion
of
man
;
amongst
others
Albert
Durens
hath
made
a
whole
book
of
the
measures
of
mans
body
,
from
the
foot
to
the
head
,
let
them
read
it
who
wil
,
they
may
have
a
prefect
knowledge
thereof
:
But
I
will
content
my selfe
and
it
may
satisfie
some
with
that
which
followeth
.
First
,
the
length
of
a
man
well
made
,
which
commonly
is
called
height
,
is
equall
to
the
distance
from
one
end
of
his
finger
to
the
other
:
when
the
armes
are
extended
as
wide
as
they
may
be
.
Secondly
,
if
a
man
have
his
feet
and
hands
extended
or
stretched
in
forme
of
S.
Andrews
Crosse
,
placing
one
foot
of
a
paire
of
Compasses
upon
his
navill
,
one
may
describe
a
circle
which
will
passe
by
the
ends
of
his
hands
and
feet
,
and
drawing
lines
by
the
termes
of
the
hands
and
feet
,
you
have
a
square
within
a
circle
.
Thirdly
,
the
breadth
of
man
,
or
the
space
which
is
from
one
side
to
another
;
the
breast
,
the
head
,
and
the
neck
,
make
the
6
part
of
all
the
body
taken
in
length
or
height
.
Fourthly
,
the
length
of
the
face
is
equall
to
the
length
of
the
hand
,
taken
from
the
small
of
the
arme
,
unto
the
extremity
of
the
longest
finger
.
Fiftly
,
the
thicknesse
of
the
body
taken
from
the
belly
to
the
back
;
the
one
or
the
other
is
the
tenth
part
of
the
whole
body
,
or
as
some
will
have
it
,
the
ninth
part
,
little
lesse
.
Sixtly
,
the
height
of
the
brow
,
the
length
of
the
nose
,
the
space
between
the
nose
and
the
chinne
,
the
length
of
the
eares
,
the
greatnesse
of
the
thumbe
,
are
perfectly
equall
one
to
the
other
.
What
would
you
say
to
make
an
admirable
report
of
the
other
parts
,
if
I
should
reckon
them
in
their
least
?
but
in
that
I
desire
to
be
excused
,
and
will
rather
extract
some
conclusion
upon
▪
that
which
is
delivered
.
In
the
first
place
,
knowing
the
proportion
of
a
man
,
it
is
easie
to
Painters
,
Image-makers
,
&c.
perfectly
to
proportionate
their
work
;
and
by
the
same
is
made
most
evident
,
that
which
is
related
of
the
images
and
statues
of
Greece
,
that
upon
a
day
diverse
workmen
having
enterprised
to
make
the
face
of
a
man
,
being
severed
one
from
another
in
sundry
places
,
all
the
parts
being
made
and
put
together
,
the
face
was
found
in
a
most
lively
and
true
proportion
.
Secondly
,
it
is
a
thing
most
cleare
,
that
by
the
help
of
proportion
,
the
body
of
Hercules
was
measured
by
the
knowledge
of
his
foot
onely
,
a
Lion
by
his
claw
,
the
Giant
by
his
thumb
,
and
a
man
by
any
part
of
his
body
.
For
so
it
was
that
Pythagoras
having
measured
the
length
of
Hercules
foot
,
by
the
steps
which
were
left
upon
the
ground
,
found
out
all
his
height
:
and
so
it
was
that
Phidias
having
onely
the
claw
of
a
Lion
,
did
figure
and
draw
out
all
the
beast
according
to
his
true
type
or
forme
,
so
the
exquisite
Painter
Timantes
,
having
painted
a
Pygmey
or
Dwarfe
,
which
he
measured
with
a
fadome
made
with
the
inch
of
a
Giant
,
it
was
sufficient
to
know
the
greatnesse
of
that
Giant
-
To
be
short
,
we
may
by
like
methode
come
easily
to
the
knowledge
of
many
fine
antiquities
touching
Statues
,
Colossus
,
and
monstrous
Giants
,
onely
supposing
one
had
found
but
one
only
part
of
them
,
as
the
head
,
the
hand
,
the
foot
or
some
bone
mentioned
in
ancient
Histories
.
Of
Statues
,
of
Colossus
,
or
huge
images
.
VItruvius
relates
in
his
second
book
,
that
the
Architect
Dinocrates
was
desirous
to
put
out
to
the
world
some
notable
thing
,
went
to
Alexander
the
great
,
and
proposed
unto
him
a
high
and
speciall
piece
of
work
which
he
had
projected
:
as
to
figure
out
the
mount
Athos
in
forme
of
a
great
Statue
,
which
should
hold
in
his
right
hand
a
Towne
capable
to
receive
ten
thousand
men
:
and
in
his
left
hand
a
vessell
to
receive
all
the
water
that
floweth
from
the
Mountaine
,
which
with
an
ingine
should
cast
into
the
Sea
.
This
is
a
pretty
project
,
said
Alexander
,
but
because
there
was
not
field-roome
thereabout
to
nourish
and
reteine
the
Citizens
of
that
place
,
Alexander
was
wise
not
to
entertaine
the
designe
.
Now
let
it
be
required
of
what
greatnesse
this
Statue
might
have
been
,
the
Towne
in
his
right
hand
,
and
the
receiver
of
water
in
his
left
hand
if
it
had
been
made
.
For
the
Statue
,
it
could
not
be
higher
than
the
Mountaine
it selfe
,
and
the
Mountaine
was
about
a
mile
in
height
plumb
or
perpendicular
;
therefore
the
hand
of
this
Statue
ought
to
be
the
10th
part
of
his
height
,
which
would
be
500
foot
,
and
so
the
breadth
of
his
hand
would
be
250
foot
,
the
length
now
multiplyed
by
the
breadth
,
makes
an
hundred
twenty
five
thousand
square
feet
,
for
the
quantitie
of
his
hand
to
make
the
towne
in
,
to
lodge
the
said
10
thousand
men
,
allowing
to
each
man
neere
about
12
foot
of
square
ground
:
now
judge
the
capacitie
of
the
other
parts
of
this
Collossus
by
that
which
is
already
delivered
.
Secondly
,
Plinie
in
his
34
book
of
his
natural
History
,
speakes
of
the
famous
Colossus
that
was
at
Rhodes
,
between
whose
legges
a
Shippe
might
passe
with
his
sailes
open
or
displayed
,
the
Statue
being
of
70
cubits
high
:
and
other
Histories
report
that
the
Sarasens
having
broken
it
,
did
load
900
Camels
with
the
mettal
of
it
,
now
what
might
be
the
greatnesse
and
weight
of
this
Statue
?
For
answer
,
it
is
usually
allowed
for
a
Camels
burthen
1200
pound
weight
,
therefore
all
the
Collosus
did
weigh
1080000
pound
weight
,
which
is
ten
hundred
and
fourescore
thousand
pound
weight
.
Now
according
to
the
former
rules
,
the
head
being
the
tenth
part
of
the
body
,
this
Statues
head
should
be
of
7
cubits
,
that
is
to
say
,
10
foot
and
a
halfe
,
and
seeing
that
the
Nose
,
the
brow
,
and
the
thumbe
,
are
the
third
part
of
the
face
,
his
Nose
was
3
foot
and
a
halfe
long
,
and
so
much
also
was
his
thumbe
in
length
:
now
the
thicknesse
being
alwayes
the
third
part
of
the
length
,
it
should
seem
that
his
thumb
was
a
foot
thick
at
the
least
.
Thirdly
,
the
said
Plinie
in
the
same
place
reports
that
Nero
did
cause
to
come
out
of
France
into
Italy
,
a
brave
and
bold
Statue-maker
called
Zenodocus
,
to
erect
him
a
Colossus
of
brasse
,
which
was
made
of
120
foot
in
height
,
which
Nero
caused
to
be
painted
in
the
same
height
.
Now
would
you
know
the
greatnesse
of
the
members
of
this
Colossus
,
the
breadth
would
be
20
foot
,
his
face
12
foote
,
his
thumb
and
his
nose
4
foot
,
according
to
the
proportion
before
delivered
.
Thus
I
have
a
faire
field
or
subject
to
extend
my selfe
upon
,
but
it
is
upon
another
occasion
that
it
was
undertaken
,
let
us
speak
therefore
a
word
touching
the
Giants
,
and
then
passe
away
to
the
matter
.
Of
monstrous
Giants
.
YOu
will
hardly
beleeve
all
that
which
I
say
touching
this
,
neither
will
I
beleeve
all
that
which
Authors
say
upon
this
subject
:
notwithstanding
you
nor
I
cannot
deny
but
that
long
ago
there
have
been
men
of
a
most
prodigious
greatnesse
;
for
the
holy
writings
witnesse
this
themselves
in
Deut
,
Chap.
3.
that
there
was
a
certaine
Giant
called
Og
,
of
the
Town
of
Rabath
,
who
had
a
bed
of
Iron
,
the
length
thereof
was
9
cubits
,
and
in
breadth
4
cubits
.
So
in
the
first
of
Kings
Chap.
17.
there
is
mention
made
of
Goliah
,
whose
height
was
a
palme
and
6
cubits
,
that
is
more
then
9
foot
,
he
was
armed
from
the
head
to
the
foot
,
and
his
Curiat
onely
with
the
Iron
of
his
lance
,
weighed
five
thousand
and
six
hundred
shekels
,
which
in
our
common
weight
,
is
more
than
233
pound
,
of
12
ounces
to
the
pound
:
Now
it
is
certaine
,
that
the
rest
of
his
armes
taking
his
Target
,
Helmet
,
Bracelets
,
and
other
Armour
together
,
did
weigh
at
the
least
5
hundred
pound
,
a
thing
prodigious
;
seeing
that
the
strongest
man
that
now
is
,
can
hardly
beare
200
pound
,
yet
this
Giant
carries
this
as
a
vesture
without
paine
.
Solinus
reporteth
in
his
5
Chap.
of
his
Historie
,
that
during
the
Grecians
warre
after
a
great
overflowing
of
the
Rivers
,
there
was
found
upon
the
sands
the
carcase
of
a
man
,
whose
length
was
33
Cubits
,
(
that
is
49
foot
and
a
halfe
)
therefore
according
to
the
proportion
delivered
,
his
face
should
be
5
foot
in
length
,
a
thing
prodigious
and
monstrous
.
Plinie
in
his
7.
book
and
16
Chap.
saith
,
that
in
the
Isle
of
Crete
or
Candie
,
a
mountaine
being
cloven
by
an
Earth-quake
,
there
was
a
body
standing
upright
,
which
had
46
Cubits
of
height
:
some
beleeve
that
it
was
the
body
of
Orion
or
Othus
,
(
but
I
think
rather
it
was
some
Ghost
or
some
delusion
)
whose
hand
should
have
beene
7
foot
,
and
his
nose
two
foot
and
a
half
long
.
But
that
which
Plutarch
in
the
l●fe
of
Sertorius
reports
of
,
is
more
strange
,
who
saith
,
that
in
Timgy
a
Morative
Towne
,
where
it
is
thought
that
the
Giant
Antheus
was
buried
,
Sertorius
could
not
beleeve
that
which
was
reported
of
his
prodigious
greatnesse
,
caused
his
sepulchre
to
be
opened
,
and
found
that
his
body
did
containe
60
Cubits
in
length
,
then
by
proportion
he
should
be
10
Cubits
or
15
foot
in
breadth
;
9
foot
for
the
length
of
his
face
,
3
foot
for
his
thumb
,
which
is
neare
the
capacitie
of
the
Colossus
at
Rhodes
.
But
behold
here
a
fine
fable
of
Symphoris
Campesius
,
in
his
book
intituled
Hortus
Gallicus
,
who
sayes
that
in
the
Kingdome
of
Sicilie
,
at
the
foot
of
a
mountaine
neare
Trepane
,
in
opening
the
foundation
of
a
house
,
they
found
a
Cave
in
which
was
●aid
a
Giant
,
which
held
in
stead
of
a
staffe
a
great
post
like
the
mast
of
a
Ship
:
and
going
to
handle
it
,
it
mouldered
all
into
ashes
,
except
the
bones
which
remained
of
an
exceeding
great
measure
,
that
in
his
head
there
might
be
easily
placed
5
quarters
of
corn
,
and
by
proportion
it
should
seeme
that
his
length
was
200
cubits
,
or
300
foot
:
if
he
had
said
that
he
had
been
300
cubits
in
length
,
then
he
might
have
made
us
beleeve
that
Noahs
Ark
was
but
great
enough
for
his
sepulchre
.
Who
can
believe
that
any
man
ever
had
20
cubits
,
or
30
foot
in
length
for
his
face
,
and
a
nose
of
10
foot
long
?
but
it
is
very
certaine
that
there
have
been
men
of
very
great
stature
,
as
the
holy
Scriptures
before
witnesse
,
and
many
Authours
worthy
of
beliefe
relate
:
Josephus
Acosta
in
his
first
book
of
the
Indian
History
,
Chap.
19.
a
late
writer
,
reporteth
,
that
at
Peru
was
found
the
bones
of
a
Giant
,
which
was
3
times
greater
than
these
of
ours
are
,
that
is
18
foot
,
for
it
is
usually
attributed
to
the
tallest
ordinary
man
in
these
our
times
but
6
foot
of
length
;
and
Histories
are
full
of
the
description
of
other
Giants
of
9
,
10
,
and
12
foot
of
height
,
and
it
hath
been
seen
in
our
times
some
which
have
had
such
heights
as
these
.
PROBLEM
.
LXXI
.
Of
the
game
at
the
Palme
,
at
Trap
,
at
Bowles
,
Paile-maile
,
and
others
.
THe
Mathematickes
often
findeth
place
in
sundry
Games
to
aid
and
assist
the
Gamesters
,
though
not
unknowne
unto
them
,
hence
by
Mathematicall
principles
,
the
games
at
Tennis
may
be
assisted
,
for
all
the
moving
in
it
is
by
right
lines
and
reflections
.
From
whence
comes
it
,
that
from
the
appearances
of
flat
or
convex
Glasses
,
the
production
and
reflection
of
the
species
are
explained
;
is
it
not
by
right
lines
?
in
the
same
proportion
one
might
sufficiently
deliver
the
motion
of
a
Ball
or
Bowle
by
Geometrical
lines
and
angles
.
And
the
first
maxime
is
thus
:
When
a
Bowle
toucheth
another
Bowle
▪
or
when
a
trapstick
striketh
the
Ball
,
the
moving
of
the
Ball
is
made
in
a
right
line
,
which
is
drawne
from
the
Centre
of
the
Bowle
by
the
point
of
contingencie
.
Secondly
,
in
all
kinde
of
such
motion
;
when
a
Ball
or
Bowle
rebounds
,
be
it
either
against
wood
,
a
wall
,
upon
a
Drumme
,
a
pavement
,
or
upon
a
Racket
;
the
incident
Angle
is
alwayes
equall
to
the
Angle
of
reflection
.
Now
following
these
maximes
,
it
is
easie
to
canclude
,
first
,
in
what
part
of
the
wood
or
wall
,
one
may
make
the
Bowle
or
Ball
go
to
reflect
or
rebound
,
to
such
a
place
as
one
would
.
Secondly
,
how
one
may
cast
a
Bowle
upon
another
,
in
such
sort
that
the
first
or
the
second
shall
go
and
meet
with
the
third
,
keeping
the
reflection
or
Angle
of
incidence
equal
.
Thirly
,
how
one
may
touch
a
Bowle
to
send
it
to
what
part
one
pleaseth
:
such
and
many
other
practices
may
be
done
.
At
the
exercises
at
Keyls
there
must
be
taken
heed
that
the
motion
slack
or
diminish
by
little
and
little
,
and
may
be
noted
that
the
Maximes
of
reflections
cannot
be
exactly
observed
by
locall
motion
,
as
in
the
beames
of
light
and
of
other
quallities
,
whereof
it
is
necessary
to
supply
it
by
industry
or
by
strength
,
otherwise
one
may
be
frustrated
in
that
respect
.
PROBLEM
.
LXXII
.
Of
the
Game
of
square
formes
.
NVmbers
have
an
admirable
secrecie
,
diversly
applied
,
as
before
in
part
is
shewed
,
and
here
I
will
say
something
by
way
of
transmutation
of
numbers
.
It
's
answered
thus
,
in
the
first
forme
the
men
were
as
the
figure
A
,
then
each
of
these
4
Souldiers
placed
themselves
at
each
Gate
,
and
removing
one
man
from
each
Angle
to
each
Gate
,
then
would
they
be
also
9
in
each
side
according
to
the
figure
B.
Lastly
,
these
4
Souldiers
at
the
Gates
take
away
each
one
his
Cumrade
,
and
placing
two
of
these
men
which
are
at
each
Gate
to
each
Angle
,
there
will
be
still
9
for
each
side
of
the
square
,
according
to
the
figure
C.
In
like
manner
if
there
were
12
men
,
how
might
they
be
placed
about
a
square
that
the
first
side
shall
have
3
every
way
,
then
disordered
,
so
that
they
might
be
4
every
way
;
and
lastly
,
being
transported
might
make
5
every
way
?
&
this
is
according
to
the
figures
,
F.
G.H
PROBLEM
.
LXXIII
.
How
to
make
the
string
of
a
Viole
sensibly
shake
,
without
any
one
touching
it
?
THis
is
a
miracle
in
musick
,
yet
easie
to
be
experimented
.
Take
a
Viole
or
other
Instrument
,
and
choose
two
strings
,
so
that
there
be
one
between
them
;
make
these
two
strings
,
agree
in
one
and
the
same
tune
:
then
move
the
Viole-bowe
upon
the
greater
string
,
and
you
shall
see
a
wonder
:
for
in
the
same
time
that
that
shakes
which
you
play
upon
,
the
other
will
likewise
sensibly
shake
without
any
one
touching
it
;
and
it
is
more
admirable
that
the
string
which
is
between
them
will
not
shake
at
all
:
and
if
you
put
the
first
string
to
another
tune
or
note
,
and
loosing
the
pin
of
the
string
,
or
stopping
it
with
your
finger
in
any
fret
,
the
other
string
will
not
shake
:
and
the
same
will
happen
if
you
take
two
Violes
,
and
strike
upon
a
string
of
the
one
,
the
string
of
the
other
will
sensibly
shake
.
Now
it
may
be
demanded
,
how
comes
this
shaking
,
is
it
in
the
occult
sympathie
,
or
is
it
in
the
strings
being
wound
up
to
like
notes
or
tunes
,
that
so
easily
the
other
may
receive
the
impression
of
the
aire
,
which
is
agitated
or
moved
by
the
shaking
or
the
trembling
of
the
other
?
&
whence
is
it
that
the
Viole-bowe
moved
upon
the
first
string
,
doth
instantly
in
the
same
time
move
the
third
string
,
and
not
the
second
?
if
the
cause
be
not
either
in
the
first
or
second
?
I
leave
to
others
to
descant
on
.
EXAMINATION
.
IN
this
Examination
we
have
something
else
to
imagine
,
than
the
bare
sympathie
of
the
Cords
one
to
another
:
for
first
there
ought
to
be
considered
the
different
effect
that
it
produceth
by
extention
upon
one
and
the
same
Cord
in
capacitie
:
then
what
might
be
produced
upon
different
Cords
of
length
and
bigness
to
make
them
accord
in
a
unisone
or
octavo
,
or
some
consort
intermediate
:
this
being
naturally
examined
,
it
will
be
facill
to
lay
open
a
way
to
the
knowledge
of
the
true
and
immediate
cause
of
this
noble
and
admirable
Phaenomeny
.
Now
this
will
sensibly
appeare
when
the
Cords
are
of
equall
length
and
greatnesse
,
and
set
to
an
unisone
;
but
when
the
Cords
differ
from
their
equalitie
,
it
will
be
lesse
sensible
:
hence
in
one
and
the
same
Instrument
,
Cords
at
a
unisone
shall
excite
or
shake
more
than
that
which
is
at
an
octavo
,
and
more
than
those
which
are
of
an
intermediate
proportionall
consort
:
as
for
the
other
consorts
they
are
not
exempted
,
though
the
effect
be
not
so
sensible
,
yet
more
in
one
than
in
another
:
and
the
experiment
will
seem
more
admirable
in
taking
2
Lutes
,
Viols
,
&c.
&
in
setting
them
to
one
tune
:
for
then
in
touching
the
Cord
of
the
one
,
it
will
give
a
sensible
motion
to
the
Cord
of
the
other
:
and
not
onely
so
but
also
a
harmony
.
PROBLEM
.
LXXIIII
.
Of
a
vessell
which
containes
three
severall
kindes
of
liquor
,
all
put
in
at
one
bung-hole
,
and
drawn
out
at
one
tap
severally
without
mixture
.
THe
vessell
is
thus
made
,
it
must
be
divided
into
three
Cells
for
to
conteine
the
three
liquors
,
which
admit
to
be
Sack
,
Claret
,
and
White-wine
:
Now
in
the
bung-hole
there
is
an
Engine
with
three
pipes
,
each
extending
to
his
proper
Cell
,
into
which
there
is
put
a
broach
or
funnell
pierced
in
three
places
,
in
such
sort
,
that
placing
one
of
the
holes
right
against
the
pipe
which
answereth
unto
him
,
the
other
two
pipes
are
stopped
;
then
when
it
is
full
,
turne
the
funnel
,
and
then
the
former
hole
will
be
stopped
,
and
another
open
,
to
cast
in
other
wine
without
mixing
it
with
the
other
.
Now
to
draw
out
also
without
mixture
,
at
the
bottome
of
the
vessell
there
must
be
placed
a
pipe
or
broach
,
which
may
have
three
pipes
;
and
a
cock
piersed
with
three
holes
so
artificially
done
,
that
turning
the
cock
,
the
whole
which
answereth
to
such
of
the
pipes
that
is
placed
at
the
bottom
,
may
issue
forth
such
wine
as
belongeth
to
that
pipe
,
&
turning
the
Cock
to
another
pipe
,
the
former
hole
wil
be
stopped
;
and
so
there
will
issue
forth
another
kinde
of
wine
without
any
mixtures
;
but
the
Cocke
may
be
so
ordered
that
there
may
come
out
by
it
two
wines
together
,
or
all
three
kindes
at
once
:
but
it
seems
best
when
that
in
one
vessell
and
at
one
Cocke
,
a
man
may
draw
severall
kindes
of
wine
,
and
which
he
pleaseth
to
drink
.
PROBLEM
.
LXXV
.
Of
burning-Glasses
.
IN
this
insuing
discourse
I
will
shew
the
invention
of
Prom●theus
,
how
to
steale
fire
from
Heaven
,
and
bring
it
down
to
the
Earth
;
this
is
done
by
a
little
round
Glasse
,
or
made
of
steele
,
by
which
one
may
light
a
Candle
,
and
make
it
flame
,
kindle
Fire-brands
to
wake
them
burne
,
melt
Lead
,
●inne
,
Gold
,
and
Silver
,
in
a
little
time
▪
with
as
great
ease
as
though
it
had
been
put
into
a
Cruzet
over
a
great
fire
.
But
this
is
nothing
to
the
burning
of
those
Glasses
which
are
hollow
,
namely
those
which
are
of
steele
well
polished
,
according
to
a
par●bolicall
or
ovall
section
.
A
sphericall
Glasse
,
or
that
which
is
according
to
the
segment
of
a
Sphere
,
burnes
very
effectually
about
the
fourth
part
of
the
Diameter
;
notwithstanding
the
Parabolie
and
Ecliptick
sections
have
a
great
effect
:
by
which
Glasses
there
are
also
diverse
figures
represented
forth
to
the
eye
.
The
cause
of
this
burning
is
the
uniting
of
the
beames
of
the
Sunne
,
which
heat
mightily
in
the
point
of
concourse
or
inflammation
,
which
is
either
by
transmissi●n
or
reflection
▪
Now
it
is
pleasant
to
behold
when
one
breatheth
in
the
point
of
concourse
,
or
throweth
small
dust
there
,
or
sprinkles
vapours
of
hot
water
in
that
place
;
by
which
the
Pyramidall
point
,
or
point
of
inflammation
is
knowne
.
Now
some
Authors
promise
to
make
Glasses
which
shall
burne
a
great
distance
off
,
but
yet
not
seen
vulgarly
produced
,
of
which
if
they
were
made
,
the
Parabolie
makes
the
greatest
eff●ct
,
and
is
g●nerally
held
to
be
the
invention
of
Archimedes
or
Pro●●us
.
Maginus
in
the
5
Chap.
of
his
Treatise
of
sphericall
Glasses
,
shewes
how
one
may
serve
himselfe
with
a
concave
Glasse
,
to
light
fire
in
the
shadow
,
or
neare
such
a
place
where
the
Sunne
shines
not
,
which
is
by
help
of
a
flat
Glasse
,
by
which
may
be
made
a
percussion
of
the
beames
of
the
Sun
into
the
concave
Glasse
,
adding
unto
it
that
it
serves
to
good
use
to
put
fi●e
to
a
Mine
,
provided
that
the
combustible
matter
be
well
applyed
before
the
concave
Glasse
;
in
which
he
saies
true
:
but
because
all
the
effect
of
the
practice
depends
upon
the
placing
of
the
Glasse
and
the
Powder
which
he
speaks
not
of
:
I
will
deliver
here
a
rule
more
generall
.
How
one
may
place
a
Burning-glasse
with
his
combust●ble
matter
in
such
sort
,
that
at
a
convenient
houre
of
the
day
,
the
Sun
shining
,
it
shall
take
fire
and
burne
:
Now
it
is
certaine
that
the
point
of
inflammation
or
burning
,
is
changed
as
the
Sun
changeth
place
,
and
no
more
nor
lesse
,
than
the
shadow
turnes
about
the
style
of
a
Dyall
;
therefore
have
regard
to
the
Suns
motion
,
and
●is
height
and
place
:
a
Bowle
of
Crystall
in
the
same
place
that
the
top
of
the
style
is
,
and
the
Powder
or
other
combustible
matter
under
the
Meridian
,
or
houre
of
12
,
1
,
2
,
3
,
&c.
or
any
other
houre
,
and
under
the
Suns
arch
for
that
day
:
now
the
Sunne
comming
to
the
houre
of
12
,
to
●
,
2
,
●
,
&c.
the
Sunne
casting
his
beames
through
the
Crystall
Bowle
,
will
fire
the
materiall
or
combustible
thing
,
which
meets
in
the
point
of
burning
:
the
like
may
be
observed
of
other
Burning-glasses
.
EXAMINATION
.
IT
is
certaine
in
the
first
part
of
this
Probleme
that
Conicall
,
●oncave
and
sphericall
Glasses
,
of
what
matter
soever
,
being
placed
to
receive
the
beames
of
the
Sun
will
excite
heat
,
and
that
heat
is
so
much
the
greater
,
by
how
much
it
is
neere
the
point
of
conc●rse
or
inflamatio●
.
But
that
Archimedes
or
Proclus
d●d
fire
or
burne
Shipps
with
such
Glasses
,
the
ancient
Histories
are
silent
,
yea
the
selves
say
nothing
:
besides
the
great
difficultie
that
doth
oppose
it
in
remotenesse
,
and
the
matter
that
the
effect
is
to
work
upon
:
Now
by
a
common
Glasse
we
fire
things
neare
at
hand
,
from
which
it
seems
very
facil
to
such
which
are
lesse
read
,
to
do
it
at
a
farre
greater
distance
,
and
so
by
re●ation
some
deliver
to
the
World
by
supposition
that
which
never
was
done
in
action
:
this
we
say
the
rather
,
not
to
take
away
the
most
excellent
and
admirable
effects
which
are
in
Burning-glasses
,
but
to
shew
the
variety
of
Antiquity
,
and
truth
of
History
:
and
as
touching
to
burne
at
a
great
distance
,
as
is
said
of
some
,
it
is
absolutely
impossible
;
and
that
the
Parabolicall
and
Ovall
Glasses
were
of
Archimedes
and
●roclus
invention
is
much
uncertaine
:
for
besides
the
construction
of
such
Glasses
,
they
are
more
difficult
than
the
obtuse
concave
ones
are
;
and
further
,
they
cast
not
a
great
heat
but
neere
at
hand
;
for
if
it
be
cast
farre
off
,
the
effect
is
little
,
and
the
heat
weake
,
or
otherwise
such
Glasses
must
be
greatly
extended
to
contract
many
beames
to
amasse
a
sufficient
quantity
of
beames
in
Parabolicall
and
Conicall
Glasses
,
the
point
of
inflammation
ought
to
concur
in
a
point
,
which
is
very
difficult
to
be
done
in
a
due
proportion
.
Moreover
if
the
place
be
farre
remote
,
as
is
supposed
before
,
such
a
Glasse
cannot
be
used
but
at
a
great
inclination
of
the
Sunne
▪
by
which
the
eff●ct
of
●urning
is
d●min●shed
,
by
reason
of
the
weaknesse
of
the
Sunne-beames
.
And
here
may
be
noted
in
the
last
part
of
this
Probleme
,
that
by
r●ason
of
obstacles
if
one
plaine
Glasse
be
not
sufficient
,
a
second
Glasse
may
be
applyed
to
help
it
:
that
so
if
by
one
simple
reflection
it
cannot
be
done
,
yet
by
a
double
reflection
the
Sun-beames
may
be
●ast
into
the
said
Caverne
or
Mine
,
and
though
the
reflected
beams
in
this
case
be
weak
▪
yet
upon
a
〈◊〉
c●mbustible
matter
it
will
not
faile
to
do
the
effect
.
PROBLEM
.
LXXVI
.
Containing
m●ny
ple●sant
Questions
by
way
of
Arithmetick●
.
J
Will
not
in●ert
i●
this
Probleme
that
which
is
drawne
from
the
●reek
Epigrams
,
but
proposing
the
Question
immediately
will
give
the
an●wer
also
,
without
●●aying
to
shew
the
manner
how
they
are
answered
;
in
this
J
will
〈◊〉
be
tied
to
the
●reek
tearms
,
w●●ch
J
account
no●
proper
to
this
place
,
nei●●er
to
my
purpose
:
●et
t●o●e
●ead
that
will
Di●phanta
S●●●●biliu●
upon
Eu●li●●
and
others
,
and
they
may
be
satisf●ed
Of
the
〈…〉
the
Mule
.
JT
〈◊〉
●hat
●he
Mule
and
the
Asse
upon
a
day
〈◊〉
a
voyage
each
of
them
carried
a
Barrell
full
of
Wine
:
now
the
las●e
Asse
f●lt
her selfe
over-loaden
,
complained
and
bowed
under
her
burthen
;
which
th●
Mule
seeing
said
unto
her
being
angry
,
(
for
it
was
in
the
time
when
beasts
spake
)
Thou
great
Asse
,
wherefore
complainest
thou
?
if
I
had
but
onely
one
measure
of
that
which
thou
carriest
,
I
should
be
loaden
twice
as
much
as
thou
art
,
and
if
J
should
give
a
measure
of
my
loading
to
thee
,
yet
my
burthen
would
be
as
much
as
thine
.
Now
how
many
measures
did
each
of
them
carry
?
Answer
,
the
Mule
did
carry
7
measures
,
and
the
Asse
5
measures
:
for
if
the
Mule
had
one
of
the
measures
of
the
Asses
loading
,
then
the
Mule
would
have
8
measures
,
which
is
double
to
4
,
and
giving
one
to
the
Asse
,
each
of
them
would
have
equall
burthens
:
to
wit
,
6
measures
apiece
.
Of
the
number
of
Souldiers
that
fought
before
old
Troy
.
HOmer
being
asked
by
He●iodus
how
many
Grecian
Souldiers
came
against
Troy
?
who
answered
him
thus
;
The
Grecians
,
said
Homer
,
made
7
fires
,
or
had
7
Kitchins
,
and
before
every
fire
,
or
in
every
Kitchin
there
were
50
broaches
turning
to
rost
a
great
quantitie
of
flesh
,
and
each
broach
had
meat
enough
to
satisfie
900
men
:
now
judge
how
many
men
there
might
be
.
Answer
,
315000.
that
is
,
three
hundred
and
fifteen
thousand
men
,
which
is
cleare
by
multiplying
7
by
50
,
and
the
product
by
900
makes
the
said
315000.
Of
the
number
of
Crownes
that
two
men
had
.
JOhn
and
Peter
had
certaine
number
of
crowns
:
John
said
to
Peter
,
If
you
give
me
10
of
your
crownes
,
I
shall
have
three
times
as
much
as
you
have
:
but
Peter
said
to
J●hn
,
If
you
give
me
10
of
your
crownes
I
shall
have
5
times
as
much
as
you
have
:
how
much
had
each
of
them
?
Answere
,
John
had
15
crownes
and
5
sevenths
of
a
crowne
,
and
Peter
had
18
crownes
,
and
4
sevenths
of
a
crowne
.
For
if
you
adde
10
of
Peters
crownes
to
those
of
Johns
,
then
should
John
have
25
crownes
and
5
sevenths
of
a
crowne
,
which
is
triple
to
that
of
Peters
,
viz.
8
▪
and
4
sevenths
:
and
John
giving
10
to
Peter
,
Peter
should
have
then
28
crownes
,
and
4
sevenths
of
a
crowne
,
which
is
Quintupla
,
or
5
times
as
much
as
John
had
left
,
viz.
5
crownes
and
5
sevenths
.
In
like
manner
two
Gamesters
playing
together
,
A
and
B
▪
after
play
A
said
to
B
,
Give
me
2
crownes
of
thy
money
,
and
I
shall
have
twice
as
much
as
thou
hast
:
and
B
said
to
A
,
Give
me
2
crownes
of
thy
money
,
and
I
shall
have
4
times
as
much
as
thou
hast
:
now
how
much
had
each
?
Answer
,
A
had
3
and
5
seventhes
,
and
B
had
4
and
6
seventhes
.
About
the
houre
of
the
day
.
SOme
one
asked
a
Mathemacian
what
a
clocke
it
was
;
who
answered
that
the
rest
of
the
day
is
foure
thirds
of
that
which
is
past
:
now
judge
what
a
clock
it
is
.
Answer
,
if
the
day
were
according
to
the
Jewes
and
ancient
Romanes
,
which
ma●e
it
alwayes
to
be
12
houres
,
it
was
then
the
●
houre
,
and
one
seventh
of
an
hou●e
,
so
there
remained
of
the
whole
day
6
,
that
is
,
6
houres
,
and
6
sevenths
of
an
hour
.
Now
if
you
take
the
1/●
of
5
●/7
it
is
●2/7
or
●
and
●
7
,
which
multipled
by
4
makes
6
and
6/7
,
which
is
the
remainder
of
the
day
,
as
before
:
but
if
the
day
had
been
24
houres
,
then
the
houre
had
been
10
of
the
clock
▪
and
two
seventhes
of
an
houre
,
which
is
found
▪
out
by
dividing
12
,
or
24
by
●
.
There
might
have
been
added
many
curious
propositions
in
this
kinde
,
but
they
would
be
too
difficult
for
the
most
part
of
people
▪
therefore
I
have
omi●ted
them
▪
Of
Pythagoras
his
Schollers
.
PYthagoras
being
asked
what
number
of
Schollers
he
had
,
answered
,
that
halfe
of
them
studied
Mathematickes
,
the
fourth
part
Physick
,
the
seventh
part
Rethorick
,
and
besides
he
had
3
women
:
now
judge
you
saith
he
,
how
many
Schollers
I
have
.
Answer
,
he
had
in
all
28
,
the
halfe
of
which
is
14
,
the
quarter
of
which
is
7
,
and
the
seventh
part
of
which
is
which
14
,
7
,
and
4
,
makes
25
,
and
the
other
3
to
make
up
the
28
,
were
the
3
women
.
Of
the
number
of
Apples
given
amongst
the
Graces
and
the
Muses
.
THe
three
Graces
carrying
Apples
upon
a
day
,
the
one
as
many
as
the
other
,
met
with
the
9
Muses
,
who
asked
of
them
some
of
their
Apples
;
so
each
of
the
Graces
gave
to
each
of
the
Muses
alike
,
and
the
distribution
being
made
,
they
found
that
the
Graces
&
the
Muses
had
one
as
many
as
the
other
:
The
question
is
how
many
Apples
each
Grace
had
,
and
how
many
they
gave
to
each
Muse
?
●o
answer
the
qeustion
,
joyne
the
number
of
Graces
and
Muses
together
which
makes
12
,
and
so
many
Apples
had
each
Grace
:
Now
may
you
take
the
double
,
triple
,
&c.
of
12
that
is
24
,
36
,
&c.
conditionally
,
that
if
each
Grace
had
but
12
,
then
may
there
be
allotted
to
each
Muse
but
one
onely
;
if
24
,
then
to
each
2
Apples
,
if
●6
,
then
to
each
Muse
3
Apples
,
and
so
the
distribution
being
made
,
they
have
a
like
number
,
that
is
one
as
many
as
the
other
.
Of
the
Testament
or
last
Will
of
a
dying
Father
.
A
Dying
Father
left
a
thousand
Crownes
amongst
his
two
children
;
the
one
being
legitimate
,
and
the
other
a
Bastard
,
conditionally
that
the
fifth
part
which
his
legittimate
Sonne
should
have
,
should
exceed
by
10
,
the
fourth
part
of
that
which
the
Bastard
should
have
:
what
was
each
〈◊〉
part
?
Answer
,
the
legitimate
Sonne
had
577
crownes
and
7/●
,
and
the
Bastard
42●
crownes
and
2/9
now
the
fifth
part
of
577
and
7
ninthes
is
1●5
,
and
5/9
,
and
the
fourth
part
of
422
and
●
is
105
and
●
which
is
lesse
then
●15
●
by
10
,
according
to
the
Will
of
the
Testator
.
Of
the
Cups
of
Croesus
.
CRoesus
gave
to
the
Temple
of
the
●ods
six
Cups
of
Gold
▪
which
weighed
together
●00
Drammes
,
but
each
cup
was
heavier
one
than
another
by
one
Dram
:
how
much
did
each
of
them
therefore
weigh
?
Answer
,
the
first
weighed
102
Drammes
and
a
halfe
;
the
second
101
Drammes
and
a
halfe
,
the
third
100
Drammes
and
●
,
the
fourth
99
a
&
halfe
,
the
fifth
98
&
a
halfe
;
and
the
sixt
Cup
weighed
97
Drammes
and
a
halfe
▪
which
together
makes
600
Drams
as
before
.
Of
Cupids
Apples
.
CVpid
complained
to
his
mother
that
the
Muses
had
taken
away
his
Apples
,
Clio
,
said
he
,
took
from
me
the
fifth
part
,
Euterp
the
twelfth
part
,
Thalia
the
eighth
part
,
M●lp●meno
the
twentieth
part
,
Erates
the
seventh
part
▪
Terpomene
the
fourth
part
,
Polyhymnia
took
away
30
,
Vrania
220
,
and
Calliope
300.
so
there
were
left
me
but
5
Appls
,
how
many
had
he
in
all
at
the
first
?
I
answer
3●60
.
There
are
an
infinite
of
such
like
questions
amongst
the
Greek
Epigrams
:
but
it
would
be
unpleasant
to
expresse
them
all
:
I
will
onely
adde
one
more
,
and
shew
a
generall
rule
for
all
the
rest
.
Of
a
Mans
Age.
A
Man
was
said
to
passe
the
sixth
part
of
his
life
in
childe-hood
,
the
fourth
part
in
his
youth
,
the
●hird
part
in
Manhood
,
and
18
yeares
besides
in
old
age
:
what
might
his
Age
be
?
the
answer
is
,
72
yeares
:
which
and
all
others
is
thus
resolved
:
multiply
1/●
▪
¼
and
⅓
▪
together
,
that
is
,
6
by
4
makes
24
,
and
that
againe
by
3
makes
72
,
then
take
the
third
part
of
72
,
which
is
24
,
the
fourth
part
of
it
,
which
is
18
,
and
the
sixth
part
of
it
which
is
12
,
these
added
together
make
54
,
which
taken
from
72
,
rests
18
this
divided
by
18
(
spoken
in
the
Question
)
gives
1
,
which
multiplied
by
the
summe
of
the
parts
,
viz.
72
,
makes
72
,
the
Answer
as
before
.
Of
the
Lion
of
Bronze
placed
upon
a
Fountaine
with
this
Epigramme
.
OVt
of
my
right
eye
if
I
let
water
passe
,
I
can
fill
the
Cisterne
in
2
dayes
:
if
I
let
it
passe
out
of
the
left
eye
,
it
will
be
filled
in
3
dayes
:
if
it
passe
out
of
my
feet
,
the
Cistern
will
be
4
dayes
a●filling
;
but
if
I
let
the
water
passe
out
of
my
mouth
,
I
can
fill
the
Cistern
then
in
6
houres
:
in
what
time
should
I
fill
it
,
if
I
poure
forth
the
water
at
all
the
passages
at
once
?
The
Greeks
(
the
greatest
talkers
in
the
world
)
variously
apply
this
question
to
divers
statues
,
and
pipes
of
Fountaines
:
and
the
solution
is
by
the
Rule
of
●
,
by
a
generall
Rule
,
or
by
●lgebra
.
They
have
also
in
their
Anthologie
many
other
questions
,
but
because
they
are
more
proper
to
exercise
,
than
to
recreate
the
spirit
,
I
passe
them
over
(
as
before
)
with
silence
.
PROBLEM
.
LXXVII
.
Divers
excellent
and
admirable
experiments
upon
Glasses
.
THere
is
nothing
in
the
world
so
beautifull
as
light
:
and
nothing
more
recreative
to
the
sight
,
than
Glasses
which
reflect
:
therefore
I
will
now
produce
some
experiments
upon
them
,
not
that
will
dive
into
their
depth
(
that
were
to
lay
open
a
mysterious
thing
)
but
that
which
may
delight
and
recreate
the
spirits
:
Let
us
suppose
therefore
these
principles
,
upon
which
is
built
the
demonstration
of
the
appar●nces
which
are
made
●n
all
sort
of
Glasses
.
First
,
that
the
rayes
or
beames
,
which
reflect
upon
a
Glasse
,
make
the
Angle
of
incident
equall
to
the
Angle
of
Reflection
,
by
the
first
Theo.
of
the
Catoptick
of
Euc.
Secondly
,
that
in
all
plain
Glasses
,
the
Images
are
seen
in
the
perpendicular
line
to
the
Glasse
,
as
far
within
the
glass
as
the
object
is
without
it
.
Thirdly
,
in
Concave
,
or
Convex
Glasses
,
the
Images
are
seen
in
the
right
line
which
passeth
from
the
object
and
through
the
Centre
in
the
Glasse
.
Theo.
17.
and
18.
And
here
you
are
to
understand
,
that
there
is
not
meant
only
those
which
are
simple
Glasses
or
Glasses
of
steele
,
but
all
other
bodies
,
which
may
represent
the
visible
Image
of
things
by
reason
of
their
reflection
,
as
Water
,
Marble
,
Mettal
,
or
such
like
.
Now
take
a
Glasse
in
your
hand
and
make
experiment
upon
that
which
followeth
.
Experiment
upon
flat
and
plaine
Glasses
.
FIrst
,
a
man
cannot
see
any
thing
in
these
Glasses
,
if
he
be
not
directly
and
in
a
perpendicular
line
before
it
,
neither
can
he
see
an
object
in
these
Glasses
,
if
it
be
not
in
such
a
place
,
that
makes
the
Angle
of
incidence
equall
to
the
Angle
of
reflexion
:
therefore
when
a
Glasse
stands
upright
,
that
is
,
perpendicular
to
the
Horizon
,
you
cannot
see
that
which
is
above
,
except
the
Glasse
be
placed
down
flat
:
and
to
see
that
on
the
right
hand
,
you
must
be
on
the
left
hand
,
&c.
Secondly
,
an
image
cannot
be
seen
in
a
Glass
if
it
be
not
raised
above
the
surface
of
it
;
or
place
a
Glasse
upon
a
wall
,
you
shall
see
nothing
which
is
upon
the
plaine
of
the
wall
,
and
place
it
upon
a
Table
or
Horizontal
Plaine
,
you
shall
see
nothing
of
that
which
is
upon
the
Table
.
Thirdly
,
in
a
plaine
Glasse
all
that
is
seene
appeares
or
seemes
to
sink
behinde
the
Glasse
,
as
much
as
the
image
is
before
the
Glasse
,
as
before
is
said
.
Fourthly
,
(
as
in
water
)
a
Glasse
lying
downe
flat
,
or
Horizontall
,
Towers
,
Trees
,
Men
,
or
any
height
doth
appeare
,
inversed
or
upside
downe
;
and
a
Glasse
placed
upright
,
the
right
hand
of
the
Jmage
seems
to
be
the
left
,
and
the
left
seems
to
be
the
right
.
Fifthly
,
will
you
see
in
a
Chamber
that
which
is
done
in
the
street
,
without
being
seen
▪
then
a
Glasse
must
be
disposed
,
that
the
line
upon
which
the
Jmages
come
on
the
Glasse
,
make
the
Angle
of
incidence
equall
to
that
Angle
of
reflexion
.
Seventhly
,
present
a
Candle
upon
a
plaine
Glasse
,
and
look
flaunting
upon
it
,
so
that
the
Candle
and
the
Glasse
be
neere
in
a
right
line
,
you
shall
see
3
,
4
,
5
,
&c.
images
,
from
one
and
the
same
Candle
.
Eightly
,
take
two
plaine
Glasses
,
and
hold
them
one
against
the
other
,
you
shall
alternately
see
them
oftentimes
one
within
the
other
,
yea
within
themselves
,
againe
and
againe
.
Ninthly
,
if
you
hold
a
plaine
Glasse
behinde
your
head
,
and
another
before
your
face
,
you
may
see
the
h●nder
part
of
your
head
,
in
that
Glasse
which
you
hold
before
your
face
.
Tenthly
,
you
may
have
a
fine
experiment
if
you
place
two
Glasses
together
,
that
they
make
an
acute
angle
,
and
so
the
lesser
the
angle
is
,
the
more
apparances
you
shall
see
,
the
one
direct
,
the
other
inversed
,
the
one
approaching
,
and
the
other
retiring
.
Eleventhly
,
it
is
a
wonder
&
astonishment
to
some
,
to
see
within
a
Glasse
an
Image
without
knowing
from
whence
it
came
,
and
it
may
be
done
many
wayes
:
as
place
a
Glass
higher
than
the
eye
of
the
beholder
,
and
right
against
it
is
some
Image
;
so
it
resteth
not
upon
the
beholder
,
but
doth
cast
the
Image
upwards
.
Then
place
another
object
,
so
that
it
reflect
,
or
cast
the
Image
downeward
to
the
eye
of
the
spectator
▪
without
perceiving
it
being
hid
behinde
something
,
for
then
the
Glasse
will
represent
a
quite
contrary
thing
,
either
that
which
is
before
the
Glasse
,
or
that
which
is
about
it
,
to
wit
,
the
other
hidden
object
.
Twelfthly
,
if
there
be
ingraved
behinde
the
backside
of
a
Glasse
,
or
drawne
any
Image
upon
it
,
it
will
appeare
before
as
an
Image
,
without
any
appearance
:
o●
portraicture
to
be
perceived
.
EXAMINATION
.
THis
12
Article
of
ingraving
an
Image
behinde
the
Glasse
,
will
be
of
no
great
consequence
▪
because
the
lineaments
will
seem
so
obscure
,
but
if
there
were
painted
some
Image
,
and
then
that
covered
according
to
the
usuall
covering
of
Glasses
behinde
,
and
so
made
up
like
an
ordinary
looking-Glasse
having
an
Image
in
the
middle
,
in
this
respect
it
would
be
sufficiently
pleasant
:
and
that
which
would
admire
the
ignorant
,
and
able
to
exercise
the
most
subtillest
,
and
that
principally
if
the
Glasse
be
in
an
obscure
place
,
and
the
light
which
is
given
to
it
be
somewhat
farre
off
.
PLace
a
Glasse
neare
the
floor
of
a
Chamber
,
&
make
a
hole
through
the
place
under
the
Glasse
,
so
that
those
which
are
below
may
not
perceive
it
,
and
dispose
a
bright
Image
under
the
hole
so
that
it
may
cast
his
species
upon
the
Glasse
,
and
it
will
cause
admiration
to
those
which
are
below
that
know
not
the
cause
;
The
same
may
be
done
by
placing
the
Image
in
a
Chamber
adjoyning
,
and
so
make
it
to
be
seen
upon
the
side
of
the
Wall
.
14
In
these
Channel-Images
which
shew
one
side
a
deaths
head
,
&
another
side
a
faire
face
:
and
right
before
some
other
thing
:
it
is
a
thing
evident
,
that
setting
a
plaine
Glasse
sidewise
to
this
Image
you
shall
see
it
in
a
contrary
thing
,
then
that
which
was
presented
before
sidewise
.
15
Lastly
,
it
is
a
fine
secret
to
present
unto
a
plaine
Glasse
writing
with
such
industry
,
that
one
may
read
it
in
the
Glasse
,
and
yet
out
of
the
Glasse
there
is
nothing
to
be
known
,
which
will
thus
happen
,
if
the
writing
be
writ
backward
:
but
that
which
is
more
strange
,
to
shew
a
kinde
of
writing
to
a
plaine
Glasse
,
it
shall
appear
another
kinde
of
writing
both
against
sense
and
forme
,
as
if
there
were
presented
to
the
Glasse
WEL
it
would
shew
it
MET
;
if
it
were
written
thus
MIV
,
and
presented
to
the
Glasse
,
it
would
appeare
thus
VIM
;
for
in
the
first
,
if
the
Glasse
ly
flat
,
then
the
things
are
inversed
that
are
perpendicular
to
the
Glass
,
if
the
Glass
and
the
object
be
upright
,
then
that
on
the
right
hand
,
is
turned
to
the
left
,
as
in
the
latter
.
And
here
I
cease
to
speak
further
of
these
plaine
Glasses
,
either
of
the
Admirable
multiplications
,
or
appearances
,
which
is
made
in
a
great
number
of
them
;
for
to
content
the
sight
in
this
particular
,
one
must
have
recourse
to
the
Cabinets
of
great
Personages
who
inrich
themselves
with
most
beautifull
ones
.
Experiments
upon
Gibbous
,
or
convex
Sphericall
Glasses
.
IF
they
be
in
the
forme
of
a
Bowle
,
or
part
of
a
great
Globe
of
Glasse
,
there
is
singular
contentment
to
contemplate
on
them
.
First
,
because
they
present
the
objects
lesse
and
more
gracious
,
and
by
how
much
more
the
Images
are
separated
from
the
Glasse
,
by
so
much
the
more
they
diminish
in
Magnitude
.
Secondly
,
they
that
shew
the
Images
plaiting
,
or
foulding
,
which
is
very
pleasant
,
especially
when
the
Glasse
is
placed
downe
,
and
behold
in
it
some
Blanching
,
feeling
,
&c.
The
upper
part
of
a
Gallerie
,
the
porch
of
a
Hall
,
&c.
for
they
will
be
represented
as
a
great
vessel
having
more
belly
in
the
middle
then
at
the
two
ends
,
and
Posts
,
and
Joists
of
Timber
will
seeme
as
Circles
.
Thirdly
,
that
which
ravisheth
the
spirits
,
by
the
eye
,
and
which
shames
the
best
perspective
Painting
that
a
Painter
can
make
,
is
the
beautifull
contraction
of
the
Images
,
that
appeare
within
the
sphericity
of
these
small
Glasses
:
for
present
the
Glasse
to
the
lower
end
of
a
Gallarie
,
or
at
the
Corner
of
a
great
Court
full
of
People
,
or
towards
a
great
street
,
Church
,
fortification
,
an
Army
of
men
,
to
a
whole
Cittie
;
all
the
faire
Architecture
,
and
appearances
will
be
seene
contracted
within
the
circuit
of
the
Glasse
with
such
varietie
of
Colours
,
and
distinctions
in
the
lesser
parts
,
that
I
know
not
in
the
world
what
is
more
agreeable
to
the
sight
,
and
pleasant
to
behold
,
in
which
you
will
not
have
an
exact
proportion
,
but
it
will
be
variable
,
according
to
the
distance
of
the
Object
from
the
Glasse
.
Exptriments
upon
hollow
,
or
Concave
sphericall
Glasses
.
I
Have
heretofore
spoken
how
they
may
burne
,
being
made
of
Glasse
,
or
Metall
,
it
remaines
now
that
I
deliver
some
pleasant
uses
of
them
,
which
they
represent
unto
our
sight
,
and
so
much
the
more
notable
it
will
be
,
by
how
much
the
greater
the
Glasse
is
,
and
the
Globe
from
whence
it
is
extracted
for
it
must
in
proportion
as
a
segment
of
some
be
made
circle
or
orbe
.
EXAMINATION
.
IN
this
we
may
observe
that
a
section
of
2.3
.
or
4.
Inches
in
diameter
,
may
be
segments
of
spheres
of
2.3
.
or
4.
foot
●
nay
of
so
many
fadome
,
for
it
is
certaine
that
amongst
those
which
comprehend
a
great
portion
of
a
lesser
sphere
,
and
those
which
comprehend
a
little
segment
of
a
great
spheere
,
whether
they
be
equall
or
not
in
section
,
there
will
happen
an
evident
difference
in
one
and
the
same
experiment
,
in
the
number
,
situation
,
quantitie
,
and
figure
of
the
Images
of
one
or
many
different
objects
,
and
in
burning
there
is
a
great
difference
.
MAginus
,
in
a
little
Tractate
that
he
had
upon
these
Glasses
,
witnesseth
of
himselfe
that
he
hath
caused
many
to
be
polished
for
sundry
great
Lords
of
Italy
,
and
Germanie
,
which
were
segments
of
Globes
of
2.3
.
and
4.
foot
diameter
;
and
I
wish
you
had
some
such
like
to
see
the
experiments
of
that
which
followeth
;
it
is
not
difficult
to
have
such
made
,
or
bought
here
in
Town
,
the
contentment
herein
would
beare
with
the
cost
.
EXAMINATION
.
TOuching
Maginus
he
hath
nothing
ayded
us
to
the
knowledge
of
the
truth
by
his
extract
out
of
Vitellius
,
but
left
it
:
expecting
it
from
others
,
rather
than
to
be
plunged
in
the
search
of
it
himselfe
,
affecting
rather
the
forging
of
the
matter
,
and
composition
of
the
Glasses
,
than
Geometrically
to
establish
their
effects
.
FIrst
therefore
in
concave
Glasses
,
the
Images
are
seene
sometimes
upon
the
surface
of
the
Glasses
,
sometimes
as
though
they
were
within
it
and
behinde
it
,
deeply
sunk
into
it
,
sometimes
they
are
seene
before
,
and
without
the
Glasse
,
sometimes
between
the
object
and
the
Glasse
;
sometimes
in
the
place
of
the
Eye
,
sometimes
farther
from
the
Glasse
then
the
object
is
:
which
comes
to
passe
by
reason
of
the
divers
concourse
of
the
beames
,
and
change
of
the
place
of
the
Images
in
the
line
of
reflection
.
EXAMINATION
.
THe
relation
of
these
appearances
passe
current
amongst
most
men
,
but
because
the
curious
may
not
receive
prejudice
in
their
experiments
,
something
ought
to
be
said
thereof
to
give
it
a
more
lively
touch
:
in
the
true
causes
of
these
appearances
,
in
the
first
place
it
is
impossible
that
the
Image
can
be
upon
the
surface
of
the
Glasse
,
and
it
is
a
principall
point
to
declare
truly
in
which
place
the
Image
is
seen
in
the
Glasse
those
that
are
more
learned
in
Opticall
knowledge
affirme
the
contrary
,
and
nature
it selfe
gives
it
a
certaine
place
according
to
its
position
being
alwayes
seen
in
the
line
of
reflection
which
Alhazen
,
Vitellius
,
and
others
full
of
grea●
knowledge
,
have
confirmed
by
their
writings
:
but
in
their
particular
they
were
too
much
occupied
by
the
authority
of
the
Ancients
who
were
not
s●fficiently
ci●cumspect
in
experience
upon
which
the
principles
of
this
sub●ect
ought
to
be
built
,
an●
searched
not
fully
into
the
true
cause
of
these
appearances
,
seeing
they
leave
unto
posterities
many
〈◊〉
in
their
writings
,
●nd
those
that
followed
them
for
the
most
part
fell
into
the
like
errors
.
As
for
the
Jmages
to
bid●
in
the
eye
▪
it
cannot
be
but
is
imp●rtinent
and
absurd
;
but
it
followeth
that
,
by
how
much
neerer
the
ob●ect
appro●cheth
to
the
Glasse
,
by
so
much
the
more
the
appearances
seem
to
come
to
the
eye
:
and
if
the
eye
be
without
the
point
of
concourse
,
and
the
object
also
;
as
long
as
the
object
approacheth
thereto
,
the
representation
of
the
Image
cometh
neere
the
eye
,
but
passing
the
point
of
concourse
it
goes
back
againe
:
these
appearances
thus
approaching
do
not
a
little
astonish
those
which
are
ignorant
of
the
cause
:
they
are
inversed
,
if
the
eye
be
without
the
point
of
concourse
untill
the
object
be
within
,
but
contrarily
if
the
eye
be
between
the
point
of
concourse
and
the
Glasse
,
then
the
Jmages
are
direct
:
and
if
the
eye
or
the
object
be
in
the
point
of
concourse
,
the
Glasse
will
be
enlightened
and
the
Jmages
confused
,
and
if
there
were
but
a
spark
of
fire
in
the
said
point
of
concourse
,
all
the
Glasse
would
seeme
a
burning
fire-brand
,
and
we
dare
say
it
would
occurre
without
chance
,
and
in
the
night
be
the
most
certaine
and
subtilest
light
that
can
be
,
if
a
candle
were
placed
there
.
And
whosoever
shall
enter
into
the
search
of
the
truth
of
new
experiments
in
this
subject
without
doubt
he
will
confirme
what
we
here
speak
of
:
&
will
finde
new
lights
with
a
conveniable
position
to
the
Glasse
,
he
will
have
reflection
of
quantities
,
of
truth
,
and
fine
secrets
in
nature
,
yet
not
known
,
which
he
may
easily
comprehend
if
he
have
but
an
indifferent
sight
,
and
may
assure
himselfe
that
the
Images
cannot
exceed
the
fight
,
nor
trouble
it
,
a
thing
too
much
absurd
to
nature
.
And
it
is
an
absolute
verity
in
this
science
,
that
the
eye
being
once
placed
in
the
line
of
reflection
of
any
object
,
and
moved
in
the
same
line
:
the
obect
is
seene
in
one
and
the
same
place
immutable
;
or
if
the
Image
and
the
eye
move
in
their
owne
lines
,
the
representation
in
the
Glasse
seemes
to
invest
it selfe
continually
with
a
different
figure
.
NOw
the
Image
comming
thus
to
the
eye
,
those
which
know
not
the
secret
,
draw
their
sword
when
they
see
an
Image
thus
to
issue
out
of
the
Glasse
,
or
a
Pistoll
which
some
one
holds
behinde
:
and
some
Glasses
will
shew
a
sword
wholly
drawne
out
,
sepa●ated
from
the
Glasse
,
as
though
it
were
in
the
aire
:
and
it
is
daily
exercised
,
that
a
man
may
touch
the
Image
of
his
hand
or
his
face
out
of
the
Glasse
,
which
comes
out
the
farther
,
by
how
much
the
Glasse
is
great
and
the
Centre
remote
.
EXAMINATION
.
NOw
that
a
Pistoll
being
presented
to
a
Glasse
behinde
a
man
,
should
come
out
of
the
Glasse
,
and
make
him
afraid
that
stands
before
,
seeming
to
shoot
at
him
,
this
cannot
be
:
for
no
object
whatsoever
presented
to
a
concave
Glasse
,
if
it
be
not
neerer
to
the
G●asse
then
the
eye
is
it
comes
not
out
to
the
sight
of
the
party
;
therefore
he
needs
not
feare
that
which
is
said
to
be
behinde
his
back
,
and
comes
out
of
the
Glasse
;
for
if
it
doth
come
out
,
it
must
then
necessarily
be
before
his
face
,
so
in
a
concave
Glasse
whose
Centre
is
farre
remote
of
a
sword
,
stick
,
or
such
like
be
presented
to
the
Glasse
,
it
shall
totally
be
seen
to
come
forth
of
the
Glasse
and
all
the
hand
that
holds
it
.
And
here
generally
note
that
if
an
Image
be
seen
to
issue
out
of
the
Glasse
to
come
towards
the
face
of
any
one
that
stands
by
,
the
object
shall
be
likewise
seen
to
thrust
towards
that
face
in
the
Glass
and
may
easily
be
knowne
to
all
the
standers
by
:
so
many
persons
standing
before
a
Glasse
,
if
one
of
the
company
take
a
sword
,
and
would
make
it
issue
forth
towards
any
o●her
that
stands
there
:
let
him
chuse
his
Image
in
the
Gl●sse
and
carry
the
sword
right
towards
it
and
the
effect
will
follow
.
In
like
manner
ones
hand
being
presented
to
the
Glosse
as
it
is
thrust
towards
the
Centre
,
s●
the
representation
of
it
comes
towards
it
,
and
so
the
hands
will
seeme
to
be
united
,
or
to
touch
one
another
.
FRom
which
may
be
concluded
,
if
such
a
Glasse
be
placed
at
the
seeling
or
planching
of
a
Hall
,
so
that
the
face
be
Horizontall
and
look
downward
;
one
may
see
under
it
as
it
were
a
man
hanging
by
the
feet
,
and
if
there
were
many
placed
so
,
one
could
not
enter
into
that
place
without
great
feare
or
scaring
:
for
one
should
see
many
men
in
the
aire
as
if
they
were
hanging
by
the
feet
.
EXAMINATION
.
TOuching
a
Glasse
tyed
at
a
seeling
or
planching
,
that
one
may
see
a
man
hang
by
the
feet
in
the
aire
,
and
so
many
Glasses
,
many
men
may
be
seen
:
without
caution
this
is
very
absurd
for
if
the
Glasse
or
Glasses
be
not
so
great
that
the
Centre
of
the
sphere
upon
which
it
was
made
,
extend
not
neere
to
the
head
of
him
that
is
under
it
,
it
will
not
pleasantly
appeare
,
and
though
the
Glasse
should
be
of
that
capacity
that
the
Centre
did
extend
so
farre
,
yet
will
not
the
Images
be
seene
to
them
which
are
from
the
Glasse
but
on●y
to
those
which
are
under
it
,
or
neere
unto
it
:
and
to
them
it
will
not
ably
appeare
,
and
it
would
be
most
admirable
to
have
a
Gallerie
vaulted
over
with
such
Glasses
which
would
wonderfully
astonish
any
one
that
enters
into
it
:
for
a●l
the
things
in
the
Gallery
would
be
seen
to
hang
in
the
aire
,
and
you
could
not
walk
without
incountering
airie
apparitions
.
SEcondly
,
in
flat
or
plaine
Glasses
the
Image
is
seen
equall
to
his
object
,
and
to
represent
a
whole
man
,
there
ought
to
be
a
Glasse
as
great
as
the
Image
is
:
In
convex
Glasses
the
Images
are
seen
alwayes
lesse
,
in
concave
Glasses
they
may
be
seen
greater
or
lesser
,
but
not
truly
proportionable
,
by
reason
the
diverse
reflexions
which
contracts
or
inlargeth
the
Species
:
when
the
eye
is
between
the
Centre
and
the
surface
of
the
Glasse
;
the
Image
appeares
sometimes
very
great
and
deformed
,
and
those
which
have
but
the
appearance
of
the
beginning
of
a
beard
on
their
chinne
,
may
cheare
up
themselves
to
see
they
have
a
great
beard
;
those
that
seeme
to
be
faire
will
thrust
away
the
Glasse
with
despight
,
because
it
will
transforme
their
beauty
:
those
that
put
their
hand
to
the
Glasse
will
seeme
to
have
the
hand
of
a
Giant
,
and
if
one
puts
his
finger
to
the
Glasse
it
will
be
seen
as
a
great
Pyramide
of
flesh
,
inversed
against
his
finger
.
Thirdly
,
it
is
a
thing
admirable
that
the
eye
being
approached
to
the
point
of
concourse
of
the
Glasse
,
there
will
be
seen
nothing
but
an
intermixture
or
confusion
:
but
retiring
back
a
little
from
that
point
,
(
because
the
rayes
do
there
meet
▪
)
he
shall
see
his
Image
inversed
,
having
his
head
below
and
his
feet
above
.
Fourthly
,
the
divers
appearances
caused
by
the
motion
of
objects
,
either
retiring
or
approaching
:
whether
they
turne
to
the
right
hand
or
to
the
left
hand
,
whether
the
Glasse
be
hung
against
a
wall
,
or
whether
it
be
placed
upon
a
Pavement
,
as
also
what
may
be
represented
by
the
mutuall
aspect
of
concave
Glasses
with
plaine
and
convex
Glasses
but
I
will
with
silence
passe
them
over
,
only
say
something
of
two
rare
experiments
more
as
followeth
.
The
first
is
to
represent
by
help
of
the
Sun
,
such
letters
as
one
would
upon
the
front
of
a
house
:
so
that
one
may
read
them
:
Maginus
doth
deliver
the
way
thus
.
Write
the
Letters
,
saith
he
,
sufficiently
bigge
,
but
inversed
upon
the
surface
of
the
Glasse
,
with
some
kinde
of
colour
,
or
these
letters
may
be
written
with
wax
,
(
the
easier
to
be
taken
out
againe
:
)
for
then
placing
the
Glasse
to
the
Sunne
,
the
letters
which
are
written
there
will
be
reverberated
or
reflected
upon
the
Wall
:
hence
it
was
perhaps
that
Pythagoras
did
promise
with
this
invention
to
write
upon
the
Moone
.
In
the
second
place
,
how
a
man
may
sundry
wayes
help
himselfe
with
such
a
Glasse
,
with
a
lighted
Torch
or
Candle
,
placed
in
the
point
of
concourse
or
inflammation
,
which
is
neare
the
fourth
part
of
the
Diameter
:
for
by
this
meanes
the
light
of
the
Candle
will
be
reverberated
into
the
Glasse
,
and
will
be
cast
back
againe
very
farre
by
parrallel
lines
,
making
so
great
a
light
that
one
may
clearly
see
that
which
is
done
farre
off
,
yea
in
the
camp
of
an
Enemie
:
and
those
which
shall
see
the
Glasse
a
farre
off
,
will
think
they
see
a
Silver
Basin
inlightened
,
or
a
fire
more
resplendent
then
the
Torch
.
It
is
this
way
that
there
are
made
certaine
Lanthorns
which
dazell
the
eyes
of
those
which
come
against
them
;
yet
it
serves
singular
well
to
enlighten
those
which
carry
them
,
accommodating
a
Candle
with
a
little
hollow
Glasse
,
so
that
it
may
successively
be
applyed
to
the
point
of
inflammation
.
In
like
manner
by
this
reflected
light
,
one
may
reade
farre
off
,
provided
that
the
letters
be
indifferent
great
,
as
an
Epitaph
placed
high
,
or
in
a
place
obscure
;
or
the
letter
of
a
friend
which
dares
not
approach
without
perill
or
suspicion
.
EXAMINATION
.
THis
will
be
scarce
sensible
upon
a
wall
remote
from
the
Glasse
,
and
but
indifferently
seen
upon
a
wall
which
is
neare
the
Glasse
,
and
withall
it
must
be
in
obscuritie
or
shadowed
,
or
else
it
will
not
be
seen
.
To
cast
light
in
the
night
to
a
place
remote
,
with
a
Candle
placed
in
the
point
of
concourse
or
inflammation
,
is
one
of
the
most
notablest
properties
which
can
be
shewne
in
a
concave
Glasse
:
for
if
in
the
point
of
inflammation
of
a
parabolicall
section
,
a
Candle
be
placed
,
the
light
will
be
reflected
by
parallel
lines
,
as
a
columne
or
Cylinder
;
but
in
the
sphericall
section
it
is
defective
in
part
,
the
beames
being
not
united
in
one
point
,
but
somewhat
scattering
:
notwithstanding
it
casteth
a
very
great
beautifull
light
.
Lastly
,
those
which
feare
to
hurt
their
sight
by
the
approach
of
Lampes
or
Candles
,
may
by
this
artifice
place
at
some
corher
of
a
Chamber
,
a
Lamp
with
a
hollow
Glasse
behinde
it
,
which
will
commodiously
reflect
the
light
upon
a
Table
,
or
to
a
place
assigned
:
so
that
the
Glasse
be
somewhat
raised
to
make
the
light
to
streeke
upon
the
Table
with
sharp
Angles
,
as
the
Sunne
doth
when
it
is
but
a
little
elevated
above
the
Horizon
,
for
this
light
shall
exceed
the
light
of
many
Candles
placed
in
the
Roome
,
and
be
more
pleasant
to
the
sight
of
him
that
useth
it
.
Of
other
Glasses
of
pleasure
.
FIrst
,
the
Columnary
and
Pyramidall
Glasses
that
are
contained
under
right
lines
,
do
represent
the
Images
as
plaine
Glasses
do
;
and
if
they
be
bowing
,
then
they
represent
the
Image
,
as
the
concave
and
convex
Glasses
do
.
Secondly
,
those
Glasses
which
are
plaine
,
but
have
ascents
of
Angels
in
the
middle
,
will
shew
one
to
have
foure
Eyes
,
two
Mouthes
,
two
Noses
,
&c.
EXAMINATION
.
TH●se
experiments
will
be
found
different
according
to
the
diverse
meeting
of
the
Glasses
,
which
commonly
are
made
scuing-wise
at
the
end
,
〈◊〉
which
there
will
be
two
divers
superficies
in
the
Glasse
,
making
the
exteriour
Angle
somewhat
raised
,
at
the
interiour
onely
one
superficies
,
which
may
be
covered
according
to
ordinary
Glasses
to
c●use
a
reflexion
,
and
so
it
will
be
but
one
Glasse
,
which
by
refraction
according
to
the
different
thicknesse
of
the
Glasse
,
and
different
Angles
of
the
scuing
forme
,
do
differently
present
the
Images
to
the
eye
,
as
foure
eyes
,
two
mouthes
,
two
noses
;
sometimes
three
eyes
one
mouth
,
and
one
nose
,
the
one
large
and
the
other
long
,
sometimes
two
eyes
onely
:
with
the
mouth
and
the
nose
deformed
,
which
the
Glasse
(
impenetrable
)
will
not
shew
.
And
if
there
be
an
interiour
solid
Angle
,
according
to
the
difference
of
it
(
as
if
it
be
more
sharp
)
there
will
be
represented
two
distinct
double
Images
,
that
is
,
two
entire
visages
and
as
the
Angle
is
open
,
by
so
much
the
more
the
double
Images
will
reunite
and
enter
one
within
another
,
which
will
present
sometimes
a
whole
visage
extended
at
large
,
to
have
foure
eyes
,
two
noses
,
and
two
mouthes
:
and
by
moving
the
Glasse
the
Angle
will
vanish
,
and
so
the
two
superficies
will
be
turned
into
one
,
and
the
duplicity
of
Images
will
also
vanish
and
appeare
but
one
onely
:
and
this
is
easily
experimented
with
two
little
Glasses
of
steel
,
or
such
like
so
united
,
that
they
make
divers
Angles
and
inclinations
.
THirdly
,
there
are
Glasses
which
make
men
seeme
pale
,
red
,
and
coloured
in
diverse
manners
,
which
is
caused
by
the
dye
of
the
Glasse
,
or
the
diverse
refraction
of
the
Species
:
and
those
which
are
made
of
Silver
,
Latine
,
Steele
,
&c.
do
give
the
Images
a
diverse
colour
also
.
In
which
one
may
see
that
the
appearances
by
some
are
made
fairer
,
younger
or
older
than
they
are
;
and
contrarily
others
will
make
them
foule
and
deformed
:
and
give
them
a
contrary
visage
:
for
if
a
Glasse
be
cut
as
it
may
be
,
or
if
many
pieces
of
Glasse
be
placed
together
to
make
a
conveniable
reflexion
:
there
might
be
made
of
a
Mole
(
as
it
were
)
a
mountaine
,
of
one
Haire
a
Tree
,
a
Fly
to
be
as
an
Elephant
,
but
I
should
be
too
long
if
I
should
say
all
that
which
might
be
said
upon
the
property
of
Glasses
.
I
will
therefore
conclude
this
discourse
of
the
properties
of
these
Glasses
with
these
foure
recreative
Problemes
following
.
PROBLEM
.
LXXVIII
.
1
How
to
shew
to
one
that
is
suspitious
,
what
is
done
in
another
Chamber
or
Roome
:
notwithstanding
the
interposition
of
the
wall
.
FOr
the
performance
of
this
,
there
must
be
placed
three
Glasses
in
the
two
Chambers
,
of
which
one
of
them
shall
be
tyed
to
the
planching
or
seeling
,
that
it
may
be
common
to
communicate
the
Species
to
each
Glasse
by
reflexion
,
there
being
left
some
hole
at
the
top
of
the
Wall
against
the
Glasse
to
this
end
:
the
two
other
Glasses
must
be
placed
against
the
two
Walls
at
right
Angles
,
as
the
figure
here
sheweth
at
B.
and
C.
Then
the
sight
at
E
by
the
line
of
incidence
FE
,
shall
fall
upon
the
Glasse
BA
,
and
reflect
upon
the
superficies
of
the
Glasse
BC
,
in
the
point
G
;
so
that
if
the
eye
be
at
G
,
it
should
see
E
,
and
E
would
reflect
upon
the
third
Glass
in
the
point
H
,
and
the
eye
that
is
at
L
,
will
see
the
Image
that
is
at
E.
in
the
point
of
the
Cath●r●
:
which
Image
shall
come
to
the
eye
of
the
suspicious
,
viz.
at
L.
by
help
of
the
third
Glasse
,
upon
which
is
made
the
second
reflexion
,
and
so
brings
unto
the
eye
the
object
,
though
a
wall
be
between
it
.
Corolarie
.
1.
BY
this
invention
of
reflections
the
besiegers
of
a
Towne
may
be
seene
upon
the
Rampart
:
notwithstanding
the
Parapet
,
which
the
besieged
may
do
by
placing
a
Glasse
in
the
hollow
of
the
Ditch
,
and
placing
another
upon
the
toppe
of
the
wall
,
so
that
the
line
of
incidence
comming
to
the
bottom
of
the
Ditch
,
make
an
Angle
equall
to
the
Angle
of
reflexion
,
then
by
this
situation
and
reflexion
,
the
Image
of
the
besiege●
〈◊〉
will
be
seen
to
him
is
upon
the
Rampart
Corolarie
2.
BY
which
also
may
be
inferred
,
that
the
same
reflexions
may
be
seen
in
a
Regular
Polygon
,
and
placing
as
many
Glasses
as
there
are
sides
,
counting
two
for
one
;
for
then
the
object
being
set
to
one
of
the
Glasses
,
and
the
eye
in
the
other
,
the
Jmage
will
be
seen
easily
.
Corolarie
3.
FArther
,
notwithstanding
the
interposition
of
many
Walls
,
Chambers
,
or
Cabinets
,
one
may
see
that
which
passeth
through
the
most
remotest
of
them
,
by
placing
of
many
Glasses
as
there
are
openings
in
the
walls
,
making
them
to
receive
the
incident
angles
equall
:
that
is
,
placing
them
in
such
sort
by
some
Geometricall
assistant
,
that
the
incident
points
may
meet
in
the
middle
of
the
Glasses
:
but
here
all
the
defect
will
be
,
that
the
Jmages
passing
by
so
many
reflexions
,
will
be
very
weak
and
scarce
observable
.
PROBLEM
.
LXXIX
.
How
with
a
Musket
to
strike
a
mark
,
not
looking
towards
it
,
as
exact
as
one
aiming
at
it
.
AS
let
the
eye
be
at
O
▪
and
the
mark
C
,
place
a
plaine
Glasse
perpendicular
as
AB
.
so
the
marke
C
shall
be
seen
in
Catheti
CA
,
viz.
in
D
,
and
the
line
of
reflexion
is
D
,
now
let
the
Musket
FE
,
upon
a
rest
▪
be
moved
to
and
fro
untill
it
be
seen
in
the
line
OD
,
which
admit
to
be
HG
,
so
giving
fire
to
the
Musket
,
it
shall
undoubtedly
strike
the
mark
.
Corolaries
.
From
which
may
be
gathered
,
that
one
may
exactly
shoot
out
of
a
Musket
to
a
place
which
is
not
seen
,
being
hindered
by
some
obstacle
,
or
other
interposition
.
AS
let
the
eye
be
at
M
,
the
mark
C
,
and
the
wall
which
keeps
it
from
being
seene
,
admit
to
be
QR
,
then
set
up
a
plaine
Glass
as
AB
,
and
let
the
Musket
by
GH
,
placed
upon
his
rest
PO.
Now
because
the
marke
C
is
seen
at
D
,
move
the
Musket
to
and
fro
,
untill
it
doth
agree
with
the
line
of
reflection
MB
,
which
suppose
at
LI
,
so
shall
it
be
truly
placed
,
and
giving
fire
to
the
Musket
,
it
shall
not
faile
to
strike
the
said
mark
at
C.
PROBLEM
.
LXXX
.
How
to
make
an
Image
to
be
seen
hanging
in
the
aire
,
having
his
head
downeward
.
TAke
two
Glasses
,
and
place
them
at
right
Angles
one
unto
the
other
,
as
admit
AB
,
and
CB
,
of
which
admit
CB
,
Ho●izontall
,
and
let
the
eye
be
at
H
,
and
the
object
or
image
to
be
DE
;
so
D
will
be
reflected
at
F
,
so
to
N
,
so
to
HE
:
then
at
G
,
so
to
●
and
then
to
H
,
and
by
a
double
reflection
ED
will
seeme
in
QR
,
the
highest
point
D
in
R
,
and
the
point
L
in
Q
inversed
as
was
said
,
taking
D
for
the
head
,
and
E
for
the
feet
;
so
it
will
be
a
man
inversed
,
which
will
seem
to
be
flying
in
the
aire
,
if
the
Jmage
had
wings
unto
it
,
and
had
secretly
〈◊〉
motion
:
and
if
the
Glasse
were
bigge
enough
to
receive
many
reflexions
,
it
would
deceive
the
sight
the
more
by
admiring
the
changing
of
colours
that
would
be
seen
by
that
motion
.
PROBLEM
.
LXXXI
.
How
to
make
a
company
of
representative
Souldiers
seeme
to
be
a
Regiment
,
or
how
few
in
number
may
be
multiplyed
to
seem
to
be
many
in
number
.
TO
make
the
experiment
upon
men
,
there
must
be
prepared
two
great
Glasses
;
but
in
stead
of
it
we
will
suppose
two
lesser
,
as
GH
.
and
FI
,
one
placed
right
against
another
perpendicular
to
the
Horizon
,
upon
a
plaine
levell
Table
:
betweene
which
Glasses
let
there
be
ranged
in
Battalia-wise
upon
the
same
Table
a
number
of
small
men
according
to
the
square
G
,
H
,
I
,
F
,
or
in
any
other
forme
or
posture
:
hen
may
you
evidently
see
how
the
said
battel
will
be
multiplyed
and
seem
farre
bigger
in
the
appearance
than
it
is
in
effect
.
Corolarie
.
BY
this
invention
you
may
make
a
little
Cabinet
of
foure
foot
long
,
and
two
foot
large
,
(
more
or
lesse
)
which
being
filled
with
Rockes
or
such
like
things
,
or
there
being
put
into
it
Silver
,
Gold
,
Stones
of
luster
,
Jewels
,
&c.
and
the
walls
of
the
said
Cabinet
being
all
covered
,
or
hung
with
plaine
glasse
;
these
visibles
will
appeare
manifoldly
increased
,
by
reason
of
the
multiplicitie
of
reflexions
,
and
at
the
opening
of
the
said
Cabinet
,
having
set
something
which
might
hide
them
from
being
seen
,
those
that
look
into
it
will
be
astonished
to
see
so
few
in
number
which
before
seemed
to
be
so
many
.
PROBLEM
.
LXXXII
.
Of
fine
and
pleasant
Dyal●
.
COuld
you
choose
a
more
ridiculous
one
than
the
natural
Dyall
written
amongst
the
Greek
Epigrams
,
upon
which
some
sound
Poet
made
verses
;
shewing
that
a
man
carrieth
about
him
alwayes
a
Dyall
in
his
face
by
meanes
of
the
Nose
and
Teeth
?
and
is
not
this
a
jolly
Dyall
?
for
he
need
not
but
open
the
mouth
,
the
lines
shall
be
all
the
teeth
,
and
the
nose
shall
serve
for
the
style
.
Of
a
Dyall
of
hearbes
.
CAn
you
have
a
finer
thing
in
a
Garden
,
or
in
the
middle
of
a
Compartemeet
,
than
to
see
the
lines
and
the
number
of
houres
represented
with
little
bushie
hearbes
,
as
of
Hysope
or
such
which
is
proper
to
be
cut
in
the
borders
;
and
at
the
top
of
the
style
to
have
a
Fanne
to
shew
which
way
the
winde
b●oweth
?
this
is
very
pleasant
and
useful
.
Of
the
Dyall
upon
the
fingers
and
the
hand
.
IS
it
nor
a
commoditie
very
agreeable
,
when
one
is
in
the
fie●d
or
in
some
vil●age
without
any
other
Dyall
,
to
see
onely
by
the
hand
what
of
the
clock
it
is
?
which
gives
it
very
neare
;
and
may
be
practised
by
the
left
hand
,
in
this
manner
.
Take
a
straw
or
like
thing
of
the
length
of
the
Index
or
the
second
finger
,
hold
this
straw
very
right
between
the
thumb
and
the
fore-finger
,
then
stretch
forth
the
hand
▪
and
turne
your
back
,
and
the
palm
of
your
hand
towards
the
Sunne
;
so
that
the
shadow
of
the
muscle
which
is
under
the
Thumb
,
touch
the
line
of
life
,
which
is
between
the
middle
of
the
two
other
great
lines
,
which
is
seen
in
the
palme
of
the
hand
,
this
done
,
the
end
of
the
shadow
will
shew
what
of
the
clock
it
is
:
for
at
the
end
of
the
first
finger
it
is
7
in
the
morning
,
or
5
in
the
evening
,
at
the
end
of
the
Ring-finger
it
is
8
in
the
morning
,
or
4
in
the
evening
,
at
the
end
of
the
little
finger
or
first
joynt
,
it
is
9
in
the
morning
,
or
3
in
the
after-noone
,
10
&
2
at
the
second
joynt
,
11
and
1
at
the
third
joynt
,
and
midday
in
the
line
following
,
which
comes
from
the
end
of
the
Index
.
Of
a
Dyall
which
was
about
an
Obeliske
at
Rome
.
WAs
not
this
a
pretty
fetch
upon
a
pavement
,
to
choose
an
Obeliske
for
a
Dyall
,
having
106
foot
in
height
,
without
removing
the
Basis
of
it
?
Plinie
assures
us
in
his
26
book
and
8
Chap.
that
the
Emperour
Augustus
having
accom●odated
in
the
field
of
Mars
an
Obeliske
of
this
height
,
he
made
about
it
a
pavement
,
and
by
the
industry
of
Man●lius
the
Mathematician
,
there
were
enchaced
markes
of
Copper
upon
the
Pavement
,
and
placed
also
an
Apple
of
Gold
upon
the
toppe
of
the
said
Obeliske
,
to
know
the
houre
and
the
course
of
the
Sunne
,
with
the
increase
and
decrease
of
dayes
by
the
same
shadow
:
and
in
the
same
manner
do
some
by
the
shadow
of
their
head
or
other
style
,
make
the
like
experiments
in
Astronomie
.
Of
Dyals
with
Glasses
.
PT●lomie
w●ites
,
as
Cardanus
reports
,
that
long
ago
there
were
Glasses
which
served
for
Dyals
,
and
presented
the
face
of
the
beholder
as
many
times
as
the
houre
ought
to
be
,
twice
if
it
were
2
of
the
clock
,
9
if
it
were
9
,
&c.
But
this
was
thought
to
be
done
by
the
help
of
water
,
and
not
by
Glasses
,
which
did
leake
by
little
and
little
out
of
the
vessell
,
discovering
anon
one
Glasse
,
then
anon
two
Glasses
,
then
3
,
4
,
5
Glasses
,
&c.
to
shew
so
many
faces
as
there
were
houres
,
which
was
onely
by
leaking
of
water
.
Of
a
Dyall
which
hath
a
Glasse
in
the
place
of
the
Style
.
WHat
will
you
say
of
the
invention
of
Mathematicians
,
which
finde
out
daily
so
many
fine
and
curious
novelties
?
they
have
now
a
way
to
make
Dyals
upon
the
wainscot
or
seeling
of
a
Chamber
,
and
there
where
the
Sunne
can
never
shine
,
or
the
beames
of
the
Sunne
cannot
directly
strike
:
and
this
is
done
in
placing
of
a
little
Glasse
in
the
place
of
the
style
which
reflecteth
the
light
,
with
the
same
condition
that
the
shadow
of
the
style
sheweth
the
houre
:
and
it
is
easie
to
make
experiment
upon
a
common
Dyall
,
changing
only
the
disposition
of
the
Dyall
,
and
tying
to
the
end
of
the
style
a
piece
of
plaine
Glasse
.
The
Almaines
use
it
much
,
who
by
this
way
have
no
greater
trouble
,
but
to
put
their
Noses
out
of
their
beds
and
see
what
a
clock
it
is
,
which
is
reflected
by
a
little
hole
in
the
Window
upon
the
wall
or
seeling
of
the
Chamber
.
EXAMINATION
.
IN
this
there
are
two
experiments
considerable
,
the
first
is
with
a
very
little
Glasse
placed
so
that
it
may
be
open
to
the
beames
of
the
Sunne
,
the
other
hath
respect
to
a
spacious
or
great
Glasse
placed
to
a
very
little
hole
so
that
the
Sun
may
shine
on
it
,
for
then
the
shadow
which
is
cast
upon
the
Dyall
is
converted
into
beames
of
the
Sunne
,
and
will
reflect
and
becast
upon
a
plain
opposite
:
and
in
the
other
it
is
a
hole
in
the
window
or
such
like
,
by
which
may
passe
the
beames
of
the
Sun
,
which
represent
the
extreamity
of
the
style
,
&
the
Glasse
representeth
the
plaine
of
the
Dyall
,
upon
which
the
beames
being
in
manner
of
shadowes
reflect
cast
upon
a
plaine
opposite
:
and
it
is
needfull
that
in
this
second
way
the
Glasse
may
be
spacious
,
as
before
,
to
receive
the
delineaments
of
the
Dyall
.
Otherwise
you
may
draw
the
lineaments
of
a
Dyall
upon
any
plaine
looking-glasse
which
reflecteth
the
Sunne-beames
,
for
the
applying
a
style
or
a
pearle
at
the
extreamitie
of
it
:
and
placed
to
the
Sunne
,
the
reflexion
will
be
answerable
to
the
delineaments
on
the
Glasse
:
but
here
note
,
that
the
Glasse
ought
to
be
great
,
and
so
the
delineaments
thereon
.
But
that
which
is
most
noble
,
is
to
draw
houre-lines
upon
the
outside
of
the
Glasse
of
a
window
,
and
placing
a
style
thereto
upon
the
outside
,
the
shadow
of
the
style
will
be
seen
within
,
and
so
you
have
the
hour
,
more
certaine
without
any
difficulty
.
Of
Dyals
with
water
.
SVch
kinde
of
Dyals
were
made
in
ancient
times
,
and
also
these
of
sand
:
before
they
had
skill
to
make
Sun-dyals
or
Dyals
with
wheeles
;
for
they
used
to
fill
a
vessell
with
water
,
and
having
experience
by
tryall
that
it
would
runne
out
all
in
a
day
,
they
did
marke
within
the
vessell
the
houres
noted
by
the
running
of
the
water
;
and
some
did
set
a
piece
of
light
board
in
the
vessell
to
swimme
upon
the
top
of
the
water
,
carrying
a
little
statue
,
which
with
a
small
stick
did
point
out
the
houre
upon
a
columne
or
wall
,
figured
with
houre-notes
,
as
the
vessell
was
figured
within
.
Now
it
seemes
a
safer
way
that
the
water
passe
out
by
drop
and
drop
,
and
drop
into
a
Cylindricall
Glasse
by
help
of
a
Pipe
:
for
having
marked
the
exterior
part
of
the
Cylinder
in
the
houre
notes
,
the
water
it selfe
which
falls
within
it
,
will
shew
what
of
the
clock
it
is
,
farre
better
than
the
running
of
sand
,
for
by
this
may
you
have
the
parts
of
the
houres
most
accurate
,
which
commonly
by
sand
is
not
had
:
and
to
which
may
be
added
the
houres
of
other
Countreys
with
greater
ease
.
And
here
note
,
that
as
soone
as
the
water
is
out
▪
of
one
of
the
Glasses
,
you
may
turne
it
over
into
the
same
againe
out
of
the
other
,
and
so
let
it
runne
anew
.
PROBLEM
.
LXXXIII
.
Of
Cannons
or
great
Artillery
.
Souldiers
,
and
others
would
willingly
see
〈◊〉
Problems
,
which
containe
:
three
or
foure
subtile
questions
:
The
first
is
,
how
to
charge
a
Cannon
without
powder
?
THis
may
be
done
with
aire
and
water
,
only
having
thrown
cold
water
into
the
Cannon
,
which
might
be
squirted
forceably
in
by
the
closure
of
the
mouth
of
the
Piece
,
that
so
by
this
pressure
the
aire
might
more
condense
;
then
having
a
round
piece
of
wood
very
just
,
and
oiled
well
for
the
better
to
slide
,
and
thrust
the
Bullet
when
it
shall
be
time
:
This
piece
of
wood
may
be
held
fast
with
some
Pole
,
for
feare
it
be
not
thrust
out
before
his
time
:
then
let
fire
be
made
about
the
Trunion
or
hinder
part
of
the
Piece
to
heat
the
aire
and
water
,
and
then
when
one
would
shoot
it
,
let
the
pole
be
quickly
loosened
,
for
then
the
aire
searching
a
greater
place
,
and
having
way
now
offered
,
will
thrust
out
the
wood
and
the
bullet
very
quick
:
the
experiment
which
we
have
in
long
trunkes
shooting
out
pellats
with
aire
only
,
sheweth
the
verity
of
this
Probleme
.
2
In
the
second
question
it
may
be
demanded
,
how
much
time
doth
the
Bull●●
of
a
Cannon
spend
in
the
aire
before
i●
falls
to
the
ground
?
THe
resolution
of
this
Question
depends
upon
the
goodnesse
of
the
Piece
&
charge
thereof
,
seeing
in
each
there
is
great
difference
.
It
is
reported
,
that
Tich●
Bra●e
,
and
the
Landsgrave
did
make
an
experiment
upon
a
Cannon
in
Germany
,
which
being
charged
and
shot
off
;
the
Bullet
spent
two
minutes
of
time
in
the
aire
before
it
fell
:
and
the
distance
was
a
Germane
mile
,
which
distance
proportionated
to
an
hours
time
,
makes
120
Italian
miles
.
3.
In
the
third
question
it
may
be
asked
,
how
it
comes
to
passe
,
that
a
Cannon
shooting
upwards
,
the
Bullet
flies
with
more
violence
than
being
shot
point-blanke
,
or
shooting
downeward
?
IF
we
regard
the
effect
of
a
Cannon
when
it
is
to
batter
a
wall
,
the
Question
is
false
,
seeing
it
is
most
evident
that
the
blowes
which
fall
Perpendicular
upon
a
wall
,
are
more
violent
than
those
which
strike
byas-wise
or
glaunsingly
.
But
considering
the
strength
of
the
blow
only
,
the
Question
is
most
true
,
and
often
experimented
to
be
found
true
:
a
Piece
mounted
at
the
best
of
the
Randon
,
which
is
neare
halfe
of
the
right
,
conveyes
her
Bullet
with
a
farre
greater
violence
then
that
which
is
shot
at
point
blanke
,
or
mounted
parallel
to
the
Horizon
.
The
common
reason
is
,
that
shooting
high
,
the
fire
carries
the
bowle
a
longer
time
in
the
aire
,
and
the
aire
moves
more
●acill
upwards
,
than
downewards
,
because
that
the
airy
circles
that
the
motion
of
the
bullet
makes
,
are
soonest
broken
.
Howsoever
this
be
the
generall
tenet
,
it
is
curious
to
finde
out
the
inequality
of
moving
of
the
aire
;
whether
the
Bullet
fly
upward
,
downeward
,
or
right
forward
,
to
produce
a
sensible
dfference
of
motion
;
&
some
think
that
the
Cannon
being
mounted
,
the
Bullet
pressing
the
powder
maketh
a
greater
resistance
,
and
so
causeth
all
the
Powder
to
be
inflamed
before
the
Bullet
is
throwne
out
,
which
makes
it
to
be
more
violent
than
otherwise
it
would
be
.
When
the
Cannon
is
otherwise
disposed
,
the
contrary
arives
,
the
fire
leaves
the
Bullet
,
and
the
Bullet
rolling
from
the
Powder
resists
lesse
:
and
it
is
usually
seene
,
that
shooting
out
of
a
Musket
charged
onely
with
Powder
,
to
shoot
to
a
marke
of
Paper
placed
Point
blanke
,
that
there
are
seene
many
small
holes
in
the
paper
,
which
cannot
be
other
than
the
graines
of
Powder
which
did
not
take
fire
:
but
this
latter
accident
may
happen
from
the
over-charging
of
the
Piece
,
or
the
length
of
it
,
or
windy
,
or
dampenesse
of
the
Powder
.
From
which
some
may
think
,
that
a
Cannon
pointed
right
to
the
Zenith
,
should
shoot
with
greater
violence
,
then
in
any
other
mount
or
forme
whatsoever
:
and
by
some
it
hath
beene
imagined
,
that
a
Bullet
shot
in
this
fashion
hath
been
consumed
,
melted
,
and
lost
in
the
Aire
,
by
reason
of
the
violence
of
the
blow
,
and
the
activity
of
the
sire
,
and
that
sundry
experiments
have
been
made
in
this
nature
,
and
the
Bullet
never
found
.
But
it
is
hard
to
believe
this
assertion
:
it
may
rather
be
supposed
that
the
Bullet
falling
farre
from
the
Piece
cannot
be
discerned
where
it
falls
:
and
so
comes
to
be
lost
.
4.
In
the
fourth
place
it
may
be
asked
,
whether
the
discharge
of
a
Cannon
b●
so
much
the
greater
,
by
how
much
it
is
longer
?
IT
seemeth
at
the
first
to
be
most
true
,
that
the
longer
the
Piece
is
,
the
more
violent
it
shoots
:
and
to
speak
generally
,
that
which
is
direction
by
a
Trunke
,
Pipe
,
or
other
concavitie
,
is
conveyed
so
much
the
more
violent
,
or
better
,
by
how
much
it
is
longer
,
either
in
respect
of
the
Sight
,
Hearing
,
Water
,
Fire
,
&c.
&
the
reason
seems
to
hold
in
Cannons
,
because
in
those
that
are
long
,
the
fire
is
retained
a
longer
time
in
the
concavitie
of
the
Piece
,
and
so
throwes
out
the
Bullet
with
more
violence
;
and
experience
lets
us
see
that
taking
Cannons
of
the
same
boare
,
but
of
diversitie
of
length
from
8
foot
to
12
,
that
the
Cannon
of
9
foot
long
hath
more
force
than
that
of
8
foot
long
,
and
10
more
than
that
of
9
,
and
so
unto
12
foote
of
length
.
Now
the
usuall
Cannon
carries
600
Paces
,
some
more
,
some
lesse
,
yea
some
but
200
Paces
from
the
Piece
,
and
may
shoot
into
soft
earth
15
or
17
foot
,
into
sand
or
earth
which
is
loose
,
22
or
24
foot
,
and
in
firme
ground
,
about
10
or
12
foot
,
&c.
It
hath
been
seen
lately
in
Germany
,
where
there
were
made
Pieces
from
8
foot
long
to
17
foot
of
like
boare
,
that
shooting
out
of
any
piece
which
was
longer
than
12
foot
;
the
force
was
diminished
,
and
the
more
in
length
the
Piece
increaseth
,
the
lesse
his
force
was
:
therefore
the
length
ought
to
be
in
a
meane
measure
,
and
it
is
often
seene
,
the
greater
the
Cannon
is
,
by
so
much
the
service
is
greater
:
but
to
have
it
too
long
or
too
short
,
is
not
convenient
,
but
a
meane
proportion
of
length
to
be
taken
,
otherwise
the
flame
of
the
fire
will
be
over-pressed
with
Aire
:
whic
hinders
the
motion
in
respect
of
substance
,
and
distance
of
getting
out
.
PROBLEM
.
LXXXIIII
.
Of
predigious
progression
and
multiplication
,
of
Creatures
,
Plants
,
Fruits
,
Numbers
,
Gold
,
Silver
,
&c.
when
they
are
alwayes
augmented
by
certaine
proportion
.
HEre
we
shall
shew
things
no
lesse
admirable
,
as
recreative
,
and
yet
so
certaine
and
easie
to
be
demonstrated
,
that
there
needs
not
but
Multiplication
only
,
to
try
each
particular
:
and
first
,
Of
graines
of
Mustard-seed
.
FIrst
,
therefore
it
is
certaine
that
the
increase
of
one
graine
of
Mustard-seed
for
20
yeares
space
,
cannot
be
contained
within
the
visible
world
,
nay
if
it
were
a
hundred
times
greater
than
it
is
:
and
holding
nothing
besides
from
the
Centre
of
the
earth
even
unto
the
firmament
,
but
only
small
grains
of
Mustard-seed
:
Now
because
this
seems
but
words
,
it
must
be
proved
by
Art
,
as
may
be
done
in
this
wise
,
as
suppose
one
Mustard-seed
sowne
to
bring
forth
a
tree
or
branch
,
in
each
extendure
of
which
might
be
a
thousand
graines
:
but
we
will
suppose
onely
a
thousand
in
the
whole
tree
,
and
let
us
proceed
to
●0
yeares
,
every
seed
to
bring
forth
yearely
a
thousand
graines
,
now
multiplying
alwayes
by
a
thousand
,
in
lesse
then
17
years
you
shall
have
to
many
graines
which
will
surpasse
the
sands
,
which
are
able
to
fill
the
whole
firmament
:
for
following
the
supposition
of
Archimedes
,
and
the
most
probable
opinion
of
the
greatness
of
the
firmament
which
●i●ho
Brahe
hath
left
us
;
the
number
of
graines
of
sand
will
be
sufficiently
expressed
with
49
Ciphers
,
but
the
number
of
graines
of
Mustard-seed
at
the
end
of
17
yeares
will
have
52
Ciphers
:
and
moreover
,
graines
of
Mustard-seed
,
are
farre
greater
than
these
of
the
sands
:
it
is
therefore
evident
that
at
the
seventeenth
yeare
,
all
the
graines
of
Mustard-seed
which
shall
successively
spring
from
one
graine
onely
,
cannot
be
contained
within
the
limits
of
the
whole
firmament
;
what
should
it
be
then
,
if
it
should
be
multiplied
againe
by
a
thousand
for
the
●8
yeare
:
and
that
againe
by
a
thousand
for
every
yeares
increase
untill
you
come
to
the
20
yeare
,
it
's
a
thing
as
cleare
as
the
day
,
that
such
a
heap
of
Mustard-seed
would
be
a
hundred
thousand
times
greater
than
the
Earth
:
and
bring
onely
but
the
increase
of
one
graine
in
20
yeares
.
Of
Pigges
.
SEcondly
,
is
it
not
a
strange
proposition
,
to
say
that
the
great
Turke
with
all
his
Revenues
,
is
not
able
to
maintaine
for
one
yeares
time
,
all
the
Pigges
that
a
Sow
may
pigge
with
all
her
race
,
that
is
,
the
increase
with
the
increase
unto
12
years
:
this
seemes
impossible
,
yet
it
is
most
true
,
for
let
us
suppose
and
put
,
the
case
,
that
a
Sow
bring
forth
but
6
,
two
Males
,
and
4
Females
,
and
that
each
Female
shall
bring
forth
as
many
every
yeare
,
during
the
space
of
12
yeares
,
at
the
end
of
the
time
there
will
be
found
above
3●
millions
of
Pigges
:
now
allowing
a
crowne
for
the
maintenance
of
each
Pigge
for
a
yeare
,
(
which
is
as
little
as
may
be
,
being
but
neare
a
halfe
of
a
farthing
allowance
for
each
day
;
)
there
must
be
at
the
least
so
many
crownes
to
maintaine
them
,
one
a
year
,
viz.
33
millions
,
which
exceeds
the
Turkes
revenue
by
much
.
Of
graines
of
Corne.
THirdly
,
it
will
make
one
astonished
to
think
that
a
graine
of
Corne
,
with
his
increase
successively
for
the
space
of
12
yeares
will
produce
in
grains
24414062●000000000000
,
which
is
able
to
load
almost
al
the
creatures
in
the
World
.
To
open
which
,
let
it
be
supposed
that
the
first
yeare
one
graine
being
sowed
brings
forth
50
,
(
but
sometimes
there
is
seen
70
,
sometimes
100
fold
)
which
graines
sowen
the
next
yeare
,
every
one
to
produce
50
,
and
so
consequently
the
whole
and
increase
to
be
sowen
every
yeare
,
until
12
yeares
be
expired
,
there
will
be
of
increase
the
aforesaid
prodigious
summe
of
graines
,
viz.
244140625000000000000
,
which
will
make
a
cubical
heap
of
6258522
graines
every
way
,
which
is
more
than
a
cubicall
body
of
31
miles
every
way
:
for
allowing
40
graines
in
length
to
each
foot
,
the
Cube
would
be
156463
foot
every
way
:
from
which
it
is
evident
that
if
there
were
two
hundred
thousand
Cities
as
great
as
London
,
allowing
to
each
3
miles
square
every
way
,
and
100
foot
in
height
,
there
would
not
be
sufficient
roome
to
containe
the
aforesaid
quantitie
of
Corne
:
and
suppose
a
bushel
of
Corne
were
equal
unto
two
Cubicke
feet
,
which
might
containe
twenty
hundred
thousand
graines
then
would
there
be
122070462500000.
bushells
,
and
allowing
30
bushels
to
a
Tunne
,
it
would
be
able
to
load
81380●0833
vessels
,
which
is
more
than
eight
thousand
one
hundred
and
thirty
eight
millions
,
ship
loadings
of
●00
Tunne
to
each
ship
a
:
quantity
so
great
that
the
Sea
is
scarce
able
to
beare
,
or
the
universal
world
able
to
finde
vessels
to
carry
it
.
And
if
this
Corne
should
be
valued
at
halfe
a
crown
the
bushel
,
it
would
amount
unto
15258807812500
pounds
sterling
,
which
I
think
exceeds
all
the
Treasures
of
all
the
Princes
,
and
of
other
particular
men
in
the
whole
world
:
and
is
not
this
good
husbandry
to
sowe
one
grain
of
Corne
;
and
to
continue
it
in
sowing
,
the
increase
only
for
12
yeares
to
have
so
great
a
profit
?
Of
the
increase
of
Sheep
.
FOurthly
,
those
that
have
great
flocks
of
Sheep
may
be
quickly
rich
,
if
they
would
preserve
their
Sheep
without
killing
or
selling
of
them
:
so
that
every
Sheep
produce
one
each
yeare
,
for
at
the
end
of
16
yeares
,
100
Sheepe
will
multiply
and
increase
unto
6553600
,
which
is
above
6
millions
,
5
hundred
53
thousand
Sheep
:
now
supposing
them
worth
but
a
crown
a
piece
,
it
would
amount
unto
1638400
pounds
sterling
,
which
is
above
1
million
6
hundred
38
thousand
pounds
,
a
faire
increase
of
one
Sheep
:
and
a
large
portion
for
a
Childe
if
it
should
be
allotted
.
Of
the
increase
of
Cod-fish
,
Carpes
,
&c.
FIfthly
,
if
there
be
any
creatures
in
the
world
that
do
abound
with
increase
or
fertilitie
,
it
may
be
rightly
attributed
to
fish
;
for
they
in
their
kindes
produce
such
a
great
multitude
of
Eggs
,
and
brings
forth
so
many
little
ones
,
that
if
a
great
part
were
not
destroyed
continually
,
within
a
●ittle
while
they
would
fill
all
the
Sea
,
Ponds
,
and
Rivers
in
the
world
;
and
it
is
easie
to
shew
how
it
would
come
so
to
passe
,
onely
by
supposing
them
to
increase
without
taking
or
destroying
them
for
the
space
of
10
or
12
yeares
:
having
regard
to
the
soliditie
of
the
waters
which
are
allotted
for
to
lodge
and
containe
these
creatures
,
as
their
bounds
and
place
of
rest
to
live
in
.
Of
the
increase
and
multiplication
of
men
.
SIxthly
,
there
are
some
that
cannot
conceive
how
it
can
be
that
from
eight
persons
(
which
were
saved
after
the
deluge
or
Noahs
flood
)
should
spring
such
a
world
of
people
to
begin
a
Monarchie
under
Nimrod
,
being
but
200
yeares
after
the
flood
,
and
that
amongst
them
should
be
raised
an
army
of
two
hundred
thousand
fighting
men
:
But
it
is
easi●y
proved
if
we
take
but
one
of
the
Children
of
Noah
,
and
suppose
that
a
new
generation
of
people
begun
at
every
30
yeares
,
and
that
it
be
continued
to
the
seventh
generation
which
is
200
yeares
;
for
then
of
one
only
family
there
would
be
produced
one
hundred
and
eleven
thousand
soules
,
three
hundred
and
five
to
begin
the
world
:
though
in
that
time
men
lived
longer
,
and
were
more
capable
of
multiplication
and
increase
:
which
number
springing
onely
from
a
simp●e
production
of
one
yearly
,
would
be
farre
greater
,
if
one
man
should
have
many
wives
,
which
in
ancient
times
they
had
:
from
which
it
is
also
that
the
Children
of
Israel
,
who
came
into
Egypt
but
onely
70
soules
,
yet
after
210
yeares
captivity
,
they
came
forth
with
their
hostes
,
that
there
were
told
six
hundred
thousand
fighting
men
,
besides
old
people
,
women
and
children
;
and
he
that
shall
separate
but
one
of
the
families
of
Joseph
,
it
would
be
sufficient
to
make
up
that
number
:
how
much
more
should
it
be
then
if
we
should
adjoyne
many
families
together
?
Of
the
increase
of
numbers
.
SEventhly
,
what
summe
of
money
shall
the
City
of
London
be
worth
,
if
it
should
be
sold
,
and
the
money
be
paid
in
a
yeare
after
this
manner
:
the
first
week
to
pay
a
pinne
,
the
second
week
2
pinnes
,
the
third
week
4
pinnes
,
the
fourth
week
8
pinnes
,
the
fifth
week
16
pinnes
.
and
so
doubling
untill
the
52
weeks
,
or
the
yeare
be
expired
.
Here
one
would
think
that
the
value
of
the
pinnes
would
amount
but
to
a
small
matter
,
in
comparison
of
the
Treasures
,
or
riches
of
the
whole
City
:
yet
it
is
most
probable
that
the
number
of
pinnes
would
amount
unto
the
sum
of
4519599628681215
,
and
if
we
should
allow
unto
a
quarter
a
hundred
thousand
pinnes
,
the
whole
would
contain
ninetie
eight
millions
,
foure
hundred
thousand
Tunne
:
which
is
able
to
load
45930
Shippes
of
a
thousand
Tunne
apiece
:
and
if
we
should
allow
a
thousand
pins
for
a
penny
,
the
summe
of
money
would
amount
unto
above
eighteen
thousand
,
eight
hundred
and
thirty
millions
of
pounds
sterling
,
an
high
price
to
sell
a
Citie
at
,
yet
certain
,
according
to
that
first
proposed
.
So
if
40
Townes
were
sold
upon
condition
to
give
for
the
first
a
penny
,
for
the
second
2
pence
,
for
the
third
4
pence
,
&c.
by
doubling
all
the
rest
unto
the
last
,
it
would
amount
unto
this
number
of
pence
,
109951●62●●76
,
which
in
pounds
is
4581298444
,
that
is
foure
thousand
five
hundred
and
fourescore
millions
of
pounds
and
more
.
Of
a
man
that
gathered
up
Apples
,
Stones
,
or
such
like
upon
a
condition
.
EIghtly
,
admit
there
were
an
hundred
Apples
,
Stones
,
or
such
like
things
that
were
plac'd
in
a
straight
line
or
right
forme
,
a
Pace
one
from
another
,
and
a
basket
being
placed
a
Pace
from
the
first
:
how
many
paces
would
there
be
made
to
put
all
these
Stones
into
the
basket
,
by
fetching
one
by
one
?
this
would
require
near
halfe
a
day
to
do
it
,
for
there
would
be
made
ten
thousand
and
ninety
two
paces
before
he
should
gather
them
all
up
.
Of
Changes
in
Bells
,
in
musicall
Instruments
,
transmutation
of
places
,
in
numbers
,
letters
,
men
or
such
like
.
NInethly
,
is
it
not
an
admirable
thing
to
consider
how
the
skill
of
numbers
doth
easily
furnish
us
with
the
knowledge
of
mysterious
and
hidden
things
?
which
simply
looked
into
by
others
that
are
not
versed
in
Arithmetick
,
do
present
unto
them
a
world
of
confusion
and
difficultie
.
As
in
the
first
place
,
it
is
often
debated
amongst
our
common
Ringers
,
what
number
of
Changes
there
might
be
made
in
5
,
6
,
7
,
8
,
or
more
Bells
:
who
spend
much
time
to
answer
their
owne
doubts
,
entring
often
into
a
Labyrinth
in
the
search
thereof
:
or
if
there
were
10
voyces
,
how
many
severall
notes
might
there
be
?
These
are
propositions
of
such
facility
,
that
a
childe
which
can
but
multiply
one
number
by
another
,
may
easily
resolve
it
,
which
is
but
only
to
multiply
every
number
from
the
unite
successively
in
each
others
product
,
unto
the
terme
assigned
:
so
the
6
number
that
is
against
6
in
the
Table
,
is
720
,
and
so
many
(
hanges
may
be
made
upon
6
Bells
,
upon
5
there
are
120
,
&c.
In
like
manner
against
10
in
the
Table
is
3618800
,
that
is
,
three
millions
,
six
hundred
twenty
eight
thousand
,
eight
hundred
:
which
shews
that
10
voices
may
have
so
many
consorts
,
each
man
keeping
his
owne
note
,
but
only
altering
his
place
;
and
so
of
stringed
Instruments
,
and
the
Gamat
may
be
varied
according
to
which
,
answerable
to
the
number
against
X
,
viz.
1124001075070399680000
notes
,
from
which
may
be
drawne
this
,
or
the
like
proposition
.
Suppose
that
7
Schollers
were
taken
out
of
a
free
Schoole
to
be
sent
to
an
Vniversitie
,
there
to
be
entertained
in
some
Colledge
at
commons
for
a
certaine
summe
of
money
,
so
that
each
of
them
have
two
meales
daily
,
and
no
longer
to
continue
there
,
then
that
sitting
all
together
upon
one
bench
or
forme
at
every
meale
,
there
might
be
a
divers
transmutation
of
place
,
of
account
in
some
one
of
them
,
in
comparison
of
another
,
and
never
the
whole
company
to
be
twice
alike
in
situation
:
how
long
may
the
Steward
entertaine
them
?
(
who
being
not
skilled
in
this
fetch
may
answere
unadvisedly
.
)
It
is
most
certaine
that
there
will
be
five
thousand
and
forty
several
1
|
a
|
1
|
2
|
b
|
2
|
6
|
c
|
3
|
24
|
d
|
4
|
120
|
e
|
5
|
7●0
|
f
|
6
|
5040
|
g
|
7
|
403●0
|
h
|
8
|
362880
|
i
|
9
|
3628800
|
k
|
10
|
39916800
|
l
|
11
|
479001600
|
m
|
12
|
6227020800
|
n
|
13
|
87178291200
|
o
|
14
|
1307674368000
|
p
|
15
|
20922789888000
|
q
|
16
|
355687537996000
|
r
|
17
|
6402375683928000
|
s
|
18
|
121645137994632000
|
t
|
19
|
2432902759892640000
|
u
|
20
|
51090957957745440000
|
w
|
21
|
1124001075070399680000
|
x
|
22
|
25852024726619192640000
|
y
|
23
|
6●0448593438860623360000
|
z
|
24
|
positions
or
changings
in
the
seatings
,
which
maks
14
years
time
wanting
10
weeks
and
3
dayes
.
Hence
from
this
mutability
of
transmutation
,
it
is
no
marvell
tha●
by
24
letters
there
ariseth
and
is
made
such
variety
of
languages
in
the
world
,
&
such
infinite
number
of
words
in
each
language
;
seeing
the
diversity
of
syllables
produceth
that
effect
,
and
also
by
the
interchanging
&
placing
of
letters
amongst
the
vowels
,
&
amongst
themselves
maketh
these
syllables
:
which
Alphabet
of
24
letters
may
be
varied
so
many
times
,
viz.
620448593438860623360000
which
is
six
hundred
twenty
thousand
,
foure
hundred
forty
eight
millions
of
millions
of
millions
five
hundred
ninety
three
thousand
,
foure
hundred
thirty
eight
milions
of
milions
,
&
more
.
Now
allowing
that
a
man
may
reade
or
speak
one
hundred
thousand
words
in
an
houre
which
is
twice
more
words
than
there
are
conteined
in
the
Psalmes
of
David
,
(
a
taske
too
great
for
any
man
to
do
in
so
short
a
time
)
and
if
there
were
foure
thousand
six
hundred
and
fifty
thousand
millions
of
men
,
they
could
not
speak
these
words
(
according
to
the
hourely
proportion
aforesaid
)
in
threescore
and
ten
thousand
yeares
;
which
variation
&
transmutation
of
letters
,
if
they
should
be
written
in
bookes
,
allowing
to
each
leaf
28000
words
,
(
which
is
as
many
as
possibly
could
be
inserted
,
)
and
to
each
book
a
reame
or
20
quire
of
the
largest
and
thinnest
printing
paper
,
so
that
each
book
being
about
15
inches
long
,
12
broad
,
and
6
thick
:
the
books
that
would
be
made
of
the
transmutation
of
the
24
letters
aforesaid
,
would
be
at
least
38778037089928788
:
and
if
a
Library
of
a
mile
square
every
way
,
of
50
foot
high
,
were
made
to
containe
250
Galleries
of
20
foot
broad
apiece
,
it
would
containe
foure
hundred
mill●ons
of
the
said
books
:
so
there
must
be
to
containe
the
rest
no
lesse
than
9●945092
such
Libraries
;
and
if
the
books
were
extended
over
the
surface
of
the
Globe
of
the
Earth
,
it
would
be
a
decuple
covering
unto
it
:
a
thing
seeming
most
incredible
that
24
letters
in
their
transmutation
should
produce
such
a
prodigious
number
,
yet
most
certaine
and
infallible
in
computation
.
Of
a
Servant
hired
upon
certaine
conditions
.
A
Servant
said
unto
his
Master
,
that
he
would
dwell
with
him
all
his
life-time
,
if
he
would
but
onely
lend
him
land
to
sowe
one
graine
of
Corne
with
all
his
increase
for
8
years
time
;
how
think
you
of
this
bargaine
?
for
if
he
had
but
a
quarter
of
an
inch
of
ground
for
each
graine
,
and
each
graine
to
bring
forth
yearely
of
increase
40
graines
,
the
whole
sum
would
amount
unto
,
at
the
terme
aforesaid
,
6553600000000
graines
:
and
seeing
that
three
thousand
and
six
hundred
millions
of
inches
do
but
make
one
mile
square
in
the
superficies
,
it
shall
be
able
to
receive
foureteene
thousand
and
foure
hundred
millions
of
graines
,
which
is
14400000000.
thus
dividing
the
aforesaid
6553600000000
,
the
Quotient
will
be
455
,
and
so
many
square
miles
of
land
must
there
be
to
sowe
the
increase
of
one
graine
of
Corne
for
8
yeares
,
which
makes
at
the
least
foure
hundred
and
twenty
thousand
Acres
of
land
,
which
rated
but
at
five
shillings
the
Acre
per
Annum
,
amounts
unto
one
hundred
thousand
pound
;
which
is
twelve
thousand
and
five
hundred
pound
a
yeare
,
to
be
continued
for
8
yeares
;
a
pretty
pay
for
a
Masters
Servant
8
yeares
service
.
PROBLEM
.
LXXXV
.
Of
Fountaines
,
Hydriatiques
,
Machinecke
,
and
other
experiments
upon
water
,
or
other
liquor
.
1.
First
how
to
make
water
at
the
foot
of
a
mountaine
to
ascend
to
the
top
of
it
,
and
so
to
descend
on
the
other
side
?
TO
do
this
there
must
be
a
Pipe
of
lead
,
which
may
come
from
the
fountaine
A
,
to
the
top
of
the
Mountaine
B
;
and
so
to
descend
on
the
other
side
a
little
lower
then
the
Fountaine
,
as
at
C.
then
make
a
hole
in
the
Pipe
at
the
top
of
the
Mountaine
,
as
at
B
,
and
stop
the
end
of
the
Pipe
at
A
and
C
;
and
fill
this
Pipe
at
B
with
water
:
&
close
it
very
carefully
againe
at
B
,
that
no
aire
get
in
:
then
unstop
the
end
at
A
,
&
at
C
;
then
will
the
water
perpetually
runne
up
the
hill
,
and
descend
on
the
other
side
,
which
is
an
invention
of
great
consequence
to
furnish
Villages
that
want
water
.
2.
Secondly
,
how
to
know
what
wine
or
other
liquor
there
is
in
a
vessell
without
opening
the
bung-hole
,
and
without
making
any
other
hole
,
than
that
by
which
it
runnes
out
at
the
top
?
IN
this
problem
there
is
nothing
but
to
take
a
bowed
pipe
of
Glasse
,
and
put
it
into
the
faucets
hole
,
and
stopping
it
close
about
:
for
then
you
shall
see
the
wine
or
liquor
to
ascend
in
this
Pipe
,
untill
it
be
just
even
with
the
liquor
in
the
vessel
;
by
which
a
man
may
fill
the
vessel
,
or
put
more
into
it
:
and
so
if
need
were
,
one
may
empty
one
vessel
into
another
without
opening
the
bung-hole
.
3.
Thirdly
,
how
is
it
that
it
is
said
that
a
vessell
holds
more
water
being
placed
at
the
foot
of
a
Mountaine
,
than
standing
upon
the
top
of
it
?
THis
is
a
thing
most
certaine
,
because
that
water
and
all
other
liquor
disposeth
it selfe
sphericaliy
about
the
Centre
of
the
earth
;
and
by
how
much
the
vessel
is
nearer
the
Centre
,
by
so
much
the
more
the
surface
of
the
water
makes
a
lesser
sphere
,
and
therefore
every
part
more
gibbous
or
swelling
,
than
the
like
part
in
a
greater
sphere
:
and
therefore
when
the
same
vessell
is
farther
from
the
Centre
of
the
earth
,
the
surface
of
the
water
makes
a
greater
sphere
,
and
therefore
lesse
gibbous
,
or
swelling
over
the
vessell
:
from
whence
it
is
evident
that
a
vessell
near
the
Centre
of
the
Earth
holds
more
water
than
that
which
is
farther
remote
from
it
;
and
so
consequently
a
vessel
placed
at
the
bottome
of
the
Mountaine
holds
more
water
,
than
being
placed
on
the
top
of
the
Mountaine
.
First
,
therefore
one
may
conclude
,
that
one
and
the
same
vessel
will
alwayes
hold
more
:
by
how
much
it
is
nearer
the
Centre
of
the
earth
.
Secondly
,
if
a
vessell
be
very
neare
the
Centre
of
the
earth
,
there
will
be
more
water
above
the
brims
of
it
,
than
there
is
within
the
vessel
.
Thirdly
,
a
vessel
full
of
water
comming
to
the
Centre
wil
spherically
increase
,
and
by
little
and
little
leave
the
vessel
;
and
passing
the
Centre
,
the
vessel
will
be
all
emptied
.
Fourthly
,
one
cannot
carry
a
Paile
of
water
from
a
low
place
to
a
higher
,
but
it
will
more
and
more
run
out
and
over
,
because
that
in
ascending
it
lies
more
levell
,
but
descending
it
swels
and
becomes
more
gibbous
.
4.
Fourthly
,
to
conduct
water
from
the
top
of
one
Mountaine
,
to
the
top
of
another
.
AS
admit
on
the
top
of
a
Mountaine
there
is
a
spring
,
and
at
the
toppe
of
the
other
Mountaine
there
are
Inhabitants
which
want
water
:
now
to
make
a
bridge
from
one
Mountaine
to
another
,
were
difficult
and
too
great
a
charge
;
by
way
of
Pipes
it
is
easie
and
of
no
great
price
:
for
if
at
the
spring
on
the
top
of
the
Mountaine
be
placed
a
Pipe
,
to
descend
into
the
valley
,
and
ascend
to
the
other
Mountlaine
,
the
water
will
runne
naturally
,
and
continually
,
provided
that
the
spring
be
somewhat
higher
than
the
passage
of
the
water
at
the
Inhabitants
.
5.
Fifthly
,
of
a
fine
Fountaine
which
spouts
water
very
high
,
and
with
great
violence
by
turning
of
a
Cock.
LEt
there
be
a
vessell
as
AB
,
made
close
in
all
his
parts
,
in
the
middle
of
which
let
CD
be
a
Pipe
open
at
D
neare
the
bottome
,
and
then
with
a
Squirt
squirt
in
the
water
at
C
,
stopped
above
by
the
cock
or
faucet
C
,
with
as
great
violence
as
possible
you
can
;
and
turne
the
cock
immediatly
.
Now
there
being
an
indifferent
quantity
of
water
and
aire
in
the
vessel
,
the
water
keeps
it selfe
in
the
bottome
,
and
the
aire
which
was
greatly
pressed
,
seeks
for
more
place
,
that
turning
the
Cock
the
water
issueth
forth
at
the
Pipe
,
and
flyes
very
high
,
and
that
especially
if
the
vessell
be
a
little
heated
:
some
make
use
of
this
for
an
Ewer
to
wash
hands
withall
,
and
therefore
putting
a
moveable
Pipe
above
C
,
such
as
the
figure
sheweth
:
which
the
water
will
cause
to
turne
very
quick
,
pleasurable
to
behold
.
6.
Sixtly
,
of
Archimedes
screw
,
which
makes
water
ascend
by
descending
.
THis
is
nothing
else
but
a
Cylinder
,
about
the
which
is
a
Pipe
in
form
of
a
screw
,
and
when
one
turnes
it
,
the
water
descends
alwayes
in
respect
of
the
Pipe
:
for
it
passeth
from
one
part
which
is
higher
to
that
which
is
lower
,
and
at
the
end
of
the
Engine
the
water
is
found
higher
than
it
was
at
the
spring
.
This
great
Enginer
admirable
in
all
Mathematicall
Arts
invented
this
Instrument
to
wash
King
Hieroies
great
vessells
,
as
some
Authors
saye
,
also
to
water
the
fields
of
Egypt
,
as
Diodorus
witnesseth
:
and
Cardanus
reporteth
that
a
Citizen
of
Milan
having
made
the
like
Engine
,
thinking
himselfe
to
be
the
first
inventer
,
conceived
such
exceeding
joy
,
that
he
be
came
mad
,
foll.
2.
Againe
a
thing
may
ascend
by
descending
,
if
a
spiral
line
be
made
having
many
circulations
or
revolutions
;
the
last
being
alwayes
lesser
than
the
first
,
yet
higher
than
the
Plaine
supposed
it
is
most
certaine
that
then
putting
a
ball
into
it
,
and
turning
the
spirall
line
so
,
that
the
first
circulation
may
be
perpendicular
,
or
touch
alwayes
the
supposed
Plain
:
the
ball
shall
in
descending
continually
ascend
,
untill
at
last
it
come
to
the
highest
part
of
the
spirall
line
,
&
so
fall
out
.
And
here
especially
may
be
noted
,
that
a
moving
body
as
water
,
or
a
Bullet
,
or
such
like
,
will
never
ascend
if
the
Helicall
revolution
of
the
screw
be
not
inclining
to
the
Horizon
:
so
that
according
to
this
inclination
the
ball
or
liquor
,
may
descend
alwayes
by
a
continuall
motion
and
revolution
.
And
this
experiment
may
be
more
usefull
,
naturally
made
with
a
thred
of
●ron
,
or
Latine
turned
or
bowed
Helically
about
a
Cylinder
,
with
some
distinction
of
distances
between
the
Heli●es
,
for
then
having
drawn
out
the
Cylinder
,
or
having
hung
or
tyed
some
weight
at
it
in
such
sort
,
that
the
water
may
easily
drop
if
one
lift
up
the
said
thred
:
these
Helices
or
revolutions
,
notwithstanding
will
remaine
inclining
to
the
Horizon
,
and
then
turning
it
about
forward
,
the
said
weight
will
ascend
,
but
backward
it
will
descend
.
Now
if
the
revolutions
be
alike
,
and
of
equallity
amongst
themselves
,
and
the
whirling
or
turning
motion
be
quicke
,
the
sight
will
be
so
deceived
,
that
producing
the
action
it
will
seeme
to
the
ignorant
no
lesse
than
a
Miracle
.
7.
Seventhly
,
of
another
fine
Fountaine
of
pleasure
.
THis
is
an
Engine
that
hath
two
wheeles
with
Cogges
,
or
teeth
as
AB
,
which
are
placed
within
an
Ovall
CD
,
in
such
sort
,
that
the
teeth
of
the
one
,
may
enter
into
the
notches
of
the
other
;
but
so
just
that
neither
aire
nor
water
may
enter
into
the
Ovall
coffer
,
either
by
the
middle
or
by
the
sides
,
for
the
wheele
must
joyne
so
neare
to
the
sides
of
the
coffer
,
that
there
be
no
vacuitie
:
to
this
there
is
an
axeltree
with
a
handle
to
each
wheele
,
so
that
they
may
be
turned
,
and
A
being
turned
,
that
turneth
the
other
wheele
that
is
opposite
:
by
which
motion
the
aire
that
is
in
E
,
&
the
water
that
is
carried
by
the
hollow
of
the
wheeles
of
each
side
,
by
continuall
motion
,
is
constrained
to
mount
and
flie
out
by
the
funnell
F
:
now
to
make
the
water
runne
what
way
one
would
have
it
,
there
may
be
applied
upon
the
top
of
the
Pipe
F
,
two
other
moveable
Pipes
inserted
one
within
another
;
as
the
figure
sheweth
.
But
here
note
,
that
there
may
acrue
some
inconveniency
in
this
Machine
seeing
that
by
quick
turning
the
Cogges
or
teeth
of
the
wheeles
running
one
against
another
,
may
neare
break
them
,
and
so
give
way
to
the
aire
to
enter
in
,
which
being
violently
inclosed
will
escape
to
occupie
the
place
of
the
water
,
whose
weight
makes
it
so
quick
:
howsoever
,
if
this
Machine
be
curiously
made
as
an
able
workeman
may
easily
do
,
it
is
a
most
sovereigne
Engine
,
to
cast
water
high
and
farre
off
for
to
quench
fires
.
And
to
have
it
to
raine
to
a
place
assigned
,
accommodate
a
socket
having
a
Pipe
at
the
middle
,
which
may
point
towards
the
place
being
set
at
the
top
thereof
,
and
so
having
great
discretion
in
turning
the
Axis
of
the
wheele
,
it
may
work
exceeding
well
,
and
continue
long
.
8.
Eightly
,
of
a
fine
watering
pot
for
gardens
.
THis
may
be
made
in
forme
of
a
Bottle
according
to
the
last
figure
or
such
like
,
having
at
the
bottome
many
small
holes
,
and
at
the
neck
of
it
another
hole
somewhat
greater
than
those
at
the
bottome
,
which
hole
at
the
top
you
must
unstop
when
you
would
fill
this
watering
pot
,
for
then
it
is
nothing
but
putting
the
lower
end
into
a
paile
of
water
,
for
so
it
will
fill
it selfe
by
degrees
:
and
being
full
,
put
your
thumb
on
the
hole
at
the
neck
to
stop
it
,
for
then
may
you
carry
it
from
place
to
place
,
and
it
will
not
sensibly
runne
out
,
som●thing
it
will
,
and
all
in
time
(
if
it
were
never
so
close
stopped
)
contrary
to
the
ancient
tenet
in
Philosophy
,
that
aire
will
not
penetrate
.
9.
Ninthly
,
how
easily
to
take
wine
out
of
a
vessell
at
the
bu●g-hole
,
without
piercing
of
a
hole
in
the
vessell
?
IN
this
there
is
no
need
but
to
have
a
Cane
or
Pipe
of
Glasse
or
such
like
,
one
of
the
ends
of
which
may
be
closed
up
almost
,
leaving
some
small
hole
at
the
end
;
for
then
if
that
end
be
set
into
the
vessell
at
the
bung-hole
,
the
whole
Cane
or
Pipe
will
be
filled
by
little
and
little
;
and
once
being
full
,
stop
the
other
end
which
is
without
and
then
pull
out
the
Cane
or
Pipe
,
so
will
it
be
ful
of
wine
,
then
opening
a
little
the
top
above
,
you
may
fill
a
Glasse
or
other
Pot
with
it
,
for
as
the
Wine
issueth
out
,
the
aire
commeth
into
the
Cane
or
Pipe
to
supply
vacuity
.
10.
Tenthly
,
how
to
measure
irregular
bodies
by
help
of
water
?
SOme
throw
in
the
body
or
magnitude
into
a
vessell
,
and
keep
that
which
floweth
out
over
,
saying
it
is
alwayes
equal
to
the
thing
cast
into
the
water
:
let
i●
is
more
nea●er
this
way
to
poure
into
a
vessell
such
a
quantity
of
water
,
which
may
be
thought
sufficient
to
cover
the
body
or
magnitude
,
and
make
a
marke
how
high
the
water
is
in
the
vessell
,
then
poure
out
all
this
water
into
another
vessell
,
and
let
the
body
or
magnitude
be
placed
into
the
first
vessel
;
then
poure
in
water
from
the
second
vessell
,
until
it
ascend
unto
the
former
marke
made
in
the
first
vessell
,
so
the
water
which
remaines
in
the
second
vessel
is
equall
to
the
body
or
magnitude
put
into
the
water
:
but
here
note
that
this
is
not
exact
or
free
from
error
,
yet
nearer
the
truth
than
any
Geometrician
can
otherwise
possibly
measure
,
and
these
bodies
that
are
not
so
full
of
pores
are
more
truly
measured
this
way
,
than
others
are
.
11.
To
finde
the
weight
of
water
.
SEeing
that
574/1000
part
of
an
ounce
weight
,
makes
a
cubicall
inch
of
water
:
and
every
pound
weight
Haverdepoize
makes
27
cubicall
inches
,
and
1
9/●
;
fere
,
and
that
●
Gallons
and
a
halfe
wine
measure
makes
a
foot
cubicall
,
it
is
easie
by
inversion
,
that
knowing
the
quantity
of
a
vessel
in
Gallons
,
to
finde
his
content
in
cubicall
feet
or
weight
:
and
that
late
famous
Geometrician
Master
Brigs
found
a
cubical
foot
of
water
to
weigh
neare
62
pound
weight
Haverdepoize
But
the
late
learned
Simon
Stevin
found
a
cubicall
foot
of
water
to
weigh
65
pound
,
which
difference
may
arise
from
the
inequalitie
of
water
;
for
some
waters
are
more
ponderous
than
others
,
and
some
difference
may
be
from
the
weight
of
a
pound
,
and
the
measure
of
a
foot
:
thus
the
weight
and
quantitie
of
a
solid
foot
settled
,
it
is
easie
for
Arithmeticians
to
give
the
contents
of
vessells
or
bodies
which
containe
liquids
.
12.
To
finde
the
charge
that
a
vessell
may
carry
as
Shippes
,
Boates
,
or
such
like
.
THis
is
generally
conceived
,
that
a
vessell
may
carry
as
much
weight
as
that
water
weigheth
,
which
is
equall
unto
the
vessell
in
bignesse
,
in
abating
onely
the
weight
of
the
vessell
:
we
see
that
a
barrel
of
wine
or
water
cast
into
the
water
,
will
not
sink
to
the
bottome
,
but
swim
easily
,
and
if
a
ship
had
not
Iron
and
other
ponderosities
in
it
,
it
might
swim
full
of
water
without
sinking
:
in
the
same
manner
if
the
vessell
were
loaden
with
lead
,
so
much
should
the
watter
weigh
:
hence
it
is
that
Marriners
call
Shippes
of
50
thousand
Tunnes
,
because
they
may
containe
one
or
two
thousand
Tunne
,
and
so
consequently
carry
as
much
.
13.
How
comes
it
that
a
Shippe
having
safely
sayled
in
the
vast
Ocean
,
and
being
come
into
the
Port
or
harbour
,
without
any
tempest
will
sink
down
right
?
THe
cause
of
this
is
that
a
vessel
may
carry
more
upon
some
kinde
of
water
than
upon
other
;
now
the
water
of
the
Sea
is
thicker
and
heavier
than
that
of
Rivers
,
Wels
,
or
Fountains
;
therefore
the
loading
of
a
vessell
which
is
accounted
sufficient
in
the
Sea
,
becomes
too
great
in
the
hurbour
or
sweet
water
.
Now
some
think
that
it
is
the
depth
of
the
water
that
makes
vessells
more
easie
to
swimme
,
but
it
is
an
abuse
;
for
if
the
loading
of
a
Ship
be
no
heavier
than
the
water
that
would
occupie
that
place
,
the
Ship
should
as
easily
swim
upon
that
water
,
as
if
it
did
swim
upon
a
thousand
fathom
deep
of
water
,
and
if
the
water
be
no
thicker
than
a
leafe
of
paper
,
and
weigheth
but
an
ounce
under
a
heavy
body
,
it
will
support
it
,
as
well
as
if
the
water
under
it
weighed
ten
thousand
pound
weight
:
hence
it
is
if
there
be
a
vessell
capable
of
a
little
more
than
a
thousand
pound
weight
of
water
,
you
may
put
into
this
vessell
a
piece
of
wood
,
which
shall
weigh
a
thousand
pound
weight
;
(
but
lighter
in
his
kinde
than
the
like
of
magnitude
of
water
:
)
for
then
pouring
in
but
a
quart
of
water
or
a
very
little
quantitie
of
water
,
the
wood
will
swim
on
the
top
of
it
,
(
provided
that
the
wood
touch
not
the
sides
of
the
vessell
:
)
which
is
a
fine
experiment
,
and
seems
admirable
in
the
performance
.
14.
How
a
grosse
body
of
mettle
may
swimme
upon
the
water
?
THis
is
done
by
extending
the
mettle
into
a
thin
Plate
,
to
make
it
hollow
in
forme
of
a
vessel
;
so
that
the
greatnesse
of
the
vessell
which
the
aire
with
it
containeth
,
be
equal
to
the
magnitude
of
the
water
,
which
weighes
as
much
as
it
,
for
all
bodies
may
swim
without
sinking
,
if
they
occupie
the
place
of
water
equal
in
weight
unto
them
,
as
if
it
weighed
12
pound
it
must
have
the
place
of
12
pound
of
water
:
hence
it
is
that
we
see
floating
upon
the
water
great
vessells
of
Copper
or
Brasse
,
when
they
are
hollow
in
forme
of
a
Caldron
.
And
how
can
it
be
otherwise
conceived
of
Islands
in
the
Sea
that
swim
and
float
?
is
it
not
that
they
are
hollow
and
some
part
like
unto
a
Boat
,
or
that
their
earth
is
very
light
and
spongeous
,
or
having
many
concavities
in
the
body
of
it
,
or
much
wood
within
it
?
And
it
would
be
a
pretty
proposition
to
shew
how
much
every
kinde
of
metall
should
be
inlarged
,
to
make
it
swim
upon
the
water
:
which
doth
depend
upon
the
proportion
that
is
between
the
weight
of
the
water
and
each
metall
.
Now
the
proportion
that
is
betweene
metalls
and
water
of
equall
magnitude
,
according
to
some
Authors
,
is
as
followeth
.
A
magnitude
of
10
pound
weight
of
water
will
require
for
the
like
magnitude
of
Gold
.
-
187
½
Lead
.
-
116
½
Silver
.
-
104
Copper
.
-
91
Iron
.
-
81
Tinne
.
-
75
From
which
is
inferred
,
that
to
make
a
piece
of
Copper
of
●0
pound
weight
to
swimme
,
it
must
be
so
made
hollow
,
that
it
may
hold
9
times
that
weight
of
water
and
somewhat
more
,
that
is
to
say
,
91
pound
:
seeing
that
Copper
and
water
of
like
magnitudes
in
their
ponderosities
,
are
as
before
,
as
●0
to
91.
15.
How
to
weigh
the
lightnesse
of
the
aire
?
PLace
a
Ballance
of
wood
turned
upside
downe
into
the
water
,
that
so
it
may
swim
,
then
let
water
be
inclosed
within
some
body
,
as
within
a
Bladder
or
such
like
,
and
suppose
that
such
a
quantitie
of
aire
should
weigh
one
pound
,
place
it
under
one
of
the
Ballances
,
and
place
under
the
other
as
much
weight
of
lightnesse
as
may
counter-ballance
and
keep
the
other
Ballance
that
it
rise
not
out
of
the
water
:
by
which
you
shall
see
how
much
the
lightnesse
is
.
But
without
any
Ballance
do
this
;
take
a
Cubicall
hollow
vessell
,
or
that
which
is
Cylindricall
,
which
may
swimme
on
the
water
,
and
as
it
sinketh
by
placing
of
weights
upon
it
,
marke
how
much
,
for
then
if
you
would
examine
the
weight
of
any
body
,
you
have
nothing
to
do
but
to
put
it
into
this
vessell
,
and
marke
how
deep
it
sinkes
,
for
so
many
pound
it
weighes
as
the
weights
put
in
do
make
it
so
to
sinke
.
16.
Being
given
a
body
,
to
marke
it
about
,
and
shew
how
much
of
it
will
sink
in
the
water
,
or
swim
above
the
water
.
THis
is
done
by
knowing
the
weight
of
the
body
which
is
given
,
and
the
quantity
of
water
,
which
weighes
as
much
as
that
body
;
for
then
certainly
it
will
sink
so
deep
,
untill
it
occupieth
the
place
of
that
quantitie
of
water
.
17.
To
finde
how
much
severall
mettle
or
other
bodies
doe
weigh
lesse
in
the
water
than
in
the
aire
:
TAke
a
Ballance
,
&
weigh
(
as
for
example
)
9
pound
of
Gold
,
Silver
,
Lead
,
or
Stone
in
the
aire
,
so
it
hang
in
aequilibrio
;
then
comming
to
the
water
,
take
the
same
quantity
of
Gold
Silver
,
Lead
,
or
Stone
,
and
let
it
softly
downe
into
it
,
and
you
shall
see
that
you
shall
need
a
lesse
counterpoise
in
the
other
Ballance
to
counter-ballance
it
:
wherefore
all
solids
or
bodies
weigh
lesse
in
the
water
than
in
the
aire
,
and
so
much
the
lesse
it
will
be
,
by
how
much
the
water
is
grosse
and
thick
,
because
the
weight
findes
a
greater
resistance
,
and
therefore
the
water
supports
more
than
aire
;
and
further
,
because
the
water
by
the
ponderositie
is
displeased
,
and
so
strives
to
be
there
againe
,
pressing
to
it
,
by
reason
of
the
other
waters
that
are
about
it
,
according
to
the
proportion
of
his
weight
.
Archimedes
demonstrateth
,
that
all
bodies
weigh
lesse
in
the
water
(
or
in
like
liquor
)
by
how
much
they
occupie
place
:
and
if
the
water
weigh
a
pound
weight
,
the
magnitude
in
the
water
shall
weigh
a
pound
lesse
than
in
the
aire
.
Now
by
knowing
the
proportion
of
water
and
mettles
,
it
is
found
that
Gold
loseth
in
the
water
the
19
part
of
his
weight
,
Copper
the
9
part
,
Quicksilver
the
15
part
,
Lead
the
12
part
,
Silver
the
10
part
,
Iron
the
8
part
,
Tinne
the
7
part
and
a
little
more
:
wherefore
in
materiall
and
absolute
weight
,
Gold
in
respect
of
the
water
that
it
occupieth
weigheth
18
,
and
¾
times
heavier
than
the
like
quantitie
of
water
,
that
is
,
as
18
¾
to
the
Quicksilver
15
times
,
Lead
11
and
⅗
,
Silver
10
and
⅔
,
Copper
9
and
1/10
,
Iron
8
and
½
,
and
Tinne
8
and
1/●
.
Contrarily
in
respect
of
greatnesse
,
if
the
water
be
as
heavy
as
the
Gold
,
then
is
the
water
almost
19
times
greater
than
the
magnitude
of
the
Gold
,
and
so
may
you
judge
of
the
rest
.
18.
How
is
it
that
a
ballance
having
like
weight
in
each
scale
,
and
hanging
in
aequilibrio
in
the
aire
,
being
placed
in
another
place
,
(
without
removing
any
weight
)
it
shall
cease
to
hang
in
aequilibrio
sensibly
:
yea
by
a
great
difference
of
weight
?
THis
is
easie
to
be
resolved
by
considering
different
mettles
,
which
though
they
weigh
equall
in
the
aire
,
yet
in
the
water
there
will
be
an
apparant
difference
;
as
suppose
so
that
in
the
scale
of
each
Ballance
be
placed
18
pound
weight
of
severall
metalls
,
the
one
Gold
,
and
the
other
Copper
,
which
being
in
aequilibrio
in
the
aire
,
placed
in
the
water
,
will
not
hang
so
,
because
that
the
Gold
los●eth
neare
the
18
part
of
his
weight
,
which
is
about
1
pound
,
and
the
Copper
loseth
but
his
9
part
,
which
is
2
pound
:
wherefore
the
Gold
in
the
water
weigheth
but
17
pound
,
and
the
Copper
16
pound
,
which
is
a
difference
most
sensible
to
confirme
that
point
.
19.
To
shew
what
waters
are
heavier
one
than
another
,
and
how
much
.
PHysicians
have
an
especiall
respect
unto
this
,
judging
that
water
which
is
lightest
is
most
healthfull
and
medicinall
for
the
body
,
&
Sea-men
know
that
the
heaviest
waters
do
beare
most
,
and
it
is
knowne
which
water
is
heaviest
thus
.
Take
a
piece
of
wax
,
and
fasten
Lead
unto
it
,
or
some
such
like
thing
that
it
may
but
precisely
swimme
,
for
then
it
is
equal
to
the
like
magnitude
of
water
,
then
put
it
into
another
vessell
which
hath
contrary
water
,
and
if
it
sinke
,
then
is
that
water
lighter
than
the
other
:
but
if
it
sinke
not
so
deep
,
then
it
argueth
the
water
to
be
heavier
or
more
grosser
than
the
first
water
,
or
one
may
take
a
piece
of
wood
,
and
marke
the
quantitie
of
sinking
of
it
into
severall
waters
,
by
which
you
may
judge
which
is
lightest
or
heaviest
,
for
in
that
which
it
sinkes
most
,
that
is
infallibly
the
lightest
,
and
so
contrarily
.
20.
How
to
make
a
Pound
of
water
weigh
as
much
as
10
,
2●
,
●0
,
or
a
hundred
pound
of
Lead
;
nay
as
much
as
a
thousand
,
or
ten
thousand
and
pound
weight
?
THis
proposition
seems
very
impossible
,
yet
water
inclosed
in
a
vessell
,
being
constrained
to
dilate
it selfe
,
doth
weigh
so
much
as
though
there
were
in
the
concavitie
of
it
a
solid
body
of
water
.
There
are
many
wayes
to
experiment
this
proposition
,
but
to
verifie
it
,
it
may
be
sufficient
to
produce
two
excellent
ones
onely
:
which
had
they
not
been
really
acted
,
little
credit
might
have
been
given
unto
it
.
The
first
way
is
thus
.
Take
a
Magnitude
which
takes
up
as
much
place
as
a
hundred
or
a
thousand
pound
of
water
,
and
suppose
that
it
were
tied
to
some
thing
that
it
may
hang
in
the
aire
;
then
make
a
Ballance
that
one
of
the
scales
may
inviron
it
,
yet
so
that
it
touch
not
the
sides
of
it
:
but
leave
space
enough
for
one
pound
of
water
:
then
having
placed
100
pound
weight
in
the
other
scale
,
throw
in
the
water
about
the
Magnitude
,
so
that
one
pound
of
water
shall
weigh
downe
the
hundred
pound
in
the
other
Ballance
.
PROBLEM
.
LXXXVI
.
Of
sundry
Questions
of
Arithmetick
,
and
first
of
the
number
of
sands
.
IT
may
be
said
incontinent
,
that
to
undertake
this
were
impossible
,
either
to
number
the
Sands
of
Lybia
,
or
the
Sands
of
the
Sea
;
and
it
was
this
that
the
Poets
sung
,
and
that
which
the
vulgar
beleeves
;
nay
,
that
which
long
ago
certaine
Philosophers
to
Gelon
King
of
Sicily
reported
,
that
the
graines
of
sand
were
innumerable
:
But
I
answere
with
Archimedes
,
that
not
only
one
may
number
those
which
are
at
the
border
and
about
the
Sea
;
but
those
which
are
able
to
fill
the
whole
world
,
if
there
were
nothing
else
but
sand
;
and
the
graines
of
sands
admitted
to
be
so
small
,
that
10
may
make
but
one
graine
of
Poppy
:
for
at
the
end
of
the
account
there
need
not
to
expresse
them
,
but
this
number
30840979456
,
and
35
Ciphers
at
the
end
of
it
.
Clavius
and
Archimedes
make
it
somewhat
more
;
because
they
make
a
greater
firmament
than
Ticho
Brahe
doth
;
and
if
they
augment
the
Vniverse
,
it
is
easie
for
us
to
augment
the
number
,
and
declare
assuredly
how
many
graines
of
sand
there
are
requisite
to
fill
another
world
,
in
comparison
that
our
visible
world
were
but
as
one
graine
of
sand
,
an
atome
or
a
point
;
for
there
is
nothing
to
do
but
to
multiply
the
number
by
it selfe
,
which
will
amount
to
ninety
places
,
whereof
twentie
are
these
,
95143798134910955936
,
and
70
Ciphers
at
the
end
of
it
:
which
amounts
to
a
most
prodigious
number
,
and
is
easily
supputated
:
for
supposing
that
a
graine
of
Poppy
doth
containe
10
graines
of
sand
,
there
is
nothing
but
to
compare
that
little
bowle
of
a
graine
of
Poppy
,
with
a
bowle
of
an
inch
or
of
a
foot
,
&
that
to
be
compared
with
that
of
the
earth
,
and
then
that
of
the
earth
with
that
o
the
firmament
;
and
so
of
the
rest
.
2.
Divers
metalls
being
melted
together
in
one
body
,
to
finde
the
mixture
of
them
.
THis
wat
a
notable
invention
of
Archimedes
,
related
by
Vitrivius
in
his
Architecture
,
where
he
reporteth
that
the
Gold-smith
which
King
Hiero
imployed
for
the
making
of
the
Golden
Crowne
,
which
was
to
be
dedicated
to
the
gods
,
had
stolen
part
of
it
,
and
mixed
Silver
in
the
place
of
it
:
the
King
suspicious
of
the
work
proposed
it
to
Archimedes
,
if
by
Art
he
could
discover
without
breaking
of
the
Crowne
,
if
there
had
been
made
mixture
of
any
other
metall
with
the
Gold
.
The
way
which
he
found
out
was
by
bathing
himselfe
;
for
as
he
entred
into
the
vessell
of
water
,
(
in
which
he
bathed
himselfe
)
so
the
water
ascended
or
flew
out
over
it
,
and
as
he
pulled
out
his
body
the
water
descended
:
from
which
he
gathered
that
if
a
Bowle
of
pure
Gold
,
Silver
,
or
other
metall
were
cast
into
a
vessell
of
water
,
the
water
proportionally
according
to
the
thing
cast
in
would
ascend
;
and
so
by
way
of
Arithmetick
the
question
lay
open
to
be
resolved
:
who
being
so
intensively
taken
with
the
invention
,
leapes
out
of
the
Bath
all
naked
,
crying
as
a
man
transported
,
I
have
found
,
I
have
found
,
and
so
discovered
it
.
Now
some
say
that
he
took
two
Masses
,
the
one
of
pure
Gold
,
and
the
other
of
pure
Silver
;
each
equall
to
the
weight
of
the
Crowne
,
and
therefore
unequall
in
magnitude
or
greatnesse
;
and
then
knowing
the
severall
quantities
of
water
which
was
answerable
to
the
Crown
,
and
the
severall
Masses
,
he
subtilly
collected
,
that
if
the
Crowne
occupied
more
place
within
the
water
than
the
Masse
of
Gold
did
:
it
appeared
that
there
was
Silver
or
other
metall
melted
with
it
.
Now
by
the
rule
of
position
,
suppose
that
each
of
the
three
Masses
weighed
18
pound
a
piece
,
and
that
the
Masse
of
Gold
did
occupie
the
place
of
one
pound
of
water
,
that
of
Silver
a
pound
and
a
halfe
▪
and
the
Crown
one
pound
and
a
quarter
only
:
then
thus
he
might
operate
the
Masse
of
Silver
which
weighed
18
pounds
,
cast
into
the
water
,
did
cast
out
halfe
a
pound
of
water
more
then
the
Masse
of
Gold
,
which
weighed
18
pound
,
and
the
Crowne
which
weighed
also
18
pound
,
being
put
into
a
vessell
full
of
water
,
threw
out
more
water
than
the
Masse
of
Gold
by
a
quarter
of
a
pound
,
(
because
of
mixt
metall
which
was
in
it
:
)
therefore
by
the
rule
of
proportion
,
if
halfe
a
pound
of
water
(
the
excesse
)
be
answerable
to
18
pound
of
Silver
,
one
quarter
of
a
pound
of
excesse
shall
be
answerable
to
9
pound
of
Silver
,
and
so
much
was
mixed
in
the
Crowne
.
Some
judge
the
way
to
be
more
facill
by
weighing
the
Crowne
first
in
the
aire
,
then
in
the
water
;
in
the
aire
it
weighed
18
pound
,
and
if
it
were
pure
Gold
,
in
the
water
it
would
weigh
but
17
pound
;
if
it
were
Copper
it
would
weigh
but
16
pound
;
but
because
we
will
suppose
that
Gold
and
Copper
is
mixed
together
,
it
will
weigh
lesse
then
17
pound
,
yet
more
than
16
pound
,
and
that
according
to
the
proportion
mixed
:
let
it
then
be
supposed
that
it
weighed
in
the
water
16
pound
and
3
quarters
,
then
might
one
say
by
proportion
,
if
the
difference
of
one
pound
of
losse
,
which
is
between
16
and
17
)
be
answerable
to
18
pound
,
to
what
shall
one
quarter
of
difference
be
answerable
to
,
which
is
between
17
and
16
¾
,
and
it
will
be
4
pound
and
a
halfe
;
and
so
much
Copper
was
mixed
with
the
Gold
.
Many
men
have
delivered
sundry
wayes
to
resolve
this
proposition
since
Archimedes
invention
,
and
it
were
tedious
to
relate
the
diversities
.
Baptista
Benedictus
amongst
his
Arithmeticall
Theoremes
,
delivers
his
way
thus
:
if
a
Masse
of
Gold
of
equall
bignesse
to
the
Crowne
did
weigh
20
pound
,
and
another
of
Silver
at
a
capacity
or
bignesse
at
pleasure
,
as
suppose
did
weigh
12
pound
,
the
Crowne
or
the
mixt
body
would
weigh
more
than
the
Silver
,
and
lesser
than
the
Gold
,
suppose
it
weighed
16
pound
which
is
4
pound
lesse
than
the
Gold
by
8
pound
,
then
may
one
say
,
if
8
pound
of
difference
come
from
12
pound
of
Silver
,
from
whence
comes
4
pound
which
will
be
6
pound
and
so
much
Silver
was
mixed
in
it
,
&c.
3.
Three
men
bought
a
quantitie
of
wine
,
each
paid
alike
,
and
each
was
to
have
alike
;
it
happened
at
the
last
partition
that
there
were
21
Barrells
,
of
which
7
were
full
,
7
halfe
full
,
and
7
empty
,
how
must
they
share
the
wine
and
vessells
,
that
each
have
as
many
vessells
one
as
another
,
&
as
much
wine
one
as
another
?
THis
may
be
answered
two
wayes
as
followeth
,
and
these
numbers
2
,
2
,
3
,
or
3
,
3
,
1
,
may
serve
for
direction
,
and
signifies
that
the
first
person
ought
to
have
3
Barrells
full
,
and
as
many
empty
ones
,
and
one
which
is
halfe
full
;
so
he
shall
have
7
vessells
and
3
Barrels
,
and
a
halfe
of
liquor
;
and
one
of
the
other
shall
in
like
manner
have
as
much
,
so
there
will
remaine
for
the
third
man
1
Barrell
full
,
5
which
are
halfe
full
,
and
1
empty
,
and
so
every
one
shall
have
alike
both
in
vessells
and
wine
.
And
generally
to
answer
such
questions
,
divide
the
number
of
vessells
by
the
number
of
persons
,
and
if
the
Quotient
be
not
an
intire
number
,
the
question
is
impossible
;
but
when
it
is
an
intire
number
,
there
must
be
made
as
many
parts
as
there
are
3
persons
,
seeing
that
each
part
is
lesse
than
the
halfe
of
the
said
Quotient
:
as
dividing
21
by
3
there
comes
7
for
the
Quotient
,
which
may
be
parted
in
these
three
parts
,
2
,
2
,
3
,
or
3
,
3
,
1
,
each
of
which
being
lesse
than
ha●fe
of
7.
4.
There
is
a
Ladder
which
stands
upright
against
a
wall
of
10
foot
high
,
the
foot
of
it
is
pulled
out
6
foot
from
the
wall
upon
the
pavement
:
how
much
hath
the
top
of
the
Ladder
descended
?
THe
answer
is
,
2
foot
:
for
by
Pythagoras
rule
the
square
of
DB
,
the
Hypothenusal
is
equall
to
the
square
of
DA
6
,
&
AB
10.
Now
if
DA
be
6
foot
,
and
AB
10
foot
,
the
squares
are
36
and
100
,
which
36
taken
from
100
rests
64
,
whose
Roote-quadrate
is
8
so
the
foot
of
the
Ladder
being
now
at
D
,
the
toppe
will
be
at
C
,
2
foot
lower
than
it
was
when
it
was
at
B.
PROBLEM
.
LXXXVII
.
Witty
suits
or
debates
between
Caius
and
Sempronius
,
upon
the
forme
of
f●gures
,
which
Geometricians
call
Isoperimeter
,
or
equall
in
circuit
or
compasse
.
MArvell
●ot
at
it
if
I
make
the
Mathematicks
take
place
at
the
Ba●●e
,
and
if
I
set
forth
here
B●rtoleus
,
who
witnesseth
of
himselfe
,
that
being
then
an
ancient
Doctor
in
the
Law
,
he
himselfe
took
upon
him
to
learne
the
elements
and
principles
of
Geometry
,
by
which
he
might
set
forth
certaine
Lawes
touching
the
divisions
of
Fields
,
Waters
,
Islands
,
and
other
incident
places
:
now
this
shall
be
to
shew
in
passing
by
,
that
these
sciences
are
profitable
and
behovefull
for
Judges
,
Counsellors
,
or
such
,
to
explaine
many
things
which
fall
out
in
Lawes
,
to
avoid
ambiguities
,
contentions
,
and
suits
often
.
1.
Incident
.
CAius
had
a
field
which
was
directly
square
,
having
24
measures
in
Circuit
,
that
was
6
on
each
side
:
Sempronius
desiring
to
fit
himselfe
,
prayed
Caius
to
change
with
him
for
a
field
which
should
be
equivalent
unto
his
,
and
the
bargaine
being
concluded
,
he
gave
him
for
counterchange
a
piece
of
ground
which
had
just
as
much
in
circuit
as
his
had
,
but
it
was
not
square
,
yet
Quadrangular
and
Rectangled
,
having
9
measures
in
length
for
each
of
the
two
longest
sides
,
and
3
in
breadth
for
each
shorter
side
:
Now
Caius
which
was
not
the
most
subtillest
nor
wisest
in
the
world
accepted
his
bargaine
at
the
first
,
but
afterward●
having
conferred
with
a
Land-measures
and
Mathematician
,
found
that
he
was
over-reached
in
his
bargaine
,
and
that
his
field
contained
36
square
measures
,
and
the
other
field
had
but
27
measures
,
(
a
thing
easie
to
be
knowne
by
multiplying
the
length
by
the
breadth
:
)
Sempronius
contested
with
him
in
suite
of
Law
,
and
argued
that
figures
which
have
equall
Perimeter
or
circuit
,
are
equall
amongst
themselves
:
my
field
,
saith
he
,
hath
equall
circuit
with
yours
,
therefore
it
is
equall
unto
it
in
quantitie
.
Now
this
was
sufficient
to
delude
a
Judge
which
was
ignorant
in
Geometricall
proportions
,
but
a
Mathematician
will
easily
declare
the
deceit
,
being
assured
that
figures
which
are
Isoperemiter
,
or
equall
in
circuit
,
have
not
alwayes
equall
capacitie
or
quantitie
:
seeing
that
with
the
same
circuit
,
there
may
be
infinite
figures
made
which
shall
be
more
and
more
capable
,
by
how
much
they
have
more
Angles
,
equall
sides
,
and
approach
nearer
unto
a
circle
,
(
which
is
the
most
capablest
figure
of
all
,
)
because
that
all
his
parts
are
extended
one
from
anothes
,
and
from
the
middle
or
Centre
as
much
as
may
be
:
so
we
see
by
an
infa●lible
rule
of
experience
,
that
a
square
is
more
capable
of
quantitie
than
a
Triangle
of
the
same
circuit
,
and
a
Pentagone
more
than
a
square
,
and
so
of
others
,
so
that
they
be
regular
figures
that
have
their
sides
equall
,
otherwise
there
might
be
that
a
regular
Triangle
,
having
24
measures
in
circuit
might
have
more
capacitie
than
a
rectangled
Parallelogram
,
which
had
also
24
measures
of
circuit
,
as
if
it
were
11
in
length
,
and
1
inbreadth
,
the
circuit
is
still
24
,
yet
the
quantitie
is
but
11.
and
if
it
had
6
every
way
,
it
gives
the
same
Perimeter
,
viz.
24.
but
a
quantitie
of
36
as
before
.
2.
Incident
.
SEmpronius
having
borrowed
of
Caius
a
sack
of
Corne
,
which
was
6
foot
high
and
2
foot
broad
,
and
when
there
was
question
made
to
repay
it
,
Sempronius
gave
Caius
back
two
sacks
full
of
Corne
,
which
had
each
of
them
6
foot
high
&
1
foot
broad
:
who
beleeved
that
if
the
sackes
were
full
he
was
repaid
,
and
it
seems
to
have
an
appearance
of
truth
barely
looked
on
.
But
it
is
most
evident
in
demonstration
,
that
the
2
sacks
of
Corn
paid
by
Sempronius
to
Caius
,
is
but
halfe
of
that
one
sack
which
he
lent
him
:
for
a
Cylinder
or
sack
having
one
foot
of
diameter
,
and
6
foot
of
length
,
is
but
the
4
part
of
another
Cylinder
,
whose
length
is
6
foot
,
and
his
diameter
is
2
foot
:
therefore
two
of
the
lesser
Cylinders
or
sackes
,
is
but
halfe
of
the
greater
;
and
so
Caius
was
deceived
in
halfe
his
Corne.
3.
Incident
.
SOme
one
from
a
common
Fountaine
of
a
City
hath
a
Pipe
of
water
of
an
inch
diameter
;
to
have
it
more
commodious
,
he
hath
leave
to
take
as
much
more
water
,
whereupon
he
gives
order
that
a
Pipe
be
made
of
two
inches
diameter
.
Now
you
will
say
presently
that
it
is
reason
to
be
so
bigge
,
to
have
just
twice
as
〈…〉
before
:
but
if
the
Magistrate
of
the
Citie
understood
Geometricall
proportions
,
he
would
soon
cause
it
to
be
amended
,
&
shew
that
he
hath
not
only
taken
twice
as
much
water
as
he
had
before
,
but
foure
times
as
much
:
for
a
Circular
hole
which
is
two
inches
diameter
is
foure
times
greater
than
that
of
one
inch
,
and
therefore
will
cast
out
four
times
as
much
water
as
that
of
one
inch
,
and
so
the
deceit
is
double
also
in
this
.
Moreover
,
if
there
were
a
heap
of
Corne
of
20
foot
every
way
,
which
was
borrowed
to
be
paid
next
yeare
▪
the
party
having
his
Corne
in
heapes
of
12
foot
every
way
,
and
of
10
foot
every
way
,
proffers
him
4
heapes
of
the
greater
or
7
heaps
of
the
lesser
,
for
his
owne
heap
of
20
every
way
,
which
was
lent
:
here
it
seems
that
the
proffer
is
faire
,
nay
with
advantage
,
yet
the
losse
would
be
neare
1000
foot
.
Infinite
of
such
causes
do
arise
from
Geometricall
figures
,
which
are
able
to
deceive
a
Judge
or
Magistrate
,
which
is
not
somewhat
seene
in
Mathematicall
Documents
.
PROBLEM
.
LXXXVIII
.
Containing
sundry
Questions
in
matter
of
Cosmography
.
FIrst
,
it
may
be
demanded
,
where
is
the
middle
of
the
world
?
I
speak
not
here
Mathematically
,
but
as
the
vulgar
people
,
who
ask
,
where
is
the
middle
of
the
world
?
in
this
sence
to
speak
absolutely
there
is
no
point
which
may
be
said
to
be
the
middle
of
the
surface
,
for
the
middle
of
a
Globe
is
every
where
:
notwithstanding
the
Holy
Scriptures
speake
respectively
,
and
make
mention
of
the
middle
of
the
earth
,
and
the
interpreters
apply
it
to
the
Citie
of
Jerusalem
placed
in
the
middle
of
Palestina
,
and
the
habitable
world
,
that
in
effect
taking
a
mappe
of
the
world
,
and
placing
one
foot
of
the
Compasses
upon
Jerusalem
,
and
extending
the
other
foot
to
the
extremity
of
Europe
,
Asia
,
and
Afric●
,
you
shall
see
that
the
Citie
of
Jerusalem
is
as
a
Centre
to
that
Circle
.
2.
Secondly
,
how
much
is
the
depth
of
the
earth
,
the
height
of
the
heavens
,
and
the
compasse
of
the
world
?
FRom
the
surface
of
the
earth
unto
the
Centre
according
to
ancient
traditions
,
is
3436.
miles
,
so
the
whole
thicknesse
is
6872
miles
,
of
which
the
whole
compasse
or
circuit
of
the
earth
is
21600
miles
.
From
the
Centre
of
the
earth
to
the
Moone
there
is
neare
56
Semidiameters
of
the
earth
,
which
is
about
192416
miles
.
unto
the
Sunne
there
is
1142
Semidiameters
of
the
earth
,
that
is
in
miles
3924912
;
from
the
starry
firmament
to
the
Centre
of
the
earth
there
is
14000
Semidiameters
,
that
is
,
48184000
miles
,
according
to
the
opinion
and
observation
of
that
learned
Ticho
Brahe
.
From
these
measures
one
may
collect
by
Arithmeticall
supputations
,
many
pleasant
propositions
in
this
manner
.
First
,
if
you
imagine
there
were
a
hole
through
the
earth
,
and
that
a
Milstone
should
be
let
fall
down
into
this
hole
,
and
to
move
a
mile
in
each
minute
of
time
,
it
would
be
more
than
two
dayes
and
a
halfe
before
it
would
come
to
the
Centre
,
and
being
there
it
would
hang
in
the
aire
.
Secondly
,
if
a
man
should
go
every
day
20
miles
,
it
would
be
three
yeares
wanting
but
a
fortnight
,
before
he
could
go
once
about
the
earth
;
and
if
a
Bird
should
fly
round
about
it
in
two
dayes
,
then
must
the
motion
be
450
miles
in
an
houre
.
Thirdly
,
the
Moone
runnes
a
greater
compasse
each
houre
,
than
if
in
the
same
time
she
should
runne
twice
rhe
Circumference
of
the
whole
earth
.
Fourthly
,
admit
it
be
supposed
that
one
should
go
20
miles
in
ascending
towards
the
heavens
every
day
,
he
should
be
above
15
years
before
he
could
attaine
to
the
Orbe
of
the
Moone
.
Fifthly
,
the
Sunne
makes
a
greater
way
in
one
day
than
the
Moone
doth
in
20
dayes
,
because
that
the
Orbe
of
the
Sunnes
circumference
is
at
the
least
20
times
greater
than
the
Orbe
of
the
Moone
.
Sixthly
,
if
a
Milstone
should
descend
from
the
p●ace
of
the
Sunne
a
thousand
miles
every
houre
,
(
which
is
above
15
miles
in
a
minute
,
farre
beyond
the
proportion
of
motion
)
it
would
be
above
163
dayes
before
it
would
fall
downe
to
the
earth
.
Seventhly
,
the
Sunne
in
his
proper
sphere
moves
more
than
seven
thousand
five
hundred
and
seventy
miles
in
one
minute
of
time
:
now
there
is
no
Bullet
of
a
Cannon
,
Arrow
,
Thunderbolt
,
or
tempest
of
winde
that
moves
with
such
quicknesse
.
Eightly
,
it
is
of
a
farre
higher
nature
to
consider
the
exceeding
and
unmoveable
quicknesse
of
the
starry
firmament
,
for
a
starre
being
in
the
Aequator
,
(
which
is
just
between
the
Poles
of
the
world
)
makes
12598666
miles
in
one
houre
which
is
two
hundred
nine
thousand
nine
hundred
and
seventy
foure
miles
in
one
minute
of
time
:
&
if
a
Horseman
should
ride
every
day
40
miles
,
he
could
not
ride
such
a
compasse
in
a
thousand
yeares
as
the
starry
firmament
moves
in
one
houre
,
which
is
more
than
if
one
should
move
about
the
earth
a
thousand
times
in
one
houre
,
and
quicker
than
possible
thought
can
be
imagined
:
and
if
a
starre
should
flye
in
the
aire
about
the
earth
with
such
a
prodigious
quicknesse
,
it
would
burne
and
consume
all
the
world
here
below
.
Behold
therefore
how
time
passeth
,
and
death
hasteth
on
:
this
made
Copernicus
,
not
unadvisedly
to
attribute
this
motion
of
Primum
mobile
to
the
earth
,
and
not
to
the
starry
firmament
;
for
it
is
beyond
humane
sense
to
apprehend
or
conceive
the
rapture
and
violence
of
that
motion
being
quicker
than
thought
;
and
the
word
of
God
testifieth
that
the
Lord
made
all
things
in
number
,
measure
,
weight
,
and
time
.
PROBLEM
.
XCII
.
To
finde
the
Bissextile
yeare
,
the
Dominicall
letter
,
and
the
letters
of
the
moneth
.
LEt
123
,
or
124
,
or
125
,
or
26
,
or
27
,
(
which
is
the
remainder
of
1500
,
or
1600
)
be
divided
by
4
,
which
is
the
number
of
the
Leape-yeare
,
and
that
which
remaines
of
the
division
shewes
the
leap-yeare
,
as
if
one
remaine
,
it
shewes
that
it
is
the
first
yeare
since
the
Bissextile
or
Leap-year
,
if
two
,
it
is
the
second
year
&c.
and
if
nothing
remaine
,
then
it
is
the
Bissextile
or
Leap-yeare
,
and
the
Quotient
shews
you
how
many
Bissextiles
or
Leap-yeares
there
are
conteined
in
so
many
yeares
.
To
finde
the
Circle
of
the
Sun
by
the
fingers
.
LEt
123
,
24
,
25
,
26
,
or
27
,
be
divided
by
28
,
(
which
is
the
Circle
of
the
Sunne
or
whole
revolution
of
the
Dominicall
letters
)
and
that
which
remaines
is
the
number
of
joynts
,
which
is
to
be
accounted
upon
the
fingers
by
Filius
esto
Dei
,
coelum
bonus
accipe
gratis
:
and
where
the
number
ends
,
that
finger
it
sheweth
the
yeare
which
is
present
,
and
the
words
of
the
verse
shew
the
Dominicall
letter
.
Example
.
DIvide
123
by
28
for
the
yeare
(
and
so
of
other
yeares
)
and
the
Quotient
is
4
,
and
there
remaineth
11
,
for
which
you
must
account
11
words
:
Filius
esto
Dei
,
&c.
upon
the
joynts
beginning
from
the
first
joynt
of
the
Index
,
and
you
shall
have
the
answer
.
For
the
present
to
know
the
Dominicall
letter
for
each
moneth
,
account
from
January
unto
the
moneth
required
,
including
January
,
and
if
there
be
8
,
9
,
7
,
or
5
,
you
must
begin
upon
the
end
of
the
finger
from
the
thumbe
and
account
,
Adam
degebat
,
&c.
as
many
words
as
there
are
moneths
,
for
then
one
shall
have
the
letter
which
begins
the
moneth
;
then
to
know
what
day
of
the
moneth
it
is
,
see
how
many
times
7
is
comprehended
in
the
number
of
dayes
,
and
take
the
rest
,
suppose
4
,
account
upon
the
first
finger
within
&
without
by
the
joynts
,
unto
the
number
of
4
,
which
ends
at
the
end
of
the
finger
:
from
whence
it
may
be
inferred
that
the
day
required
was
Wednesday
,
Sunday
being
attributed
to
the
first
joynt
of
the
first
finger
or
Index
:
and
so
you
have
the
present
yeare
,
the
Dominicall
letter
,
the
letter
which
begins
the
Moneth
,
and
all
the
dayes
of
the
Moneth
.
PROBLEM
.
XCIII
.
To
finde
the
New
and
Full
Moone
in
each
Moneth
.
ADde
to
t●e
Epact
for
the
yeare
,
the
Moneth
from
March
,
then
subtract
that
surplus
from
30
,
and
the
rest
is
the
day
of
the
Moneth
that
it
will
be
New
Moone
,
and
adding
unto
it
14
,
you
shall
have
that
Full
Moone
.
Note
THat
the
Epact
is
made
alwayes
by
adding
11
unto
30
,
and
if
it
passe
30
,
subtract
30
,
and
adde
11
to
the
remainder
,
and
so
ad
infinitum
:
as
if
the
Epact
were
12
,
adde
11
to
it
makes
23
for
the
Epact
next
year
,
to
which
adde
11
makes
34
,
subtract
30
,
rests
4
the
Epact
for
the
yeare
after
,
and
15
for
the
yeare
following
that
,
and
26
for
the
next
,
and
7
for
the
next
,
&c.
PROBLEM
.
XCIV
.
To
finde
the
Latitude
of
●
Countrey
.
THose
that
dwell
between
the
North-Pole
and
the
Tropicke
of
Cancer
,
have
their
Spring
and
Summer
between
the
10
of
March
,
and
the
13
of
September
:
and
therefore
in
any
day
between
that
time
,
get
the
sunnes
distance
by
instrumentall
observation
from
the
zenith
at
noone
,
and
adde
the
declination
of
the
sun
for
that
day
to
it
:
so
the
Aggragate
sheweth
such
is
the
Latitude
,
or
Poles
height
of
that
Countrey
.
Now
the
declination
of
the
sunne
for
any
day
is
found
out
by
Tables
calculated
to
that
end
:
or
Mechanically
by
the
Globe
,
or
by
Instrument
it
may
be
indifferently
had
:
and
here
note
that
if
the
day
be
between
the
13
of
September
and
the
10
of
March
,
then
the
sunnes
declination
for
that
day
must
be
taken
out
of
the
distance
of
the
sunne
from
the
zenith
at
noone
:
so
shall
you
have
the
Latitude
,
as
before
.
PRBOLEM
XCV
.
Of
the
Climates
of
countreys
,
and
to
finde
in
what
C●imate
any
countrey
is
under
.
CLimates
as
they
are
taken
Geographically
signifie
nothing
else
but
when
the
l●ngt●
of
the
longest
day
of
any
place
,
is
half
an
houre
longer
,
or
shorter
than
it
is
in
another
place
(
and
so
of
the
sh●rtest
day
)
and
this
account
to
begin
from
the
Equinoctia●l
Circle
,
seeing
all
Countreys
under
it
have
the
shortest
and
longest
day
that
can
be
but
12
houres
;
But
all
other
Countreys
that
are
from
the
Equinoctiall
Circle
either
towards
the
North
or
South
of
it
unto
the
Poles
themselves
,
are
said
to
be
in
some
one
Climate
or
other
,
from
the
Equinoctiall
to
either
of
the
Poles
Circles
,
(
which
are
in
the
Latitude
of
66
degr.
30
m.
)
between
each
of
which
Polar
Circles
and
the
Equinoctial
Circle
there
is
accounted
24
Climates
,
which
differ
one
from
another
by
halfe
an
hours
time
:
then
from
each
Polar
Circle
,
to
each
Pole
there
are
reckoned
6.
other
Climates
which
differ
one
from
another
by
a
moneths
time
:
so
the
whole
earth
is
divided
into
60
Climates
,
30
being
allotted
to
the
Northerne
Hemisphere
,
and
30●
to
the
Southerne
Hemispheare
.
And
here
note
,
that
though
these
Climats
which
are
betweene
the
Equinoctiall
and
the
Polar
Circles
are
equall
one
unto
the
other
in
respect
of
time
,
to
wit
,
by
halfe
an
houre
;
yet
the
Latitude
,
breadth
,
or
internall
,
conteined
between
Climate
and
Climate
,
is
not
equall
:
and
by
how
much
any
Climate
is
farther
from
the
Equinoctiall
than
another
Climate
,
by
so
much
the
lesser
is
the
intervall
between
that
Climate
and
the
next
:
so
those
that
are
nearest
the
Equinoctial
are
largest
,
and
those
which
are
farthest
off
most
contracted
:
and
to
finde
what
Climate
any
Countrey
is
under
:
subtract
the
length
of
an
Equinoctiall
day
,
to
wit
,
12
houres
from
the
length
of
the
longest
day
of
that
Countrey
;
the
remainder
being
doubled
shews
the
Climate
:
So
at
London
the
longest
day
is
neare
16
houres
and
a
halfe
;
12
taken
from
it
there
remaines
4
houres
and
a
halfe
,
which
doubled
makes
9
halfe
houres
,
that
is
,
9
Climates
;
so
London
is
in
the
9
climate
.
PROBLEM
.
XCVI
.
Of
Longitude
and
Latitude
of
the
Earth
and
of
the
Starres
.
LOngitude
of
a
Countrey
,
or
place
,
is
an
arcke
of
the
Aequator
conteined
between
the
Meridian
of
the
Azores
,
and
the
Meridian
of
the
place
,
and
the
greatest
Longitude
that
can
be
is
360
degrees
.
Note
.
That
the
first
Meridian
may
be
taken
at
pleasure
upon
the
Terrestriall
Globe
or
Mappe
,
for
that
some
of
the
ancient
Astronomers
would
have
it
at
Hercules
Pillars
,
which
is
at
the
straights
at
Gibraltar
:
Ptolomy
placed
it
at
the
Canary
Isl●nds
,
but
now
in
these
latter
times
it
is
held
to
be
neare
the
Azores
.
But
why
it
was
first
placed
by
Ptolomy
at
the
Canary
Islands
,
were
because
that
in
his
time
these
Islands
were
the
farthest
westerne
parts
of
the
world
that
was
then
discovered
.
And
why
it
reteines
his
place
now
at
Saint
Michaels
neare
the
Azores
,
is
that
because
of
many
accurate
observations
made
of
late
by
many
expert
Navigators
and
Mathematicians
,
they
have
found
the
Needle
there
to
have
no
variation
,
but
to
point
North
and
South
:
that
,
is
to
each
Pole
of
the
world
:
and
why
the
Longitude
from
thence
is
accounted
Eastwards
,
is
from
the
motion
of
the
Sunne
Eastward
,
or
that
Ptolomy
and
others
did
hold
it
more
convenient
to
begin
from
the
Westerne
part
of
the
world
and
so
account
the
Longitude
Eastward
from
Countrey
to
Countrey
that
was
then
knowne
;
till
they
came
to
the
Easterne
part
of
Asia
,
rather
than
to
make
a
beginning
upon
that
which
was
unknowne
:
and
having
made
up
their
account
of
reckoning
the
Longitude
from
the
Westerne
part
to
the
Eastern
part
of
the
world
knowne
,
they
supposed
the
rest
to
be
all
sea
,
which
since
their
deaths
hath
been
found
almost
to
be
another
habitable
world
.
To
finde
the
Longitude
of
a
Countrey
.
IF
it
be
upon
the
Globe
,
bring
the
Countrey
to
the
Brasen
Meridian
,
and
whatsoever
degree
that
Meridian
cuts
in
the
Equinoctiall
,
that
degree
is
the
Longitude
of
that
Place
:
if
it
be
in
a
Mappe
,
then
mark
what
Meridian
passeth
over
it
,
so
have
you
the
Longitude
thereof
,
if
no
Meridian
passe
over
it
,
then
take
a
paire
of
Compasses
,
and
measure
the
distance
betweene
the
Place
and
the
next
Meridian
,
and
apply
it
to
the
divided
parallel
or
Aequator
,
so
have
you
the
Longitude
required
.
Of
the
Latitude
of
Countreys
.
LAtitude
of
a
Countrey
is
the
distance
of
a
Countrey
from
the
Equinoctiall
,
or
it
is
an
Arke
of
the
Meridian
conteined
between
the
Zenith
of
the
place
and
the
Aequator
;
which
is
two-fold
,
viz.
either
North-Latitude
or
South-Latitude
,
either
of
which
extendeth
from
the
Equinoctiall
to
either
Pole
,
so
the
greatest
Latitude
that
can
be
is
but
90
degrees
:
If
any
Northern
Countrey
have
the
Artick
Circle
verticall
,
which
is
in
the
Latitude
of
66.
gr.
30.
m.
the
Sun
will
touch
the
Horizon
in
the
North
part
thereof
,
and
the
longest
day
will
be
there
then
24
houres
,
if
the
Countrey
have
lesse
Latitude
than
66.
degrees
30.
m.
the
Sun
will
rise
and
set
,
but
if
it
have
more
Latitude
than
66.
gr.
30
m.
it
will
be
visible
for
many
dayes
,
and
if
the
Countrey
be
under
the
Pole
,
the
Sun
will
make
a
Circular
motion
above
the
Earth
,
and
be
visible
for
a
half
yeare
:
so
under
the
Pole
there
will
be
but
one
day
,
and
one
night
in
the
whole
yeare
.
To
finde
the
latitude
of
Countreys
.
IF
it
be
upon
a
Globe
,
bring
the
place
to
the
Brasen
Meridian
,
and
the
number
of
degrees
which
it
meeteth
therewith
,
is
the
Latitude
of
the
place
.
Or
with
a
paire
of
Compasses
take
the
distance
between
the
Countrey
and
the
Equinoctiall
,
which
applied
unto
the
Equinoctiall
will
shew
the
Latitude
of
that
Countrey
;
which
is
equall
to
the
Poles
height
;
if
it
be
upon
a
Mappe
.
Then
mark
what
parallel
passeth
over
the
Countrey
and
where
it
crosseth
the
Meridian
,
that
shall
be
the
Latitude
:
but
if
●o
parallel
passeth
over
it
,
then
take
the
distance
betweene
the
place
and
the
next
parallel
,
which
applied
to
the
divided
Meridian
from
that
parallel
will
shew
the
Latitude
of
that
place
.
To
finde
the
distance
of
places
.
IF
it
be
upon
a
Globe
:
then
with
a
paire
of
Compasses
take
the
distance
betweene
the
two
Places
,
and
apply
it
to
the
divided
Meridian
or
Aequator
,
and
the
number
of
degrees
shall
shew
●e
distance
;
each
degree
being
60.
miles
.
●f
it
be
in
a
Mappe
(
according
to
Wrights
pro●ection
)
take
the
distance
with
a
paire
of
Com●asses
between
the
two
places
,
and
apply
this
distance
to
the
divided
Meridian
on
the
Mappe
right
against
the
two
places
;
so
as
many
degrees
as
is
conteined
between
the
feet
of
the
Compasses
so
much
is
the
distance
between
the
two
places
.
If
the
distance
of
two
places
be
required
in
a
particular
Map
then
with
the
Compasses
take
the
distance
between
the
two
places
,
and
apply
it
to
the
scale
of
Miles
,
so
have
you
the
distance
,
if
the
scale
be
too
short
,
take
the
scale
between
the
Compasses
,
and
apply
that
to
the
two
places
as
often
as
you
can
,
so
have
you
the
distance
required
.
Of
the
Longitude
,
Latitude
,
Declination
,
and
distance
of
the
Starres
.
THe
Declination
of
a
starre
is
the
nearest
distance
of
a
Star
from
the
Aequator
;
the
Latitude
of
a
Starre
is
the
nearest
distance
of
a
Sarre
from
the
Ecliptick
:
the
Longitude
of
a
Starre
is
an
Ark
of
the
Ecliptick
conteined
between
the
beginning
of
Aries
,
and
the
Circle
of
the
Starres
Latitude
,
which
is
a
circle
drawne
from
the
Pole
of
the
Ecliptick
unto
the
starre
,
and
so
to
the
Ecliptick
.
The
distance
between
two
Sarres
in
Heaven
is
taken
by
a
Crosse-staffe
or
other
Instrument
,
and
upon
a
Globe
it
is
done
by
taking
between
the
feet
of
the
Compasses
the
two
Starres
,
and
applying
it
to
the
Aequator
,
so
have
you
the
distance
betweene
those
two
starre●
.
How
is
it
that
two
Horses
or
other
creatures
being
foaled
or
brought
forth
into
the
world
at
one
and
the
same
time
,
that
after
certaine
dayes
travell
the
one
lived
more
dayes
than
the
other
,
notwithstanding
they
dyed
together
in
one
and
the
sam●
moment
also
?
THis
is
easie
to
be
answered
:
let
one
of
them
travell
toward
the
West
and
the
other
towards
the
East
:
then
that
which
goes
towards
the
West
followeth
the
Sunne
:
and
therefore
shall
have
the
day
somewhat
longer
than
if
there
had
been
no
travell
made
,
and
that
which
goes
East
by
going
against
the
Sunne
,
shall
have
the
day
shorter
,
and
so
in
respect
of
travell
though
they
dye
at
one
and
the
selfe
same
houre
and
moment
of
time
,
the
one
shall
be
older
than
the
other
.
From
which
consideration
may
be
inferred
that
a
Christian
,
a
Jew
,
and
a
Saracen
,
may
have
their
Sabbaths
all
upon
one
and
the
same
day
though
notwithstanding
the
Saracen
holds
his
Sabath
upon
the
Friday
,
the
Jew
upon
the
Saturday
,
and
the
Christian
upon
the
Sunday
:
For
being
all
three
resident
in
one
place
,
if
the
Saracen
and
the
Christian
begin
their
travell
upon
the
Saturday
,
the
Christian
going
West
,
and
the
Saracen
Eastwards
,
shall
compasse
the
Globe
of
the
earth
,
thus
the
Christian
at
the
conclusion
shall
gaine
a
day
,
and
the
Saracen
shall
lose
a
day
,
and
so
meet
with
the
Jew
every
one
upon
his
owne
Sabbath
.
Certaine
fine
observations
.
1
UNder
the
Equinoctiall
the
Needle
hangs
in
equilibrio
,
but
in
these
parts
it
inclines
under
the
Horizon
,
and
being
under
the
Pole
it
is
thought
it
will
hang
verticall
.
2
In
these
Countreys
which
are
without
the
Tropicall
Circles
,
the
Sunne
comes
East
and
West
every
day
for
a
halfe
yeare
,
but
being
under
the
Equinoctiall
the
Sun
is
never
East
,
nor
West
▪
but
twice
in
the
yeare
,
to
wit
,
the
10.
of
March
and
the
13
of
September
.
3
If
a
ship
be
in
the
Latitude
of
23
gr.
30
m.
that
is
,
if
it
have
either
of
the
Tropicks
verticall
:
then
at
what
time
the
Sunnes
Altitude
is
equall
to
his
distan●e
from
any
of
the
Equinoctiall
points
,
then
t●e
Sunne
is
due
East
or
West
.
4
If
a
ship
be
betweene
the
Equinoctiall
and
either
of
the
Tropicks
,
the
Sunne
will
come
twice
to
one
point
of
the
compasse
in
the
forenoone
,
that
is
,
in
one
and
the
same
position
.
5
Vnder
the
Equinoctiall
neare
Guinea
there
is
but
two
sorts
of
windes
all
the
year
,
6
moneths
a
Northerly
winde
,
and
6
moneths
a
Southerly
winde
,
and
the
flux
of
the
Sea
is
accordingly
.
6
If
two
ships
under
the
Equinoctiall
be
100.
leagues
asunder
,
and
should
sayle
Northerly
untill
they
were
come
under
the
Articke
circle
,
they
should
then
be
but
50
leagues
asunder
.
7
Those
which
have
the
Artick
circle
,
verticall
:
when
the
Sunne
is
in
the
Tropick
of
Cancer
,
the
Sun
setteth
not
,
but
toucheth
the
western
part
of
the
Horizon
.
8
If
the
complement
of
the
Sunnes
height
at
noon
be
found
equall
to
the
Sunnes
Declination
for
that
day
,
then
the
●quinoctiall
is
verticall
:
or
a
shippe
making
such
an
observation
,
the
Equinoctiall
is
in
the
Zenith
,
or
direct
over
them
,
by
which
Navigators
know
when
they
crosse
the
line
,
in
their
travels
to
the
Indies
,
or
other
parts
.
9
The
Sunne
being
in
the
Equinoctiall
,
the
extremity
of
the
stile
in
any
Sunne-dyall
upon
a
plaine
,
maketh
a
right
line
,
otherwise
it
is
Eclipticall
,
Hyperbolicall
,
&c.
10
When
the
shadow
of
a
man
,
or
other
thing
upon
a
Horizontall
〈◊〉
is
equall
unto
it
in
length
,
then
is
the
Sunne
in
the
middle
point
between
the
Horizon
and
the
Zenith
,
that
is
,
45
degrees
high
.
PROBLEM
.
XCVII
.
To
make
a
Triangle
that
shall
have
three
right
Angles
.
OPen
the
C●passes
at
p●easure
:
and
upon
A
,
describe
an
Arke
BC.
then
at
the
same
opening
,
place
one
of
the
feet
in
B
,
and
describe
the
Ark
AC
.
Lastly
,
place
one
of
the
feet
of
the
Compasses
in
C.
and
describe
the
Arke
AB·
so
shall
you
have
the
sphericall
Aequilaterall
Triangle
ABC
,
right
angled
at
A
,
at
B
,
and
at
C.
that
is
,
each
angle
comprehended
9●
.
degrees
:
which
can
never
be
in
any
plaine
Triangle
,
whether
it
be
Equilaterall
,
Isocelse
,
scaleve
,
Orthogonall
,
or
Opigonall
.
PROBLEM
.
XCVIII
.
To
divide
a
line
in
as
many
equall
parts
as
one
will
,
without
compasses
,
or
without
seeing
of
it
.
THis
Proposition
hath
a
fallacie
in
it
,
&
cannot
be
practised
but
upon
a
Maincordion
:
for
the
Mathematicall
line
which
proceeds
from
the
flux
of
a
point
,
cannot
be
divided
in
that
wise
:
One
may
have
therefore
an
Instrument
which
is
called
Maincordion
,
because
there
is
but
one
cord
:
and
if
you
desire
to
divide
your
line
into
3
parts
,
run
your
finger
upon
the
frets
untill
you
sound
a
third
in
musick
:
if
you
would
have
the
fourth
part
of
the
line
,
then
finde
the
fourth
sound
,
a
fifth
,
&c.
so
shall
you
have
the
answer
.
PROBLEM
.
XCIX
.
To
draw
a
line
which
shall
incline
to
another
line
,
yet
never
meet
:
against
the
Axiome
of
Parallels
.
THis
is
done
by
help
of
a
Conoeide
line
,
produced
by
a
right
line
upon
one
&
the
same
plaine
,
held
in
great
account
amongst
the
Ancients
,
and
it
is
drawne
after
this
manner
.
Draw
a
right
line
infinitely
,
and
upon
some
end
of
it
,
as
at
I
,
draw
a
perpendicular
line
I
A.
augment
it
to
H.
then
from
A.
draw
lines
at
pleasure
to
intersect
the
line
I.
M.
in
each
of
which
lines
from
the
right
line
,
IM
.
transferre
IH
.
viz.
KB
.
LC.OD.PE.QF.MG
.
then
from
those
points
draw
the
line
H.B.C.D.E.F.G.
which
will
not
meet
with
the
line
IM
.
and
yet
incline
nearer
and
nearer
unto
it
.
PROBLEM
.
C.
To
observe
the
variation
of
the
compasses
,
or
needle
in
any
places
.
FIrst
describe
a
Circle
upon
a
plaine
,
so
that
the
Sun
may
shine
on
it
both
before
noone
and
afternoone
:
in
the
centre
of
which
Circle
place
a
Gn●●on
or
wire
perpendicular
as
AB
,
and
an
houre
before
noone
marke
the
extremitie
of
the
shadow
of
AB
,
which
suppose
it
be
at
C.
describe
a
Circle
at
that
semidiamiter
CDF
.
then
after
noone
mark
when
the
top
of
the
shadow
of
AB
.
toucheth
the
Circle
,
which
admit
in
D
;
divide
the
distance
CD
into
two
equall
parts
,
which
suppose
at
E.
draw
the
line
EAF
.
which
is
the
Meridian
line
,
or
line
of
North
&
South
:
now
if
the
Arke
of
the
Circle
CD
.
be
divided
into
degrees
.
place
a
Needle
GH
,
upon
a
plaine
set
up
in
the
Centre
,
and
marke
how
many
degrees
the
point
of
the
Needle
G
,
is
from
E.
so
much
doth
the
Needle
vary
from
the
North
in
that
place
.
PROBLEM
.
CI.
How
to
finde
at
any
time
which
way
the
wind
is
in
ones
Chamber
,
without
going
abroad
?
VPon
the
Plancking
or
floore
of
a
Chamber
,
Parlor
,
or
Hall
,
that
you
intend
to
have
this
device
,
let
there
come
downe
from
the
top
of
the
house
a
hollow
post
,
in
which
place
an
Iron
rod
that
it
ascend
above
the
house
10
,
or
6
foot
with
a
vane
or
a
scouchen
at
it
to
shew
the
winds
without
:
and
at
the
lower
end
of
this
rod
of
Iron
,
place
a
Dart
which
may
by
the
moving
of
the
vane
with
the
wind
without
,
turne
this
Dart
which
is
within
:
about
which
upon
the
plaister
must
be
described
a
Circle
divided
into
the
32
points
of
the
Mariners
Compasse
pointed
and
distinguished
to
that
end
,
then
may
it
be
marked
by
placi●
to
Compasse
by
it
;
for
having
noted
the
North
point
,
the
East
,
&c
▪
it
is
easie
to
note
all
the
rest
of
the
points
:
and
so
at
any
time
comming
into
this
Roome
,
you
have
nothing
to
do
but
to
look
up
to
the
Dart
,
which
will
point
you
out
what
way
the
winde
bloweth
at
that
instant
.
PROBLEM
.
CII
.
How
to
draw
a
parallel
sphericall
line
with
great
ease
?
FIrst
draw
an
obscure
line
GF
.
in
the
middle
of
it
make
two
points
AB
,
(
which
serves
for
Centres
then
place
one
foot
of
the
Compasses
in
B
,
and
extend
the
other
foot
to
A
,
and
describe
the
semicircle
AC
.
then
place
one
foot
of
the
Compasses
in
A
,
and
extend
the
other
foot
to
C
,
and
describe
the
semicircle
CD
.
Now
place
the
Compasses
in
B
,
and
extend
the
other
foot
unto
D
,
and
describe
the
semicircle
DF
,
and
so
ad
infinitum
;
which
being
done
neatly
,
that
there
be
no
right
line
seene
nor
where
the
Compasses
were
placed
,
will
seeme
very
strange
how
possibly
it
could
be
drawne
with
such
exactnes
,
to
such
which
are
ignorant
of
that
way
.
PROBLEM
.
CIII
.
To
measure
an
in
accessible
distance
,
as
the
breadth
of
a
River
with
the
help
of
ones
hat
onely
.
THe
way
of
this
is
easie
:
for
having
ones
hat
upon
his
head
,
come
neare
to
the
bank
of
the
River
,
and
holding
your
head
upright
(
which
may
be
by
putting
a
small
stick
to
some
one
of
your
buttons
to
prop
up
the
chin
)
pluck
downe
the
brim
or
edge
of
your
hat
untill
you
may
but
see
the
other
side
of
the
water
,
then
turne
about
the
body
in
the
same
posture
that
it
was
before
towards
some
plaine
,
and
marke
where
the
sight
by
the
brimme
of
the
hat
glaunceth
on
the
ground
▪
for
the
distance
from
that
place
to
your
standing
,
is
the
breadth
of
the
River
required
.
PROBLEM
.
CIIII.
How
to
measure
a
height
with
two
strawes
or
two
small
stickes
.
TAke
two
strawes
or
two
stickes
which
are
one
as
long
as
another
,
and
place
them
at
right
Angles
one
to
the
other
,
as
AB
.
and
AC
.
then
holding
AB
.
parallel
to
the
ground
,
place
the
end
A
to
the
eye
at
A.
and
looking
to
the
other
top
BC.
at
C.
by
going
backward
or
forward
untill
you
may
see
the
top
of
the
Tower
or
tree
,
which
suppose
at
E.
So
the
distance
from
your
standing
to
the
Tower
or
Tree
,
is
equall
to
the
height
thereof
above
the
levell
of
the
eye
:
to
which
if
you
adde
your
owne
height
you
have
the
whole
height
.
Otherwise
.
TAke
an
ordinary
square
which
Carpenters
or
other
workemen
use
,
as
HKL
.
and
placing
H.
to
the
eye
so
that
HK
.
be
levell
,
go
back
or
come
nearer
untill
that
by
it
you
may
see
the
top
M.
for
then
the
distance
from
you
to
the
height
is
equall
to
the
height
.
PROBLEM
.
CV
.
How
to
make
statues
,
letters
,
bowles
,
or
other
things
which
are
placed
in
the
side
of
a
high
building
,
to
be
seen
below
of
an
equall
bignesse
.
LEt
BC.
be
a
Pillar
7
yards
high
,
and
let
it
be
required
that
three
yards
above
the
levell
of
the
eye
A
,
viz.
at
B.
be
placed
a
Globe
,
and
9
yards
above
B.
be
placed
another
,
&
22.
yards
above
that
be
placed
another
Globe
:
how
much
shall
the
Diameter
of
these
Globes
be
,
that
at
the
eye
,
at
A
,
they
may
all
appeare
to
be
of
one
and
the
same
Magnitude
:
It
is
thus
done
,
first
draw
a
line
as
AK
.
and
upon
K.
erect
a
perpendicular
KX
.
divide
this
line
into
27
parts
▪
and
according
to
AK
.
describe
an
Arke
KY
.
then
from
K
▪
in
the
perpendicular
KX
,
account●
▪
par●s
,
viz
at
L.
which
shall
represent
the
former
three
yardes
,
and
draw
the
line
LA.
from
L
,
in
the
said
perpendicular
reckon
the
diameter
of
the
lesser
Globe
of
what
Magnitude
it
is
intended
to
be
:
suppose
SL
,
and
draw
the
line
SA
.
cutting
the
Arke
VK
.
in
N.
then
from
K.
in
the
perpendicular
account
9
yards
,
which
admit
at
T.
draw
TA
,
cutting
YK.
in
O
transferre
the
Arke
MN
,
from
A
to
P.
and
draw
AP.
which
will
cut
the
perpendicular
in
V.
so
a
line
drawne
from
the
middle
of
VF
.
unto
the
visuall
lines
AI
,
and
AV
,
shall
be
the
diameter
of
the
next
Globe
:
Lastly
,
account
from
K.
in
the
perpendicular
XK
.
22
parts
,
and
draw
the
line
WA
.
cutting
YK
in
Q.
then
take
the
Arke
MN
,
and
transferre
it
from
Q
to
R
and
draw
AR
▪
which
will
cut
the
perpendicular
in
X
so
the
line
which
passeth
by
the
meddle
of
XW
.
perpendicular
to
the
visuall
line
AW
,
and
AX.
be
the
Diameter
of
the
third
Globe
,
to
wit
5
,
6.
which
measures
transferred
in
the
Pillar
BC.
which
sheweth
the
true
Magnitude
of
the
Globes
1
,
2
,
3.
from
this
an
Architect
or
doth
proportion
his
Images
,
&
the
foulding
of
the
Robes
which
are
most
deformed
at
the
eye
below
in
the
making
,
yet
most
perfect
when
it
is
set
in
his
true
height
above
the
eye
.
PROBLEM
.
CVI.
How
to
disg●is●
or
disfigure
an
Image
,
as
a
head
,
an
arme
,
a
whole
body
,
&c.
so
that
it
hath
no
proportion
the
eares
to
become
long
:
the
nose
as
that
of
a
swan
,
the
mouth
as
a
coaches
entrance
,
&c
yet
the
eye
placed
at
a
certaine
point
will
be
seen
in
a
direct
&
exact
proportion
.
I
Will
not
strive
to
set
a
Geometricall
figure
here
,
for
feare
it
may
seeme
too
difficult
to
understand
,
but
I
will
indeavour
by
discourse
how
Mechanically
with
a
Candle
you
may
perceive
it
sensible
:
first
there
must
be
made
a
figure
upon
Paper
,
such
as
you
please
,
according
to
his
just
proportion
,
and
paint
it
as
a
Picture
(
which
painters
know
well
enough
to
do
)
afterwards
put
a
Candle
upon
the
Table
,
and
interpose
this
figure
obliquely
,
between
the
said
Candle
and
the
Bookes
of
Paper
,
where
you
desire
to
have
the
figure
disguised
in
such
sort
that
the
height
passe
athwart
the
hole
of
the
Picture
:
then
will
it
carry
all
the
forme
of
the
Picture
upon
the
Paper
,
but
with
deformity
;
follow
these
tracts
and
marke
out
the
light
with
a
Coles
black
head
or
Ink
:
and
you
have
your
desire
.
To
finde
now
the
point
where
the
eye
must
see
it
in
his
naturall
forme
:
it
is
accustomed
according
to
the
order
of
Perspective
,
to
place
this
point
in
the
line
drawne
in
height
,
equall
to
the
largenesse
of
the
narrowest
side
of
the
deformed
square
,
and
it
is
by
this
way
that
it
is
performed
.
PROBLEM
.
CVII
.
How
a
Cannon
after
that
it
hath
shot
,
may
be
covered
from
the
battery
of
the
enemy
.
LEt
the
mouth
of
a
Cannon
be
I
,
the
Cannon
M.
his
charge
NO
,
the
wheele
L
,
the
axletree
PB
.
upon
which
the
Cannon
is
placed
,
at
which
end
towards
B
,
is
placed
a
pillar
AE·
supported
with
props
D
,
C
,
E
,
F
,
G
▪
about
which
the
Axeltree
turneth
:
now
the
Cannon
being
to
shoot
,
it
retires
to
H
,
which
cannot
be
directly
because
of
the
Axletree
,
but
it
make
a
segment
of
a
circle
▪
and
hides
himselfe
behind
the
wal
QR
,
and
so
preserves
it selfe
from
the
Enemies
battery
,
by
which
meanes
one
may
avoid
many
inconveniences
which
might
arise
:
and
moreover
,
one
man
may
more
easily
replace
it
againe
for
another
shot
by
help
of
poles
tyed
to
the
wall
,
or
other
help
which
may
multiply
the
strength
.
PROBLEM
.
CVIII
.
How
to
make
a
Lever
,
by
which
one
man
may
alone
place
a
Cannon
upon
his
carriage
,
or
raise
what
other
weight
he
would
.
FIrst
place
two
thick
boards
upright
,
as
the
figure
sheweth
,
pierced
with
holes
,
alike
opposite
one
unto
another
as
CD
,
and
EF
:
&
let
L
,
and
M
,
be
the
two
barres
of
Iron
which
passeth
through
the
holes
GH
,
and
F
,
K
,
the
two
supports
,
or
props
,
AB
.
the
Cannon
,
OP
,
the
Lever
,
RS
,
the
two
notches
in
the
Lever
,
and
Q
,
the
hooke
where
the
burthen
or
Cannon
is
tyed
to
.
The
rest
of
the
operation
is
●cill
,
that
the
youngest
schollers
or
learners
cannot
faile
to
performe
it
:
to
teach
Minerva
were
in
vaine
,
and
it
were
to
Mathematicians
injury
in
the
succeeding
Ages
.
PROBLEM
.
CIX
.
How
to
make
a
Clock
with
one
onely
wheele
.
MAke
the
body
of
an
ordinary
Dyall
,
and
divide
the
houre
in
the
Circle
into
12.
parts
:
make
a
great
wheele
in
height
above
the
Axletree
,
to
the
which
you
shall
place
the
cord
of
your
counterpoize
▪
so
that
it
may
descend
,
that
in
1●
houres
of
time
your
Index
or
Needle
may
make
one
revolution
,
which
may
be
knowne
by
a
watch
which
you
may
have
by
you
:
then
put
a
balance
which
may
stop
the
course
of
the
Wheele
,
and
give
it
a
regular
motion
,
and
you
shall
see
an
effect
as
just
from
this
as
from
a
Clock
with
many
wheeles
.
PROBLEM
.
CX
.
How
by
help
of
two
wheeles
to
make
a
Childe
to
draw
up
alone
a
hogshead
of
water
at
a
time
:
and
being
drawne
up
shall
cast
out
it selfe
into
another
vessell
as
one
would
have
it
.
LEt
R
be
the
Pit
from
whence
water
is
to
be
drawne
,
P
the
hook
to
throw
out
the
water
when
it
is
brought
up
(
this
hook
must
be
moveable
)
let
AB
be
the
Axis
of
the
wheele
SF
,
which
wheele
hath
divers
forkes
of
Iron
made
at
G
,
equally
fastened
at
the
wheele
;
let
I
,
be
a
Card
,
which
is
drawne
by
K
,
to
make
the
wheele
S
,
to
turne
,
which
wheele
S
,
beares
proportion
to
the
wheele
T
,
as
8
to
●
.
let
N
be
a
Chaine
of
Iron
to
which
is
tyed
the
vessel
O
,
and
the
other
which
is
in
the
Pit
:
E●
is
a
piece
of
wood
which
hath
a
mortes
in
1
,
and
●
,
by
which
the
Cord
I
,
passeth
,
tyed
at
the
wall
,
as
KH
,
and
the
other
piece
of
timber
of
the
little
wheele
as
M
,
mortified
in
likewise
for
the
chaine
to
passe
through
:
draw
the
Cord
I
,
by
K
,
and
the
wheele
will
turne
,
&
so
consequently
the
wheele
T
,
which
will
cause
the
vessell
O
to
raise
;
which
being
empty
,
draw
the
cord
againe
by
Y
,
and
the
other
vessell
which
is
in
the
pit
●ill
come
out
by
the
same
reason
.
This
is
an
invention
which
will
save
labour
if
practised
;
but
here
is
to
be
noted
that
the
pit
must
be
large
enough
,
to
the
end
that
it
conteine
two
great
vessels
to
passe
up
and
downe
one
by
another
▪
PROBLEM
.
CXI
.
To
make
a
Ladder
of
Cords
,
which
may
be
carryed
in
ones
pocket
:
by
which
one
may
easily
mount
up
a
Wall
,
or
Tree
alone
.
TAke
two
Pullies
A
,
&
D
,
unto
that
of
A
,
let
there
be
fastened
a
Cramp
of
Iron
as
B
;
and
at
D
,
let
there
be
fastened
a
staffe
of
a
foot
and
a
halfe
long
as
F
,
then
the
Pully
A
:
place
a
hand
of
Iron
,
as
E
,
to
which
tie
a
cord
of
an
halfe
inch
thick
(
which
may
be
of
silk
because
it
is
for
the
pocket
:
)
then
strive
to
make
fast
the
Pully
A
,
by
the
help
of
the
Crampe
of
Iron
B
,
to
the
place
that
you
intend
to
scale
;
and
the
staffe
F
,
being
tyed
at
the
Pully
D
,
put
it
between
your
legges
as
though
you
would
sit
upon
it
:
then
holding
the
Cord
C
in
your
hand
,
you
may
guide
your selfe
to
the
place
required
▪
which
may
be
made
more
facill
by
the
multiplying
of
Pullies
.
This
secret
is
most
excellent
in
Warre
,
and
for
lovers
,
its
supportablenesse
avoids
suspition
.
PROBLEM
.
CXII
.
How
to
make
a
Pumpe
whose
strength
is
marvelous
by
reason
of
the
great
weight
of
water
that
it
is
able
to
bring
up
at
once
,
and
so
by
continuance
.
LEt
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
,
be
the
height
of
the
case
about
two
or
three
foot
high
,
and
broader
according
to
discretion
:
the
rest
of
the
Case
or
concavity
let
be
O
:
let
the
sucker
of
the
Pumpe
which
is
made
,
be
just
for
the
Case
or
Pumpes
head
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
,
&
may
be
made
of
wood
or
brasse
of
4
inches
thick
,
having
a
hole
at
E
,
which
descending
raiseth
up
the
cover
P
,
by
which
issueth
forth
the
water
,
&
ascending
or
raising
up
it
shuts
it
or
makes
it
close
:
RS
,
is
the
handle
of
the
sucker
tyed
to
the
handle
TX
,
which
works
in
the
post
VZ
.
Let
A
,
B
,
C
,
D
,
be
a
piece
of
Brasse
,
G
the
piece
which
enters
into
the
hole
to
F
,
to
keep
out
the
Aire
.
H
,
I
,
K
,
L
,
the
piece
tyed
at
the
funnell
or
pipe
:
in
which
playes
the
Iron
rod
or
axis
G
,
so
that
it
passe
through
the
other
piece
MN
,
which
is
tyed
with
the
end
of
the
pipe
of
Brasse
.
Note
,
that
the
lower
end
of
the
Cisterne
ought
to
be
rested
upon
a
Gridiron
or
Iron
Grate
▪
which
may
be
tyed
in
the
pit
,
by
which
means
lifting
up
and
putting
downe
the
handle
,
you
may
draw
ten
times
more
water
than
otherwise
you
could
.
PROBLEM
.
CXIII
.
How
by
meanes
of
a
Cisterne
,
to
make
water
of
a
Pit
continually
to
ascend
without
strength
,
or
the
assistance
of
any
other
Pumpe
.
LEt
IL
,
be
the
Pit
where
one
would
cause
water
to
ascend
continually
to
●●ach
office
of
a
house
or
the
places
which
are
separated
from
it
:
let
there
be
made
a
receive●
as
A
,
well
closed
up
with
lead
or
other
matter
that
aire
enter
not
in
,
to
which
fasten
a
pipe
of
lead
as
at
E
,
which
may
have
vent
at
pleasure
,
then
let
there
be
made
a
Cisterne
as
B
,
which
may
be
communicative
to
A
,
by
helpe
of
the
pipe
G
,
from
which
Cistern
B
,
may
issue
the
water
of
pipe
D
,
which
may
descend
to
H
,
which
is
a
little
below
the
levell
of
the
water
of
the
pit
as
much
as
is
GH
.
to
the
end
of
which
shall
be
soldered
close
a
Cock
which
shall
cast
out
the
water
by
KH
.
Now
to
make
use
of
it
,
let
B
be
filled
full
of
water
,
and
when
you
would
have
it
run
turne
the
Cock
,
for
then
the
water
in
B
,
will
descend
by
K.
and
for
feare
that
there
should
be
vacuity
,
nature
which
abhors
it
,
will
labour
to
furnish
and
supply
that
emptinesse
out
of
the
spring
F
,
and
that
the
Pit
dry
not
,
the
Pipe
ought
to
be
small
of
an
indifferent
capacity
according
to
the
greatnesse
or
smalnesse
of
the
spring
.
PROBLEM
.
CXIIII
.
How
out
of
a
fountaine
to
cast
the
water
very
high
:
different
from
a
Probleme
formerly
delivered
.
LEt
the
fountaine
be
BD
,
of
a
round
forme
(
seeing
it
is
the
most
capable
and
most
perfect
figure
)
place
into
it
two
pipes
conjoyned
as
EA
,
and
HC
,
so
that
no
Aire
may
enter
in
at
the
place
of
joyning
:
let
each
of
the
Pipes
have
a
cock
G
,
&
L
:
the
cocke
at
G
,
being
closed
,
open
that
at
I
▪
&
so
with
a
squirt
force
the
water
through
the
hole
at
H
,
then
close
the
Cocke
at
A
,
&
draw
out
the
squirt
,
and
open
the
cock
at
G.
the
Aire
being
before
rarified
will
extend
his
dimensions
,
and
force
the
water
with
such
violence
,
that
it
will
amount
above
the
height
of
one
or
two
Pipes
:
and
so
much
the
more
by
how
much
the
Machine
is
great
:
this
violence
will
last
but
a
little
while
if
the
Pipe
have
too
great
an
opening
,
for
as
the
Aire
approacheth
to
his
naturall
place
,
so
the
force
will
diminish
.
PROBLEM
.
CXV
.
How
to
empty
the
water
of
a
Cisterne
by
a
Pipe
which
shall
have
a
motion
of
it selfe
.
LEt
AB
,
be
the
vessell
;
CDE
,
the
Pipe
:
HG
,
a
little
vessell
under
the
greater
,
in
which
one
end
of
the
Pipe
is
,
viz.
C
,
and
let
the
other
end
of
the
Pipe
E.
passing
through
the
bottome
of
the
vessell
at
F
,
then
as
the
vessell
filleth
so
will
the
Pipe
,
and
when
the
vessell
,
shall
be
full
as
farre
as
PO
,
the
Pipe
will
begin
to
runne
at
E
,
of
his
owne
accord
,
and
never
cease
untill
the
vessell
be
wholly
empty
.
PROBLEM
CXVI
.
How
to
squirt
or
spout
out
a
great
height
,
so
that
one
pot
of
water
shall
last
a
long
time
.
LEt
there
be
prepared
two
vessels
of
Brasse
,
Lead
,
or
of
other
matter
of
equal
substance
as
are
the
two
vessels
AB
,
and
BD
,
&
let
them
be
joyned
together
by
the
two
Pillars
MN
,
&
EF
:
then
let
there
be
a
pipe
HG
.
which
may
passe
through
the
cover
of
the
vessell
CD
,
and
passe
through
AB
,
into
G
,
making
a
little
bunch
or
rising
in
the
cover
of
the
vessell
AB
,
so
that
the
pipe
touch
it
not
at
the
bottome
:
then
let
there
be
soldered
fast
another
Pipe
IL
,
which
may
be
separated
from
the
bottome
of
the
vessell
,
and
may
have
his
bunchie
swelling
as
the
former
without
touching
the
bottome
:
as
is
represented
in
L
,
and
passing
through
the
bottome
of
AB
,
may
be
continued
unto
I
,
that
is
to
say
,
to
make
an
opening
to
the
cover
of
the
vessell
AB
,
&
let
it
have
a
little
mouth
as
a
Trumpet
:
to
that
end
to
receive
the
water
.
Then
there
must
further
be
added
a
very
smal
Pipe
which
may
passe
through
the
bottome
of
the
vessell
AB
,
as
let
it
be
OP
,
and
let
there
be
a
bunch
;
or
swelling
over
it
as
at
P
,
so
that
it
touch
not
also
the
bottome
:
let
there
be
further
made
to
this
lesser
vessell
an
edge
in
forme
of
a
Basin
to
receive
the
water
,
which
being
done
poure
water
into
the
Pipe
IL
,
untill
the
vessell
CD
,
be
full
,
then
turne
the
whole
Machine
upside
downe
that
the
vessell
CD
,
may
be
uppermost
,
and
AB
,
undermost
;
so
by
helpe
of
the
pipe
GH
,
the
water
of
the
vessell
CD
,
will
runne
into
the
vessel
AB
,
to
have
passage
by
the
pipe
PO.
This
motion
is
pleasant
at
a
feast
in
filling
the
said
vessel
with
wine
,
which
will
spout
it
out
as
though
it
were
from
a
boyling
fountaine
,
in
the
forme
of
a
threed
very
pleasant
to
behold
.
PROBLEM
.
CXVIII
.
How
to
practise
excellently
the
reanimation
of
simples
,
in
case
the
plants
may
not
be
transported
to
be
replanted
by
reason
of
distance
of
places
.
TAke
what
simple
you
please
,
burne
it
and
take
the
ashes
of
it
,
and
let
it
be
calcinated
two
houres
between
two
Creusets
wel
luted
,
and
extract
the
salt
:
that
is
,
to
put
water
into
it
in
moving
of
it
;
then
let
it
settle
:
and
do
it
two
or
three
times
,
afterwards
evaporate
it
,
that
is
,
let
the
water
be
boyled
in
some
vessel
,
untill
it
be
all
consumed
:
then
there
will
remaine
a
salt
at
the
bottome
,
which
you
shall
afterwards
sowe
in
good
Ground
wel
prepared
:
such
as
the
Theatre
of
husbandry
sheweth
,
and
you
shall
have
your
desire
.
PROBLEM
.
CVIII
.
How
to
make
an
infalliable
perpetuall
motion
.
M●xe
5.
or
6.
ounces
of
☿
with
is
equall
weight
of
♃
,
grinde
it
together
with
10.
or
12
ounces
of
sublimate
dissolved
in
a
celler
upon
a
Marble
the
space
of
foure
dayes
,
and
it
will
become
like
Oile
,
Olive
,
which
distill
with
fire
of
chaffe
or
driving
fire
,
and
it
will
sublime
dry
substance
,
then
put
water
upon
the
earth
(
in
forme
of
Lye
)
which
will
be
at
the
bottom
of
the
Limbeck
,
and
dissolve
that
which
you
can
;
filter
it
,
then
distill
it
,
and
there
will
be
produced
very
subtill
Antomes
,
which
put
into
a
bottle
close
stopped
,
and
keep
it
dry
,
and
you
shall
have
your
desire
,
with
astonishment
to
all
the
world
,
and
especially
to
those
which
have
travelled
herein
without
fruit
.
PROBLEM
.
CXIX
.
Of
the
admirable
invention
of
making
the
Philosophers
Tree
,
which
one
may
see
with
his
eye
to
grow
by
little
and
little
.
TAke
two
ounces
of
Aqua
fortis
,
and
dissolve
in
it
halfe
an
ounce
of
fine
silver
refined
in
a
Cappell
:
then
take
an
ounce
of
Aqua
fortis
,
and
two
drams
of
Quick-silver
:
which
put
in
it
,
and
mixe
these
two
dissolved
things
together
,
then
cast
it
into
a
Viall
of
halfe
a
pound
of
water
,
which
may
be
well
stopped
;
for
then
every
day
you
may
see
it
grow
both
in
the
Tree
and
in
the
branch
.
This
liquid
serves
to
black
haire
which
is
red
,
or
white
,
without
fading
untill
they
fall
,
but
here
is
to
be
noted
that
great
care
ought
to
be
had
in
anointing
the
haire
,
for
feare
of
touching
the
flesh
:
for
this
composition
is
very
Corrosive
or
searching
,
that
as
soone
as
it
toucheth
the
flesh
it
raiseth
blisters
,
and
bladders
very
painfull
.
PROBLEM
.
CXX
.
How
to
make
the
representation
of
the
great
world
?
DRaw
salt
Niter
out
of
salt
Earth
▪
which
is
found
along
the
Rivers
side
,
and
at
the
foot
of
Mountaines
,
where
especially
are
Minerals
of
Gold
and
Silver
:
mix
that
Niter
well
cleansed
with
♃
,
then
calcinate
it
hermetically
▪
then
put
it
in
a
Limbeck
and
let
the
receiver
be
of
Glasse
,
well
luted
,
and
alwayes
in
which
let
there
be
placed
leaves
of
Gold
at
the
bottome
,
then
put
fire
under
the
Limbeck
untill
vapours
arise
which
will
cleave
unto
the
Gold
;
augment
your
fire
untill
there
ascend
no
more
,
then
take
away
your
receiver
,
and
close
it
hermetically
,
and
make
a
Lampe
fire
under
it
untill
you
may
see
presented
in
it
that
which
nature
affords
us
:
as
Flowers
,
Trees
,
Fruits
,
Fountaines
,
Sunne
,
Moone
,
Starres
,
&c.
Behold
here
the
forme
of
the
Limbeck
,
and
the
receiver
:
A
represents
the
Limbeck
,
B
stands
for
the
receiver
.
PROBLEM
.
CXXI
.
How
to
make
a
Cone
,
or
a
Pyramidall
body
move
upon
a
Table
without
springs
or
other
Artificiall
meanes
:
so
that
it
shall
move
by
the
edge
of
the
Table
without
falling
?
THis
proposition
is
not
so
thornie
and
subtile
as
it
seemes
to
be
,
for
putting
under
a
Cone
of
paper
a
Beetle
or
such
like
creature
,
you
shall
have
pleasure
with
astonishment
&
admiration
to
those
which
are
ignorant
in
the
cause
:
for
this
animall
will
strive
alwayes
to
free
herself
from
the
captivity
in
which
she
is
in
by
the
imprisonment
of
the
Cone
:
for
comming
neere
the
edge
of
the
Table
she
will
returne
to
the
other
side
for
feare
of
falling
.
PROBLEM
CXXII
.
To
cleave
an
Anvill
with
the
blow
of
a
Pistoll
.
THis
is
proper
to
a
Warrier
,
and
to
performe
it
,
let
the
Anvill
be
heated
red
hot
as
one
can
possible
,
in
such
sort
that
all
the
solidity
of
the
body
be
softned
by
the
fire
:
then
charge
the
Pistoll
with
a
bullet
of
silver
,
and
so
have
you
infallibly
the
experiment
.
PROBLEM
.
CXXIII
.
How
to
r●st
a
Capon
carried
in
a
Budget
at
a
Saddle-bowe
,
in
the
space
of
riding
5
or
6
miles
?
HAving
made
it
ready
and
larded
it
,
stuffe
●t
with
Butter
;
then
heat
a
piece
of
steele
which
may
be
formed
round
according
to
the
length
of
the
Capon
,
and
big
enough
to
fill
the
Belly
of
it
,
and
then
stop
it
with
Butter
;
then
wrap
it
up
well
and
inclose
it
in
a
Box
in
the
Budget
,
and
you
shall
have
your
desire
:
it
is
said
that
Count
Mansfield
served
himse●fe
with
no
others
,
but
such
as
were
made
ready
in
this
kinde
,
for
that
it
loseth
none
of
its
substance
,
and
it
is
dressed
very
equally
.
PROBLEM
.
CXXIV
.
How
to
make
a
Candle
burne
and
continue
three
times
as
long
as
otherwise
it
would
?
VNto
the
end
of
a
Candle
half●burned
stick
a
farthing
lesse
or
more
,
to
make
it
hang
perpendicular
in
a
vessel
of
water
,
so
that
it
swimme
above
the
water
;
then
light
it
,
and
it
will
susteine
it self
&
float
in
this
manner
;
and
being
placed
into
a
fountaine
,
pond
,
or
lake
that
runnes
slowly
,
where
many
people
assemble
,
it
will
cause
an
extreme
feare
to
those
which
come
therein
in
the
night
,
knowing
not
what
it
is
.
PROBLEM
.
CXXV
.
How
out
of
a
quantitie
of
wine
to
extract
that
which
is
most
windy
,
and
evill
,
that
it
hurt
not
a
sick
Person
?
TAke
two
vials
in
such
sort
that
they
be
of
like
greatnesse
both
in
th●
belly
and
the
neck
;
fill
one
of
them
of
wine
,
and
the
other
of
water
:
let
the
mouth
of
that
which
hath
the
water
be
placed
into
the
mouth
of
that
which
hath
the
wine
,
so
the
water
shall
be
uppermost
,
now
because
the
water
is
heavier
than
the
wine
,
it
will
descend
into
the
other
Viall
,
and
the
wine
which
is
lowest
,
because
it
is
highest
will
ascend
above
to
supply
the
place
of
the
water
,
and
so
there
will
be
a
mutuall
interchange
of
liquids
,
and
by
this
penetration
the
wine
wil
lose
her
vapors
in
passing
through
the
water
.
PROBLEM
CXXVI
.
How
to
make
two
Marmouzets
,
one
of
which
shall
light
a
Candle
,
and
the
other
put
it
out
?
Upon
the
side
of
a
wall
make
the
figure
of
a
Marmouzet
or
other
animall
or
forme
,
and
right
against
it
on
the
other
wall
make
another
;
in
the
mouth
of
each
put
a
pipe
or
quill
so
artificially
that
it
be
not
perceived
;
in
one
of
which
place
salt
peter
very
fine
,
and
dry
and
pulverised
;
and
at
the
end
set
a
little
match
of
paper
,
in
the
other
place
sulphur
beaten
smal
,
then
holding
a
Candle
lighted
in
your
hand
,
say
to
one
of
these
Images
by
way
of
command
,
Blow
out
the
Candle
;
then
lighting
the
paper
with
the
candle
,
the
salt-peter
wil
blow
out
the
Candle
immediatly
,
and
going
to
the
other
Image
(
before
the
match
of
the
Candle
be
out
)
touch
the
sulphur
with
it
and
say
,
Light
the
Candle
,
&
it
will
immediatly
be
lighted
,
which
will
cause
an
admiration
to
those
which
see
the
action
,
if
it
be
wel
done
with
a
secret
dexterity
.
PROBLEM
.
XXVII
.
How
to
keepe
wine
fresh
as
if
it
were
in
a
celler
though
it
were
in
the
heat
of
Summer
,
and
without
Ice
or
snow
,
yea
though
it
were
carried
at
a
saddles
bow
,
and
exposed
to
the
Sun
all
the
day
?
SEt
your
wine
in
a
viall
of
Glasse
;
and
place
it
in
a
Box
made
of
wood
,
Leather
,
or
such
like
:
about
which
vial
place
Salt-peeter
,
and
it
will
preserve
it
and
keep
it
very
fresh
:
this
experiment
is
not
a
little
commodious
for
those
which
are
not
neare
fresh
waters
,
and
whose
dwellings
are
much
exposed
to
the
Sunne
.
PUOBLEM
.
CXXVIII
.
To
make
a
Cement
which
indureth
or
lasteth
as
marble
,
which
resisteth
aire
and
water
without
ever
disjoyning
or
uncementing
?
TAke
a
quantity
of
strong
and
gluing
Morter
well
beaten
,
mixe
with
this
as
much
new
slaked
Lime
,
and
upon
it
cast
Oile
of
Olive
or
Linseed-Oile
,
and
it
will
become
hard
as
Marble
being
applyed
in
time
.
PROBLEM
.
CXXIX
.
How
to
melt
metall
very
quickly
,
yea
in
a
shell
upon
a
little
fire
.
MAke
a
bed
upon
a
bed
of
metall
with
pouder
of
Sulphur
,
of
Salt-peeter
,
and
saw-dust
alike
;
then
put
fire
to
the
said
pouder
with
a
burning
Charcole
,
and
you
shall
see
that
the
metall
will
dissolve
incontinent
and
be
in
a
Masse
.
This
secret
is
most
excellent
,
and
hath
been
practised
by
the
reverend
father
Mercen●●
of
the
order
of
the
Minims
.
PROBLEM
.
CXXX
.
How
to
make
Iron
or
steele
exceeding
hard
?
QVench
your
Blade
or
other
Instrument
seven
times
in
the
blood
of
a
male
Hog
mixt
with
Goose-grease
,
and
at
each
time
dry
it
at
the
fire
before
you
wet
it
:
and
it
will
become
exceeding
hard
,
and
not
brittle
,
which
is
not
ordinary
according
to
other
temperings
and
quenchings
of
Iron
:
an
experiment
of
small
cost
,
often
proved
,
and
of
great
consequence
for
Armorie
in
warlike
negotiations
.
PRBOLEM
CXXXI
.
To
preserve
fire
as
long
as
you
will
,
imitating
the
inextinguible
fire
of
Vestales
.
AFter
that
you
have
extracted
the
burning
spirit
of
the
salt
of
♃
,
by
the
degrees
of
fire
,
as
is
required
according
to
the
Art
of
Chymistrie
,
the
fire
being
kindled
of
it selfe
,
break
the
Limbeck
,
and
the
Irons
which
are
found
at
the
bottome
will
flame
and
appeare
as
burning
Coles
as
soone
as
they
feele
the
aire
;
which
if
you
promptly
inclose
in
a
viall
of
Glasse
,
and
that
you
stop
it
exactly
with
some
good
Lute
:
or
to
be
more
assured
it
may
be
closed
up
with
Hermes
wax
for
feare
that
the
Aire
get
not
in
.
Then
will
it
keep
more
than
a
thousand
yeares
(
as
a
man
may
say
)
yea
at
the
bottome
of
the
Sea
;
and
opening
it
at
the
end
of
the
time
,
as
soone
as
it
feeles
the
Aire
〈◊〉
takes
fi●e
▪
with
which
you
may
light
a
Match
.
This
secret
merits
to
be
travelled
after
and
put
in
practice
,
for
that
it
is
not
common
,
and
full
of
astonishment
,
seeing
that
all
kinde
of
fire
lasteth
but
as
long
as
his
matter
lasteth
,
and
that
there
is
no
matter
to
be
found
that
will
so
long
in●●●e
.
Artificiall
fire-Workes
:
Or
the
manner
of
making
of
Rockets
and
Balls
of
fire
,
as
well
for
the
Water
,
as
for
the
Aire
;
with
the
composition
of
Starres
,
Golden-rain
,
Serpen●s
,
Lances
,
Whee●s
of
fire
and
such
like
,
pleasant
and
Recreative
.
Of
the
composition
for
Rockets
.
IN
the
making
of
Rockets
,
the
chiefest
thing
to
be
regarded
is
the
composition
that
they
ought
to
be
filled
with
;
forasmuch
as
that
which
is
proper
to
Rockets
which
are
of
a
lesse
sort
is
very
improper
to
those
which
are
of
a
more
greater
forme
;
for
the
fire
being
lighted
in
a
great
concave
,
which
is
filled
with
a
quick
composition
,
burnes
with
great
violence
;
contrarily
,
a
weak
composition
being
in
a
small
concave
,
makes
no
effect
:
therefore
we
shall
here
deliver
in
the
first
place
rules
and
directions
,
which
may
serve
for
the
true
composition
,
or
matter
with
which
you
may
charge
any
Rocket
,
from
Rockets
which
are
charged
but
with
one
ounce
of
Powder
unto
great
Rockets
which
requireth
for
their
charge
10
pound
of
Powder
,
as
followeth
.
For
Rockets
of
one
ounce
.
Vnto
each
pound
of
good
musket
Powder
smal
beaten
,
put
two
ounces
of
smal
Cole
dust
,
and
with
this
composition
charge
the
Rocket
.
For
Rockets
of
2
or
3
ounces
.
Vnto
every
foure
ounces
and
a
halfe
of
powder
dust
,
adde
an
ounce
of
Salt-peter
,
or
to
every
4
ounces
of
powder
dust
,
adde
an
ounce
of
Cole
dust
.
For
Rockets
of
4
ounces
.
Vnto
every
pound
of
Powder
dust
adde
4
ounces
of
Salt
peter
&
one
ounce
of
Cole
dust
:
but
to
have
it
more
slow
,
unto
every
10.
ounces
of
good
dust
powder
adde
3
ounces
of
Salt-peter
,
and
3
ounces
of
Cole
dust
.
For
Rockets
of
5
or
6
ounces
.
Vnto
every
pound
of
Powder
dust
,
adde
3
ounces
and
a
halfe
of
Salt
peter
,
and
2
ounces
and
a
halfe
of
Coledust
,
as
also
an
ounce
of
Sulphur
,
and
an
ounce
of
fyle
dust
.
For
Rockets
of
7
or
8
ounces
.
Vnto
every
pound
of
Powder
dust
adde
4
ounces
of
Salt
peter
,
and
3
ounces
of
Sulphur
.
Of
Rockets
of
10
or
12
ounces
.
Vnto
the
precedent
composition
adde
halfe
an
ounce
of
Sulphur
,
and
it
will
be
sufficient
.
For
Rockets
of
14
or
15
ounces
.
Vnto
every
pound
of
Powder
dust
adde
4
ounces
of
Salt
peter
,
or
Cole
dust
2
¼
ounces
of
Sulphur
and
file
dust
of
1
¼
ounce
.
For
Rockets
of
1
,
pound
.
Vnto
every
pound
of
Powder
dust
adde
3
ounces
of
Cole
dust
,
and
one
ounce
of
Sulphur
.
Of
Rockets
of
2
,
pound
.
Vnto
every
pound
of
Powder
dust
adde
9
½
ounces
of
Salt
peter
,
of
Cole
dust
2
1/●
ounces
,
filedust
1
●/2
ounce
,
and
of
Sulphur
¾
of
ounce
.
For
Rockets
of
3
,
pound
.
Vnto
every
pound
of
Salt
peter
adde
6
ounces
of
Cole
dust
,
and
of
Sulpher
4
,
ounces
.
For
Rockets
of
4
,
5
,
6
,
or
7
,
pound
.
Vnto
every
pound
of
Salt
peter
adde
5
ounces
of
Cole
dust
,
and
2
½
ounces
of
Sulphur
.
For
Rockets
of
8
,
9
,
or
10
pound
.
Vnto
every
pound
of
Salt
peter
,
adde
5
½
ounces
of
Cole
dust
,
and
of
Sulphur
2
½
ounces
.
Here
note
that
in
all
great
Rockets
,
there
is
no
Powder
put
,
because
of
the
greatnesse
of
the
fire
which
is
lighted
at
once
,
which
causeth
too
great
a
violence
,
therefore
ought
to
be
filled
with
a
more
weaker
composition
.
Of
the
making
of
Rockets
and
other
Fireworkes
.
FOr
the
making
of
Rockets
of
sundry
kindes
,
divers
moulds
are
to
be
made
,
with
their
Rolling
pins
,
Breaths
,
Chargers
,
&c.
as
may
be
seen
here
in
the
figure
.
And
having
rolled
a
Case
of
paper
upon
the
Rolling
pin
for
your
mould
,
fill
it
with
the
composition
belonging
to
that
mould
as
before
is
delivered
:
now
may
you
load
it
on
the
top
,
with
Serpents
,
Reports
,
Stars
,
or
Golden
Raine
:
the
Serpents
are
made
about
the
bignesse
of
ones
little
finger
,
by
rolling
a
little
paper
upon
a
small
stick
,
and
then
tying
one
end
of
it
,
and
filling
it
with
the
mixt
composition
somewhat
close
,
and
then
tying
the
other
end
.
The
reports
are
made
in
their
paper-Cases
as
the
Serpents
,
but
the
Paper
somewhat
thicker
to
give
the
greater
report
.
These
are
filled
with
graine-Powder
or
halfe
Powder
and
halfe
composition
,
and
tying
both
ends
close
,
they
are
finished
.
The
best
kinde
of
starres
are
made
with
this
mixture
following
;
unto
every
4
ounces
of
Salt-peter
,
adde
2
ounces
of
Sulphur
,
and
to
it
put
1.
ounce
of
Powder-dust
,
and
of
this
composition
make
your
starres
,
by
putting
a
little
of
it
within
a
small
quantity
of
towe
;
and
then
tying
it
up
in
the
form
of
a
ball
as
great
as
an
Hasel-Nut
or
a
little
Wal-nut
,
through
which
there
must
be
drawne
a
little
Primer
to
make
it
take
fire
.
Touching
the
making
of
the
Golden
Raine
,
that
is
nothing
but
filling
of
Quilles
with
the
composition
of
your
Rockets
somewhat
hard
.
Now
if
the
head
of
a
Rocket
be
loaded
with
a
thousand
of
those
Quilles
,
it
s
a
goodly
sight
to
see
how
pleasantly
they
●pread
themselves
in
the
Aire
,
and
come
downe
like
streames
of
Gold
much
like
the
falling
downe
of
Snow
being
agitated
by
some
turbulent
winde
.
Of
recreative
fires
.
1
PHil●strates
saith
,
that
if
wine
in
a
platter
be
placed
upon
a
receiver
of
burning
Coles
,
to
exhale
the
spirit
of
it
,
and
be
inclosed
within
a
Cupboard
or
such
like
place
,
so
that
the
Aire
may
not
go
in
,
nor
out
,
and
so
being
shut
up
for
30
yeares
,
he
that
shall
open
it
,
having
a
wax
Candle
lighted
,
and
shall
put
it
into
the
Cubboard
there
will
appeare
unto
him
the
figure
of
many
cleare
starres
.
2
If
Aquavitae
have
Camphire
dissolved
in
it
;
and
be
evaporated
in
a
close
Chamber
,
where
there
is
but
a
Charcole
fire
,
the
first
that
enters
into
the
Chamber
with
a
Candle
lighted
,
will
be
extremely
astonished
,
for
all
the
Chamber
will
seeme
to
be
full
of
fire
very
subtile
,
but
it
will
be
of
little
continuance
.
3
Candles
which
are
deceitful
are
made
of
halfe
powder
,
covered
over
with
Tallow
,
and
the
other
halfe
is
made
of
cleane
Tallow
,
or
Waxe
,
with
an
ordinary
week
;
this
Candle
being
lighted
,
and
the
upper
halfe
consumed
,
the
powder
will
take
fire
,
not
without
great
noise
and
astonishment
to
those
which
are
ignorant
of
the
cause
.
4
A
dozen
or
twenty
smal
Serpents
placed
secretly
under
a
Candlestick
that
is
indifferent
big
,
which
may
have
a
hole
passe
through
the
socket
of
it
to
the
Candle
,
through
which
a
piece
of
primer
may
be
placed
,
and
setting
a
smal
C●ndle
in
the
socket
to
burne
according
to
a
time
limited
:
which
Candlestick
may
be
set
on
a
side
Table
without
suspition
to
any
;
then
when
the
Candle
is
burned
,
that
it
fires
the
primer
,
that
immediately
will
fire
all
the
Serpents
,
which
overthrowing
the
Candlestick
will
flye
here
and
there
,
intermixing
themselves
,
sometimes
in
the
Aire
,
sometimes
in
the
Planching
,
one
amongst
another
,
like
the
crawling
of
Serpents
,
continuing
for
a
pretty
while
in
this
posture
,
and
in
extinguishing
every
one
will
give
his
report
like
a
Pistoll
;
This
will
not
a
little
astonish
some
,
thinking
the
house
will
be
fired
,
though
the
whole
powder
together
makes
not
an
ounce
,
and
hath
no
strength
to
do
such
an
effect
.
How
to
make
fire
run
up
and
downe
,
forward
and
backward
TAke
small
Rockets
,
and
place
the
taile
of
one
to
the
head
of
the
other
upon
a
Cord
according
to
your
fancie
,
as
admit
the
Cord
to
be
ABCDEFG
.
give
fire
to
the
Rocket
at
A
,
which
will
flye
to
B
,
which
will
come
back
againe
to
A
,
and
fire
another
at
C
,
that
will
flie
at
D
,
which
will
fire
another
there
,
and
fl●e
to
E
,
and
that
to
F
,
and
so
from
F
,
to
G
,
and
at
G
,
may
be
placed
a
pot
of
fire
,
viz.
GH
.
which
fired
will
make
good
sport
▪
bec●u●e
the
Serpents
which
are
in
it
will
variously
●ntermix
themselves
in
the
Aire
,
and
upon
the
ground
,
and
every
one
will
extinguish
with
a
report
:
and
here
may
you
note
that
upon
the
Rockets
may
be
placed
fierie
Dragons
,
Combatants
,
or
such
like
to
meet
one
another
,
having
lights
placed
in
the
Concavity
of
their
bodies
which
will
give
great
grace
to
the
action
.
How
to
make
Wheels
of
fire
.
TAke
a
Hoop
,
and
place
two
Lath●
acrosse
one
the
other
;
upon
the
crossing
of
which
make
a
hole
,
so
that
it
may
be
placed
upon
a
pin
to
turne
easily
,
as
the
figure
Q.
sheweth
upon
the
sides
of
which
hoope
or
round
Circle
place
your
Rockets
,
to
which
you
may
place
Lances
of
fire
between
each
Rocket
:
let
this
wheele
be
placed
upon
a
standard
as
here
is
represented
,
and
place
a
piece
of
Primer
from
one
Lance
to
another
,
then
give
fire
at
G
,
which
will
fire
F
,
that
B
,
that
will
fire
D
,
that
C
,
and
that
will
fire
the
Rocket
at
A
▪
then
immediatly
the
wheel
will
begin
to
move
,
and
represent
unto
the
spectators
a
Circle
of
changeable
fire
,
and
if
pots
of
fire
be
tied
to
it
,
you
will
have
fine
sport
in
the
turning
of
the
wheele
and
casting
out
of
the
Serpents
.
Of
night-Combatants
.
CLubbes
,
Targets
,
Faulchons
,
and
Maces
charged
with
severall
fires
,
do
make
your
night-Combatants
,
or
are
used
to
make
place
amongst
a
throng
of
people
.
The
Clubbes
at
the
ends
are
made
like
a
round
Panier
with
small
sticks
,
filled
with
little
Rockets
in
a
spirall
forme
glu●d
and
so
placed
that
they
fire
but
one
after
another
;
the
Ma●es
are
of
divers
fashions
,
some
made
oblong
at
the
end
,
some
made
of
a
sp●rall
forme
,
but
all
made
hollow
to
put
in
several
composition
,
and
are
boared
in
divers
places
,
which
are
for
sundry
Rockets
,
and
Lances
of
weak
composition
to
be
fired
at
pleasure
:
The
Faulchons
are
made
of
wood
in
a
bowing
forme
like
the
figure
A
,
having
their
backes
large
to
receive
many
Rockets
,
the
head
of
one
neare
the
neck
of
another
,
glued
and
fastned
well
together
,
so
that
one
being
spent
another
may
be
fired
.
〈◊〉
Targets
are
made
of
wooden
thinne
boards
,
which
are
channeled
in
spiral
lines
to
containe
primer
to
fire
the
Rockets
one
after
another
,
which
is
all
covered
with
thinne
covering
of
wood
,
or
Pastboard
,
boared
with
holes
spirally
also
;
which
Rockets
must
be
glued
and
made
fast
to
the
place
of
the
Channels
:
Now
if
two
men
,
the
one
having
a
Target
in
his
hand
,
and
the
other
a
Falchon
,
or
Mace
of
fire
,
shall
begin
to
fight
,
it
will
appeare
very
pleasant
to
the
Spectators
:
for
by
the
motion
of
fighting
,
the
place
will
seem
to
be
ful
of
streames
of
fire
:
and
there
may
be
adjoyned
to
each
Target
a
Sunne
or
a
burning
Comet
with
Lances
of
fire
,
which
will
make
them
more
beautifull
and
resplendent
in
that
acti●n
.
Of
standing
Fires
.
SVch
as
are
used
for
recreation
,
are
Collossus
,
Statues
,
Arches
,
Pyramides
,
Chariots
,
Chaires
of
triumph
and
such
like
,
which
may
be
accommodated
with
Rockets
of
fire
,
and
beautified
with
sundry
other
artificiall
fires
,
as
pots
of
fire
for
the
Aire
which
may
cast
forth
several
figures
,
Scutchions
,
Rockets
of
divers
sorts
,
Starres
,
Crownes
,
Leaters
,
and
such
like
,
the
borders
of
which
may
be
armed
with
sundry
Lances
of
fire
,
of
small
flying
Rockets
with
reports
,
flames
,
of
small
birds
of
Cypres
,
Lan●hornes
of
fire
,
Candles
of
divers
uses
,
and
colours
in
burning
:
and
whatsoever
the
fancie
of
an
ingenious
head
may
allude
unto
.
Of
Pots
of
fire
for
the
Aire
,
which
are
throwne
out
of
one
Case
one
after
another
of
a
long
continuance
.
MAke
a
long
Trunk
as
AG
,
and
by
the
side
AH
,
let
there
be
a
Channel
which
may
be
fiered
with
slow
primer
or
composition
;
then
having
charged
the
Trunk
AG
,
with
the
Pots
of
fire
for
the
Aire
at
IGEC
,
and
make
the
Trunk
AG
,
very
fast
unto
a
Post
as
IK
,
give
fire
at
the
top
as
at
A
,
which
burning
downewards
will
give
fire
to
C
,
and
so
throw
out
that
Pot
in
the
Aire
,
which
being
spent
,
in
the
meane
time
the
fire
wil-burne
from
B
to
D
,
and
so
fire
E
,
and
throw
it
out
also
into
the
Ayre
,
and
so
all
the
rest
one
after
another
will
be
throwne
out
:
and
if
the
Pots
of
fire
for
the
Aire
which
are
cast
out
,
be
filled
with
diverse
Fire-workes
,
they
will
be
so
much
the
more
pleasant
to
the
beholders
.
These
Trunks
of
fire
doe
greatly
adorne
a
Fireworke
,
and
may
conveniently
be
placed
at
each
angle
of
the
whole
worke
.
Of
Pots
of
fire
for
the
ground
.
MAny
Pots
of
fire
being
fired
together
do
give
a
fine
representation
,
and
recreation
to
the
spectators
,
and
cause
a
wonderfull
shout
amongst
the
common
people
which
are
standers
by
;
for
those
Pots
being
filled
with
Balles
of
fire
and
flying
Serpents
for
the
Aire
,
they
will
so
intermix
one
within
another
,
in
flying
here
and
there
a
little
above
the
ground
,
and
giving
such
a
volley
of
reports
that
the
Aire
will
rebound
with
their
noise
,
and
the
whole
place
be
filled
with
sundry
streames
of
pleasant
fire
;
which
serpents
will
much
occupie
these
about
the
place
to
defend
themselves
in
their
upper
parts
,
when
they
will
no
lesse
be
busied
by
the
balls
of
fire
,
which
seeme
to
annoy
their
feet
.
Of
Balles
of
f●re
.
THese
are
very
various
according
to
a
mans
fancy
;
some
of
which
are
made
with
very
small
Rockets
,
the
head
of
one
tyed
to
the
neck
of
another
:
the
ball
being
made
may
be
covered
over
with
pitch
except
the
hole
to
give
fire
to
it
;
this
Ball
will
make
fine
sport
amongst
the
standers
by
,
which
will
take
all
a
fire
,
and
rolle
sometimes
this
way
,
sometimes
that
way
,
between
the
legs
of
those
that
are
standers
by
▪
if
they
take
not
heed
,
for
the
motion
will
be
very
irregular
,
and
in
the
motion
will
cast
forth
severall
fires
with
reports
.
In
the
second
kind
there
may
be
a
channell
of
Iron
placed
in
divers
places
in
spirall
manner
,
against
which
may
be
placed
as
many
small
petards
of
paper
as
possible
may
be
,
the
Channell
must
be
full
of
slow
comp●sition
,
and
may
be
covered
a●
the
former
,
and
made
fit
with
his
Rockets
in
the
middle
:
this
Ball
may
be
shot
out
of
a
morter
Peece
,
or
charged
on
the
top
of
a
Roc●et
:
for
in
its
motion
it
will
flye
here
and
there
,
and
give
many
reports
in
the
Aire
:
because
of
the
discharge
of
the
petards
.
Of
fire
upon
the
Water
.
Places
which
are
〈◊〉
upon
Rivers
or
great
Ponds
,
are
proper
to
make
Recreative
fres
on
:
and
if
it
be
required
to
make
some
of
consequence
,
such
may
conveniently
be
made
upon
two
Bo●ts
,
upon
which
may
be
built
two
Beasts
,
Turrets
,
Pagins
,
Castles
,
or
such
like
,
to
receive
or
hold
the
diversity
of
Fire
workes
that
may
be
made
within
it
,
in
which
may
play
〈◊〉
fires
,
Petards
,
&c.
and
cast
out
many
simple
Granadoes
,
Balls
of
fire
to
burne
in
the
water-Serpents
and
other
things
,
and
often
times
these
boates
in
their
incounters
may
hang
one
in
another
,
that
so
the
Combatants
with
the
Targets
,
and
Maces
may
fight
;
which
will
give
great
▪
content
to
the
eyes
of
those
which
are
lookers
on
,
and
in
the
conclusion
fire
one
another
,
(
for
which
end
they
were
made
:
)
by
which
the
dexterity
of
the
one
may
be
knowne
in
respect
of
the
other
,
and
the
triumph
and
victory
of
the
fight
gotten
.
Of
Balles
of
fire
which
move
upon
the
water
.
THese
may
be
made
in
forme
of
a
Ball
stuffed
with
other
little
Balls
,
glued
round
about
and
filled
with
composition
for
the
water
,
which
fiered
,
will
produce
marvellous
and
admirable
effects
,
for
which
there
must
be
had
little
Cannons
of
white
Iron
,
as
the
ends
of
small
funnels
;
these
Iron
Cannons
may
be
pierced
in
sundry
places
,
to
which
holes
,
may
be
set
small
Balles
ful
of
composition
for
the
water
which
small
Balls
must
be
peirced
deep
and
large
,
and
covered
with
Pitch
,
except
the
hole
:
in
which
hole
must
be
first
placed
a
little
quantitie
of
grain-Powder
;
and
the
rest
of
the
hole
filled
up
with
composition
;
and
note
further
that
these
Iron
Cannons
,
must
be
filled
with
a
slow
composition
;
but
such
which
is
proper
to
burne
in
the
water
:
then
must
these
Cannons
with
their
small
Balls
be
put
so
together
that
it
may
make
a
Globe
,
and
the
holes
in
the
Cannons
be
answerable
to
the
hollow
Balls
,
and
all
covered
over
with
Pitch
and
Tallow
;
afterwards
pierce
this
Ball
against
the
greatest
Cannon
(
to
which
all
the
lesser
should
answer
)
unto
the
composition
,
then
fire
it
,
and
when
it
begins
to
blow
,
throw
it
into
the
water
,
so
the
fire
comming
to
the
holes
will
fire
the
graine
Powder
,
the
which
will
cause
the
Balls
to
separate
and
fly
here
and
there
,
sometimes
two
at
a
time
,
sometimes
three
,
sometimes
more
,
which
will
burne
within
the
water
with
great
astonishment
and
content
to
those
which
see
it
.
Of
Lances
of
fire
.
STanding
Lances
of
fire
,
are
made
commonly
with
hollow
wood
,
to
containe
sundry
Petards
,
or
Rockets
,
as
the
figure
here
sheweth
,
by
which
is
easie
to
invent
others
occording
to
ones
fancy
.
These
Lances
have
wooden
handles
,
that
so
they
may
be
fastned
at
some
Post
,
so
that
they
be
not
overthrowne
in
the
flying
out
of
the
Rockets
or
Petards
:
there
are
lesser
sorts
of
Lances
whose
cases
are
of
three
or
foure
fouldings
of
Paper
of
a
foote
long
,
and
about
the
bignesse
of
ones
finger
,
which
are
filled
with
a
composition
for
Lances
.
But
if
these
Lances
be
filled
with
a
composition
,
then
(
unto
every
4
ouncs
of
powder
add●
2
ounces
of
Salt-Peter
,
and
unto
that
adde
1
ounce
of
Sulphur
)
it
will
make
a
brick
fire
red
before
it
be
halfe
spent
,
if
the
Lance
be
fiered
and
held
to
it
:
and
if
20
such
Lances
were
placed
about
a
great
Rocket
and
shot
to
a
house
or
ship
,
it
would
produce
a
mischievous
effect
.
How
to
shoot
a
Rocket
Horizontall
,
or
otherwise
.
VNto
the
end
of
the
Rocket
place
an
Arrow
which
may
not
be
too
heavy
,
but
in
stead
of
the
feathers
let
that
be
of
thinne
white
tinne
plate
,
and
place
it
upon
a
rest
,
as
here
you
may
see
by
the
Figure
,
then
give
fire
unto
it
,
and
you
may
see
how
serviceable
it
may
be
.
To
the
head
of
such
Rockets
,
may
be
placed
Petards
,
Balls
of
fire
,
Granadoes
,
&c.
and
so
may
be
applyed
to
warlike
affaires
.
How
a
Rocket
burning
in
the
water
for
a
certaine
time
,
at
last
shall
fly
up
in
the
Aire
with
an
exceeding
quickness
.
TO
do
this
,
take
two
Rockets
,
the
one
equall
to
the
other
,
and
joyne
them
one
unto
another
in
the
middle
at
C.
in
such
sort
that
the
fire
may
easily
passe
from
one
to
another
:
it
being
thus
done
,
tye
the
two
Rockets
at
a
stick
in
D
,
and
let
it
be
so
long
and
great
that
it
may
make
the
Rockets
in
the
water
hang
,
or
lye
upright
:
then
take
a
pack-thread
and
tye
it
at
G.
and
let
it
come
double
about
the
stick
DM
.
at
〈◊〉
and
at
that
point
hang
a
Bullet
of
some
weight
as
K.
for
then
giving
fire
at
A.
it
will
burne
to
B.
by
a
small
serpent
filled
there
and
tyed
at
the
end
,
and
covered
so
that
the
water
injure
it
hot
,
which
will
fire
the
Rocket
BD
,
and
so
mounting
quick
out
of
the
water
by
the
loose
tying
at
C.
and
the
Bullet
at
the
pack-thread
,
will
leave
the
other
Rocket
in
the
water
:
and
so
ascend
like
a
Rocket
in
the
Aire
,
to
the
admiration
of
such
as
know
not
the
secrecie
.
Of
the
framing
of
the
parts
of
a
Fire-Worke
,
together
,
that
the
severall
workes
may
fire
one
after
another
.
CAuse
a
frame
to
be
made
as
ABCD.
of
two
foot
square
every
way
,
or
thereabouts
(
according
to
the
quantity
of
your
severall
workes
)
then
may
you
at
each
angle
have
a
great
Lance
of
fire
to
stand
,
which
may
cast
out
Pots
of
fire
as
they
consume
:
upon
the
ledges
AB.BC.
and
CD
.
may
be
placed
small
Lances
of
fire
about
the
number
of
30
or
60
,
some
sidewise
,
and
others
upright
,
between
these
Lances
may
be
placed
Pots
of
fire
sloping
outwards
,
but
made
very
fast
,
and
covered
very
close
,
that
they
chance
not
to
fire
before
they
should
;
then
upon
the
ledges
RE.
FG.HI
.
and
AD
may
be
placed
your
soucisons
,
and
behinde
all
the
work
may
be
set
your
Boxes
of
Rockets
,
in
each
of
which
you
may
place
6
,
9
,
●2
.
or
20
small
Rockets
:
Now
give
fire
at
A.
(
by
help
of
a
piece
of
primer
going
from
one
Lance
to
another
)
all
the
Lances
will
instantly
at
once
be
lighted
,
and
as
soone
as
the
Lance
at
A
is
consumed
,
it
will
fire
the
Channell
which
is
made
in
the
ledge
of
the
frame
which
runnes
under
the
Pots
of
fire
,
and
as
the
fire
goes
along
burning
,
the
Pots
will
be
cast
forth
,
and
so
the
rank
of
Pots
upon
the
sides
of
the
frame
AB.BC.
and
CD
.
being
spent
,
the
soucisons
will
begin
to
play
being
fiered
also
by
a
Channel
which
runnes
under
them
,
upon
the
ledges
AD
,
HI●G
,
and
RE.
then
when
the
Soucisons
are
spent
upon
the
last
ledge
RE.
there
may
be
a
secret
Channel
in
the
ledge
CD
which
may
fire
the
Box
of
Rockets
at
K.
and
may
fire
all
the
rest
one
after
another
,
which
Boxes
may
be
all
charged
with
severall
Fire-Workes
:
for
the
Rockets
of
the
first
Box
may
be
loaden
with
Serpents
,
the
second
with
Stars
,
the
third
with
Reports
,
the
fourth
with
Golden
raine
,
and
the
fifth
with
small
flying
Serpents
;
these
mounting
one
after
another
and
flying
to
and
fro
will
much
inlighten
the
Aire
in
their
ascending
,
but
when
these
Rockets
discharge
themselves
above
,
then
will
there
be
a
most
pleasant
representation
,
for
these
fires
will
dilate
themselves
in
divers
beautifull
formes
,
some
like
the
branching
of
Trees
,
others
like
fountaines
of
water
gliding
in
the
Aire
,
others
like
flashes
of
lightning
,
others
like
the
glittering
of
starres
,
giving
great
contentment
,
and
delight
to
those
which
behold
them
;
But
if
the
worke
be
furnished
also
with
Balons
(
which
is
the
chiefest
in
recreative
Fire-works
)
then
shall
you
see
ascending
in
the
Aire
but
as
it
were
onely
a
quill
of
fire
,
but
once
the
Balon
taking
fire
,
the
Aire
will
seeme
more
than
100.
foot
square
full
of
crawling
,
and
flying
Serpents
,
which
will
extinguish
with
a
volley
of
more
than
500
reports
:
and
so
fill
the
Aire
and
Firmament
with
their
rebounding
clamour
.
The
making
of
which
with
many
other
rare
and
excellent
Fire-workes
,
and
other
practises
,
not
onely
for
recreation
,
but
also
for
service
:
you
may
finde
in
a
book
intituled
Artificiall
Fire-workes
,
made
by
Mr.
Malthas
(
a
master
of
his
knowledge
)
and
are
to
be
sold
by
William
Leake
,
at
the
Crowne
in
Fleet-street
,
between
the
two
Temple-Gates
.
Conclusion
.
In
this
Booke
we
have
nothing
omitted
what
was
materiall
in
the
originall
,
but
have
abundantly
augmented
it
in
sundry
experiments
:
And
though
the
examinations
are
not
so
full
,
and
manifold
;
yet
(
by
way
of
brevitie
)
we
have
expressed
fully
their
substance
,
to
avoid
prolixitie
,
and
so
past
by
things
reiterated
.
FINIS
.
Printed
or
sold
by
William
Leak
,
at
the
Crowne
in
Fleetstreet
neere
the
Temple
,
these
Books
following
.
-
YOrk's
Heraldry
,
Folio
-
A
Bible
of
a
very
fair
large
Roman
letter
,
4●
-
Orlando
F●rios●
Folio
.
-
Callu
learned
Readings
on
the
Scat.
21.
Hen.
80.
Cap
5
of
Sewer●
-
Perkins
on
the
Laws
of
England
.
-
Wi●kinsons
Office
of
She●●●fs
.
-
Vade
Mecum
,
of
a
Justice
of
Peace
.
-
The
book
of
Fees
.
-
Peasons
Law
.
-
Mirrour
of
Just●ce
.
-
Topicks
in
the
Laws
of
England
.
-
Sken
de
significatione
Verborum
.
-
Delaman's
use
of
the
Horizontal
Quadrant
.
-
Wilby's
2d
set
of
Musique
,
345
and
6
Parts
.
-
Corderius
in
English
.
-
D●ctor
Fulk's
Meteors
.
-
Malthus
Fire-workes
.
-
Nyes
Gunnery
&
Fire-workes
-
C●to
Ma●or
with
Annotations
,
by
Wil.
Austin
Esquire
.
-
Mel
Helliconium
,
by
Alex.
Ross●
-
Nosce
teipsum
,
by
Sr
John
Davis
-
Animadversions
on
Lil●i●s
Grammer
.
-
The
History
of
Vienna
,
&
Paris
-
Lazarillo
de
Tormes
.
-
Hero
and
L●ander
,
by
G.
Chapman
and
Christoph.
Marlow
.
-
Al●ilia
or
Philotas
loving
folly
.
-
Bishop
Andrews
Sermons
.
-
Adams
on
●eter
.
-
Posing
of
the
Accidence
.
-
Am●dis
de
Gaule
.
-
Guillieliam's
Heraldry
.
-
Herberts
Travels
.
-
Bacc●s
Tales
.
-
Man
become
guilty
,
by
John
Francis
Sen●●t
,
and
Englished
by
Henry
Earl
of
Monmouth
.
-
The
Ideot
in
4
books
;
the
first
and
second
of
Wisdom
;
the
third
of
the
Mind
,
the
fourth
of
S●●tick
Experiments
of
the
Ballance
.
-
The
life
and
Reign
of
Hen.
the
Eighth
,
written
by
the
L.
Herbet
-
Cornwallis
Essays
,
&
Paradoxes
.
-
Clenards
greek
G●ammar
80
-
A●laluci●
,
or
the
house
of
light
:
A
discourse
written
in
the
year
1651
,
by
SN
.
a
modern
Speculator
.
-
A
Tragedy
written
by
the
most
learned
Hugo
Grotius
called
,
Christus
Patience
,
and
translated
into
Engl.
by
George
Sand
▪
-
The
Mount
of
Olives
:
or
Sollitary
Devotions
,
by
Henry
Vaughan
Silurist
With
an
excellent
discourse
of
Man
in
glory
,
written
by
the
Reverend
Anselm
Arch
Bishop
of
Canterbury
.
-
The
Fort
Royall
of
Holy
Scriptures
by
I.
H.
PLAYES
.
-
Hen.
the
Fourth
.
Philaster
.
-
The
wedding
.
The
Hollander
.
-
Maids
Tragedie
.
King
&
no
K.
-
The
gratefull
Servant
.
-
The
strange
Discovery
.
-
Othello
;
the
Moor
of
Venice
.
-
The
Merchant
of
Venice
.