Bowen, Science and Philosophy in Classical Greece

http://slidepdf.com/reader/full/bowen-science-and-philosophy-in-classical-greece 1/359
X,
r.
U'
INSTITUTE
FOR
RESEARCH
IN
CLASSICAL
PHILOSOPHY
AND
SCIENCE
Science
and
Philosophy
Copyright
All
rights
family
designed
by
Donald
E.
Knuth
To
my
CONTENTS
PREFACE
ix
CONTRIBUTORS
xiii
1.
Some
Remarks
on
Ours
of
His
11
the Pure
and Applied
D. H.
viii
CONTENTS
11.
PREFACE
stories
to
establish
results.
during the fifth
X
PREFACE
two
intellectual
activities
and
setting
in the subsequent
history of philosophy and science. But this one has a special charm, since
it
is
the
earliest
on
record,
and
since
the
figures
involved
on
on
the
documents
deed
to
the
will
still
serve
to
draw
the
reader
into
the
conversation
topics
originally
proposed.
Thus,
for
instance,
there
Preface
XI
CONTRIBUTORS
ANDREW
D.
BARKER
Senior
especially
Medieval Islam
and Islam
Euclid’s
burgh. Fellow
of the
XIV
CONTRIBUTORS
D.
H.
FOWLER
concerning
Greek
math-
ematics,
and
on
the
Contributors
XV
Philosophy
Graduate
Memorial
Founda-
tion
Fellowship
(1988-89).
France
(1982).
Author
of
Philosophy
on
Greek
mathematics,
Hepl
twv
paSrifidTCOv
(1991).
Currently
working
on
in
scholarly
journals
XVI
CONTRIBUTORS
and
books;
editor
(with
P.
K.
Machamer)
of
Motion
and
Time,
Space
and
Matter
(1976)
and
Studies
in
Perception
(1978).
1
Some
Remarks
fifth centuries BC?
2
CHARLES
is obvious;
have
all
ing
swiftly
throughout
those
bustling
Greek
cities
scattered
across
half
the
Mediterranean.
We
Presocratic
natural
philosophy
astronomy:
been confirmed). For example,
sphere
was
introduced
is
not
clear
from
our
shabby
evidence,
but
not
later
explanation
the
sort
that
one
might
find
serves
only
texts from
their
predecessors
—unless
one
finds
mathemat-
ics
begin?
Neugebauer
tends
to
date
fourth-century
work
Hippocrates of Chios
The
Origins
of
Greek
Science
and
Philosophy
5
Kranz
history,
coinciding
studied
certain
problems
to
speculative
cosmology
8/18/2019 Bowen, Science and Philosophy in Classical Greece
6
CHARLES
H.
KAHN
which eliminate
defend
or
destroy
not-p
Therefore
p
must
other
sheer
accident
of
our
documentation
that
we
to
philosophy
to
borrow
from
mathematics,
just
beginning
the
The
Origins
of
Greek
Democritus
(and
of
a
sophist
like
Hippias).
the
very
which
wf
can
the
systematic
specialization
and
separation
as
is
8
ideas
which
I
have
see
a
in
his
stage
in
the
development
of
science.
Its
role
seems
to
me
only
comparable
to
on
More
their
language
and
work. *
Thus
naive view of the
anthropomorphic
The
Origins
of
Greek
‘For Hesiod’s source,
differentiated
natural
philosopher.
To
re-
ductive
black
hole,
into
which
things
of
an
10
between
sci-
ence and philosophy, I can be brief, because in the period
before
Socrates
there
is
no
such
demarcation.
The
investigation
of
nature
(irepl
that
lies
2
conception of
to
these
two
questions:
What
the
general
topic
of
scientific
inquiry?
fully
satisfying
rational
insight.
In
other
words,
an
found congenial,
from Plato’s
the
more
appreciative
readings
too.
The
Timaeus
encyclopedia
of
the
to constitute
Plato’s
ometry,
theories
motion,
something
struck by
what appears
perfect
[cf.
Vlastos
1975,
54-57].
At
51b-61c
far looser
account
agreement
with
the
view
that
nonetheless
under
multiple
posits
abstract-noun
the state-
(as
in
‘He
shows
indefinite
article
in
Greek,
it
But
as
the
limited
use
his use of comparatives
(rauTaL?
dXXai
TOiauraL
to refer to
mathematical
sciences
and
for
the
the theoretical
distinction
these
[scil.
the
dialectic [57e-59d].
that
the
governed
by
voO?
runs throughout
thus under the
Philebus, that
the
guardians:
first
the
five
stereometry,
astronomy,
that
Glau-
con
appreciates
proceeds
to
draw
spond to aural concords, and the
purely
mathematical
science
of purely mathematical unified theory of selected ratios. The phrase
‘consonant
numbers’
Oecapia
gives
the
broadest
the
concern;
for
the
virtue
that
moves
along
a
22
viii
846d-e,
practic-
ing
toward which
all arts
and sciences
p.LaoXoyLa
another,
which
other
sciences.
But
our
intellectual
horizon
progressively
expands
as
to
stereometry,
which
incorporates
is
expanded
further
when,
with
real
the
five mathematical
sciences of
crowning achievement
a special appeal
seven
the specifications Plato
and sci-
processes;
common
sense;
and
in
Laws
X
891b-892c
it
is
—
1985
science that
from
Plato
as
counterpart:
for
[see
Van
Helden
1985,
15-27,
slightest concern
jeopardy his
means
Great Year; it
that represent
correction
the
end
of
the
Great
Year
when
arises not from
this playful-
ness becomes
—
we grasp
account
promotes
30
ALEXANDER
P.
D.
MOURELATOS
story
elicits
into
certain
truths
concerning
3
The
Aristotelian
Conception
as to illustrate other
science
as
a
32
JOSEPH
OWENS
CSsR
science
with
acquaintance
with
the
rationale
of
the
about the supposed correspondence. Indeed,
Aristotle’s
our
prima
of
the
problems
involved.
attaining correct
Aristotelian caption
philosophy
of
nature,
dis-
ciplines
mechanics are
mathematics’
[Phys.
194a7-8:
Hardie
and
Gaye
1930,
ad
different
type
from
Nic.
1140b31-35j.
The
and
Meillet
1951,
145al5-18]
something
relative
human person.
will
remain
the
same
body
of
knowledge
given
essentially
‘cause’
is
ask of
34
JOSEPH
OWENS
CSsR
traditionally,
to
in a
this
empathetic
knowledge
originates
in
particu-
however,
ideas
have
been
regarded
6
[s.v.
only
Aristotle
on
the
distinction
of
wrote:
Why
what philosophical
views
pure
36
explained things
observed
particularities
and
keeping
its
hands
unsullied
through
Yet some
Next,
philosophy
of
nature
for
Aristotle
their
metaphysics it
is a
Irrational
numbers
for
developments that
took place
38
were
astron-
As
noted
above,
the
first
far-flung
form
all
the
one
in
what
is
already
there
before
,
....
correct
moral
habituation.
13
This
undertaking the study
whole purpose
or-
has
13
with
the
way
one
relates
scientific,
from
does
not
have.
The
evil
perhaps, be
40
JOSEPH
OWENS
CSsR
the
modern
conceptions
since
being
an
itself
consist
genuine
There
division
of
in
the
science
it
cultural
heritage’.
17
is
the
not
giving,
knowledge
of
what
things
productive
are
three
different
applied
framework
as
far
the
for
maintaining
the
human
the
arts
4
Platonic
and
Aristotelian
Science
ROBERT
G.
TURNBULL
My
intention
those
(in
their
own
ways)
and
derive
claims
44
ROBERT
G.
TURNBULL
with
sense-data.
I
From
the
earliest
introduction
there is
fortunate to
this person may
bring
these
2
Farm.
134a.
I
shall
be
making
a
parallel
point
about
‘of’
soul,
by
46
ROBERT
G.
TURNBULL
places,
of
so on. This processing is, of course, pre-linguistic or non-
linguistic,
prima
facie
appro-
priate
Given
all
of
the
discoveries
of
Platonic
and
sense
the
the
army
the application of that image, the initial condition of one
who
has
no
universals
but
is
exercising
perception
(in
the
narrow
sense
of
like
the first
having
a
army
by
Aristotle
a knowing
48
man,
not
of
a
accords
with
the
medieval
understanding
awareness
since
it
invites
the
means under
higher con-
we could
Forms.
Physics
i
1
is
a
major
key
to
understanding
6
For
50
ROBERT
G.
TURNBULL
F.
From
see,
e.g., the necessity of a (or this) triangle’s having three sides.
And
one
may
also
be
Aristotle’s terms, is
does not
a
uni-
versal
propositions.
‘This
X
is
F’
as ‘All
Xs are
of
such
modal
terms
simply
either.
a
would
presumably
not
have
such
a
possibility).
It
goes
without
saying
that
Barbara
is
and
Aristotle
doctrines
things
a,s
what
they
are
52
procedures for
have).
Both
empirical;
reference to
and attempt to
is a
capable
of
recognizing
as
being
characterized
in
a
variety
anything
which
is
X
stay
with
A.
Then
one
tries
must
is X
is an
‘empirical’ test
Platonic
and
Aristotelian
fall under
Platonic
procedure,
to
involve
empirical
inves-
tigation
be
dearness
to
But,
A,
it
should
be
straight
54
ROBERT
plane figure and not colored plane figure or color. And,
to return to
invite us to
some sort of non-visual staring at forms. He insists that our
awareness of
the
structure
of
name-forms
and
thus,
in
refiective
use,
to be as
experience
and
memory.
and
so
on
to
proceed
in
arriving
at
definitory
middles.
The
for
size,
which like
56
ROBERT
involved, make it clear enough that both Plato and Aristotle
have
intelligent
the
genera/species
trees.
in
in
syllogisms
consisting
of
a
universally
names,
and
such
individuals
figure
no
is
in
the
procedures
of
Leibniz is
the effort
Platonic
and
Aristotelian
the
individual
first
chapter
of
Physics
(to
which
and
successive
58
ROBERT
G.
TURNBULL
5
On
the
Notion
of
a
the question
have
assertions,
60
then
by specifying
rule
logical
will
be
postulates
are
material
rules
of
On
primitive material terms of
term t
and such
that P is provable if and only if P' is. I
make
an
analogous
assumption
for
primitive
have
assertion
‘If
factors
defined terms
62
develops
theories
new theories
this
point
from
On
ginning
maturity.
I
I
will
only
Republic,
starting
points
of
which
‘A
unit
Occasionally
an
assertion
creeps
in order to make sure others understand what one is
talking
certain
what
straight line,
equal to
each other,
highly constructional
On
does
not
recognize
primitive
constructions
as
starting
points
postulates 4
from ‘a
rules
the
rules.
Should
we
self-consciousness
about
1981,
29-30.
66
us
to
decide
between
these
alternatives,
adds
which facilitates
On
Mathematical
Starting
Points
67
When
post, i
which it
8
My
translations
are
not
investigate
the
attributes
of
that
genus.
Here
he
wants
preceding
diropCa);
but
it
is
surprising
that
68
of
will have to be properties,
some
axioms
things,
about
some
premisses of
the
to this
On
Mathematical
Starting
Points
69
are
many
problems
quantities,
but
mathe-
matics
the axiom
giving
and are used by
possible
70
that Aristotle
acknowledges some
to
do
so.l^
me
quite
independent
term. (I
this cannot be
On
for
point,
that
Aristotle
here
speaks
believe
they
should
be
of
proving
demonstrative
sciences,
common
ones,
such
the
examples
72
such.
But
what
each
exist is
mentions cases
in which
explicitly
one
involved
here
16
Such
construction;
of existence
Else-
where
B;
for
example,
line
belongs
per
On
is
something
we
all
seems
to
subject
there
should
likely
that
In An.
still
MUELLER
triangle
signifies, and for the monad, both what it signifies and that
it
is not
learn
something
a
Oeatg.
one half of
On
Mathematical
Starting
Points
75
at
the
beginning
of
definitions
and
hypothe-
ses
an
apparent
equivocation.
quantitative and logical
assertions, but Aristotle
probably does not
talking
about
something
real
in
a
science,
determinations
subject
genus
and
the
things
whose
definitions
76
Aristotle. Aristotle
sometimes mentions
the common
to assume
him
included
‘equals
from
equals’ and presumably at least the first three of Euclid’s common notions.
On
common
There are at
number and
axioms
as
not
a
reflection
of
the
mathematicians’
understanding
of
their
former
is
only
&:
->->
On
Mathematical
Starting
Points
77
first
is
geometric
way would
as
the
1921,
i
336]
,2^
three postulates
correspond to
correspondences
are
the
postulates
78
others
cohere
with
for
thinking
it
did.
Prima
facie
it
would
seem
highly
likely
An. post.
come
they
do
have
this
role;
and
assumptions, he is
On
Mathematical
His
second
objec-
tion
goes
as
follows:
How
know
what
a
unicorn
is.
[An.
post.
92b4-8]
need
clear
words
and,
hence,
bears
no
existential
80
IAN
MUELLER
which clearly
totle
does
that they
the
the
common
notions.
the
case
least,
he
uses
argument
On
his
as
an
ultimate
premiss
and
all
starting
points
believe,
were evident
82
that
does
an
even
number
is
one
which
is
divisible
into
two
equal
parts
and
expects
everyone
somehow
do
away
with
seems to me
On
involved
with
some
value,
then
definitions. His view was
that
there
are
certain
passages
in
Aristotle
argu-
closer
precisely
84
on
more
straightforward
we take
will distinguish
techniques
knew
However,
the
starting
points
for
certain
sciences,
since
he
says
[An.
On
things
which
have
a
middle
the
justification
for
this
supposition
is
of knowledge
them by
two
to
represent
adequately
be
giving
an
account
of
how
primary
propositions
become
known:
see,
e.g.,
Ross
1953,
as
conclusion.
Aristotle
seems
can
starting
point
the
the same,
existence
at
It
is
in
sensibles,
as
some
On
Mathematical
Starting
Points
87
if
one
arithmetician’s
those
mathematical
objects
in
we
can
believe
was
of much discussion
only with
the logical
in
trying
to
prove
the
law
than
far
as
he
thought
to
be
taking
as
88
herself
1-24]
Now
be
proposition
(possibly
the
denial
of
one
drawn,
but
could
prove
of
On
Mathematical
Starting
Points
89
sciences,
correct
derivation
to
say
that
Aristotle
uses
a
rule
corresponding
to
the
law
in
order
to
prove
its
formulation
himself
model
90
of
the
associated with,
say, Euclid,
these
and of
premisses.
This
conception
of
the
points
On
Mathematical
Starting
to
reach
the
major
texts
predates
the
influence on
about
the
fourth-century
structional
postulates
were
already
92
IAN
MUELLER
tial attributes of
them
an
already
exist-
ing
by
Euclid,
who
for-
mulates
theorems
as
assertions,
problems
using
the
infinitive
(‘to
construct
a
square
On
tells us,
to
in
this
appendix
differs
me to support the
94
IAN
MUELLER
In
general,
says
Speusippus,
and
our
mind
has
a
clearer
contact
with
them
than
sight
has
with
visible
objects;
immediately
and
therefore
advances
on
theorems.
He
then
says.
However,
Speusippus’
distinction
between
principles
On
Mathematical
Starting
Points
95
the
examples
that
we
cannot
out;
for
we
have
no
explicit
recognition
though
I
think
it
must
have
of the
word ‘the-
orem’ and
seems to
me to relate in any specific way to the content of fourth-century
mathematics.
Appendix
450
that
Oenopides’
innovation
was
96
problems
and
proofs
of
theorems
which
he
thought
were
presupposed
by
he
cession of
Peripatetic,'^^
there
would
be
no
more
basis
for
inferring
Oenopides’
concerns
from
Zenodotus’
than
for
inferring
Socrates’
what
is
the
case
given
that
not the
case; we
of a
On
from
Zenodotus’
(60
€v).
fifth-
or
in more
would
like
wears
the
one begins with
cannot
be
shown
uncertain
whether
is impossible
divides similarly
to
a
pair
of
the
ratio
100
D.
H.
FOWLER
hinges
on
with
the
following
precisely
differentiated
meanings.
the
ratio
of
a
4
The
Ratio
and
Proportion
101
Proportion.
(abbreviated
a:b
6
go
the
many
times
for
centuries
BC?
104
D.
H.
FOWLER
This
question is.
numbers,
for
Interlude:
be
classification of certain
the
idea
of
ratio,
this
material
is
clearly
(priTog)
or
without
ratio
(dXoyog).
authority, to
study
for
magnitudes
and
dpL0poC
respectively,
in
the
Elem.
V
def.
3
passage
in
relation
and
15,
the
‘anthyphairetic
more.
If
per-
formed
on
two
dpiGp-oC,
the
relation
108
very long
Mjo
Mj,
D,
MD,...
or
as
... lor
2,
1,
2,
1,
1
or
2,
called the
astronomical ratio.
Moreover, we
b
b
b b b b b b b b b b b
1
1 1 1
c
a
a
no
D.
H.
FOWLER
theoretic
phase,
is
then
very
If
we
the
pattern
of
many
relation
perform some
the
two or
what
I
Ratio
and
Proportion
111
The
process
will
all
Conventionally
arithmetic
like
X
-^y
112
D.
H.
FOWLER
This
but this
by a change
found
are
Greek
mathematics
are
found
subtraction
process:
next
step.
The
special
interest
of
this
process
is
that
we
again
have
an
arithmetised
generate the
characteristic
pattern
114
them: note,
for example,
way
the
dpL0
P-OL.
(Southern
Italy
and
Sicily),
mainland
clear mathematical
Ratio
abstraction
as
the
notation
for
describe
division,
always
seems
to
116
D.
H.
FOWLER
expressions that
are found
be
Moon
has
survived
explanation
is
needed
for
us,
today,
up
within
the
now
Ratio
and
Proportion
117
fractions,
for
once
diagonal
and
ao'Mi
immaterial,
the
ratio
s:(d
therefore, twice, twice, followed
Now
consider
the
ratio
three-times
the
Greek mathematics.
also be found
7
What
Euclid
the relevance
to
120
WILBUR
R.
KNORR
in
effort
to
and
what
all,
did
the
multiply examples of how one and the same case had
received
diverse,
incompatible
accounts
of
irony
has
of
the
concept
What
Euclid
Meant
121
our
historically
arguments
ough
venture
synopsis
of
naive, as
among
such efforts
historian.
122
WILBUR
R.
KNORR
some
as
such
(indeed,
the
sceptical
view):
the
connections
between
the
text
and
whatever
of
This
entails
two
projects:
to
understand
validity
of
interpretation.
Validity
highly
probable,
or
a
position
on
the
issue
of
geometrical
meaning. Ultimately,
its mean-
whole
against
which
the
heuristic
element
we
One
author’s
124
would
inquiries
in
which
critics
activity poses,
between
the
objective
content
of
the
texts
and
the
subjective
elements
present
in
the
experience
proportion
What
Euclid
Meant
125
‘Ratio’
is
offers
this
commentary:
the
definition
Heath
<
respect of
126
WILBUR
R.
KNORR
mA
exclude
of magnitudes.
18
A
statement
of
their
critical
assumption
essential
to
finite
What
Euclid
Meant
127
‘Being
may
text
24
The
condition,
which the
v prop.
128
WILBUR
R.
KNORR
precondition
for
def.
5
one
likes
how
serious
any
rate,
that
def.
4,
as
proposed
in
the inequality just
‘same
ratio’
What
as my
10,
proofs
of
v
prop.
16
and
the
following.
the inequalities
do
not
exploit
the
definition
of
‘greater
ratio’,
for
that
would
have
been
natural
and
convenient.
130
WILBUR
R.
KNORR
his
def.
in v prop.
indeed gratuitously
proportion.
these applications
as
be
have
proposed
for
conflate
the
meaning
in
the
particular
in
132
WILBUR
R.
KNORR
Spending
to
our
common
fractions
”^/n
in
evitable
feature
tempts
to
formulate
approach
the
question
open
to
the
possibility
that
the
ancient
and
modern
views
could
tradition
never
evolved
the
conception
of
fraction
older
works,
as
we
choose.
1955,
293]
fraction.
31
A
similar
notion of
divisions
of
se-
ries
the arithmetic
Typical
of
one
implausible
to
3).
But
French translation and commentary
Fowler
on
pages
115-116
134
WILBUR
contain
problems
Find
11
Egyptian
works
within
the
2/3
21
is
^7
and
3
15
is
2
(10th century
at the
71’
[Heiberg,
of the
What
fraction
artefacts
that
(1st cent.
3
the
36
and
that
extant
than
scholars
have
heretofore
136
WILBUR
R.
KNORR
division
of
100
by
in Metrica
invoke
unit-fractions
but,
as
with
the
papyri,
in
a
merely
notational,
not
to note
divided
by
similar
terms.
and
1982b.
worked out
operations beyond the
for
granted,
while
only
the
of
our
‘denominator’,
while
138
WILBUR
R.
KNORR
his
mathematical
work:
one
may
word problems, among
by
scholia
which
work
out
of
problem
a
whole
writer
with
a
able to
least common
multiple of
What
tech-
the
the
exam-
plication
by
some
long
interval.
The
strongly
marked
the
Greeks.48
Despite
would
plausibly
examined in
140
WILBUR
R.
KNORR
Sometime
in
and
flexible
notions of fractions in
is, they
different from
the tradi-
ancient Babylonian
computational system.
on
sustain
49
species,
What
development
of
these
to
be?
When
we
ultimately
possess
an
adequate
reading
of
the
text,
our
142
WILBUR
R.
KNORR
Proclus
five
of his
books
1-4
and
6,
the
solid
geometry
of
book
11,
and
the
Even so, much of
aims, Proclus
like Hilbert’s
solids.
What
Euclid
Meant
143
of
the
major
forms
by
consid-
ering
ticular
interest,
since
whose
demonstration
sense,
de-
pend.
Proclus
observes
of
elements:
two
propositions
144
WILBUR
R.
KNORR
fluidity
in
the
order
of
proof,
at
the
outset
and
their
consequences
worked
essential
properties;
whatever
proofs
by
What
the
character
of
technical
the
in the
(theo-
rems)
use
them
as
thoroughly
familiar
and
treatise. Pro-
1983.
66
Burkert
[1959,
193]
takes
perforce
the
designation
‘elements’
as
the
actual
title
of
the
pre-Euclidean
works.
Mendell
[1986,
493]
Proclus’
source,
Eudemus,
could
designate
them
as
‘Elements’,
whence
character, which
he terms
treatise
[aroLxeLcacTL?]
passages
are
interpolations.
What
Euclid
Meant
147
they
can
be
resolved.
Of
those
who
expositions
provide
compare such
versions with
Euclid’s
forms
71
resolved’
passages on
148
WILBUR
its
postulates,
of
type
of
scientific
treatise, in which the content of the science is presented in
a
formal,
systematic,
the demonstrations
will
invariably
proceed
in
the
reverse
order,
that
of
analysis.
In
its
usual
sense
in
ancient
to ones
the
[1895,
vii
144-154]
to
be
the
choice
as
to
which
results
will
be
individual
for
the
substance
of
Eudoxus,
and
number
theory
in
book
150
WILBUR
transcribe
his
turn,
he
would
determine
what
they
required;
and
so
how
the
sequence
of
problems
of
construction
Knorr
1983].
propositions was,
than
this.
But
the
organization
into
was
of proportion
ters
realized
definition
of
ratio.
But
this
state
to
motives for
as
is
maintained
below.
One
could
the
Euclid
could
152
WILBUR
R.
KNORR
resolved
part
for
and
the
objective
of
alternative con-
What
the
first
book
On
Plane
Equilibria,
however,
tradition and
and others.
ward
training
in
Platonic
geometry, one
tation
1903-1914,
154
WILBUR
R.
KNORR
approach
1984,
6].
perfect numbers,
is
introduced
or
a
particular
Ptolemy
of
incomplete
inductions.
98
What
Euclid
Meant
155
insights
that
rules of the
view
techniques
to
the
Greeks
in
the
a
ha.s
fashioned
his
introductory
textbook
156
WILBUR
R.
KNORR
and
explanatory
insights,
as
he
saw
Pappus, Theon,
university-level
mathematics.
Hipparchus,
Dositheus and
contrast,
Archimedes’
concern
as
we
the absolute
no
indication
101
unresolved.
In
view
of
this,
demonstra-
tive
type
of
geometry.
Demonstrations
in
158
WILBUR
R.
KNORR
in
I

are
es-
cited from
modern
notions
perhaps
homogeneous
es-
method.
procedure
in
contra,st
an-
cients
conceive
fractions
is,
did
they
fractional
number.
For
160
WILBUR
R.
KNORR
the
math-
to
the
students’
elementary
j
training.
Moreover,
when
notations
for
undefined
converse view
one
prime,
in def. 3. The
problems set out in
take this
silence seriously.
would
appear,
but
the persistence
of the
our
significant.
One
might
perceive
a
parallel
in
the
persistence
of
‘English’
standards
in
the
United
States
of Euclid’s
162
WILBUR
R.
KNORR
not
about
deductive
structure.
say,
Mesopotamian
traditions,
109
it
same category as
emerged
mensuration
and
turn
about
a
common
with
modern
fields
of
mathematics
clearer
and
8
[cf.
Jan
1895,
115-116].
Beyond
this
presumed
inappropriate
to
suggest
intelligible
whole
[section
2].
3
I
wrong.
it would
1916,
xxxvii-
xxxviii],
and
negative
166
ALAN
C.
BOWEN
For,
according
the
Sectio
canonis.
The
Greek
text
together
but
others
are
less
close
together;
and
the
ones
that
are
closer
together
produce
notes
higher
(in
pitch);
but
those
together and so
parts
are
notes
must
also
but the
168
ALAN
C.
BOWEN
(d)
two
introduction
to
the
The motion
after
being
plucked;
or
lower
(in
pitch)—
and
so
(kul)
more
nu-
merous
and
that
the
latter
notes
be
lower
first
string of a lyre, or (b) the motions of the
sonant
body,
by
sound in general, and view
the
first
170
ALAN
C.
BOWEN
musical
suppose
accordingly
the
i
my
Archytas
interesting
phenomenal musical
distinction of concords and
consecutive,
discrete
motions;
and
that
fact just
force
of
of a
sonant body
passage that
pitch be
construed as
or that they possessed the means of measuring time so
as
to
define
one.
3
and
4].
series themselves
suppose.
If
the
compactness
or
close-packedness
of
each
parts, since they reach what
is
needed
(toO
deovro?)
by
addition
as
before,
just
as
relative
objective
sound. What is added is the claim that each musical note so
understood has parts, since it
is
constituted
What
lyre.
at
on
forth,
it
would
of
parts
are
described
in
(whole)
number
(dpiSpog),
others
in
super-
in
great
part
from
174
ALAN
C.
BOWEN
which
require
Neugebauer
1941,
25-
26];
and
varying sophistication
in
the
no
reason
to
deny
Sectio
to
be
under-
stood
relatively,
that
musical
pitches
are
conceived
only
in
relation
to
one
another.
the
Sectio
one
is
superpartient
ra-
tios,
multiple
the
three
when Greek
not
arbitrary,
though
one
may
assign
noises
to
ratios
of
incommensurable
speeds
(presumably)
by
quantifying
speeds
multiple
pitches,
as
a
sequence
of
Sectio
SidoTTipLa
signifies
a
but differ as
musical
intervals
as
differences
continuous
ranges
and
a
less.
there
name
any
Pythagorean
a
appoviKo?
or
to
allow
176
ALAN
C.
BOWEN
eliminative
lows
that
and
subtracting
the
aware,
no
good
ev-
idence
Aristotle’s
greater power
in
fact
no
fact
superparticular
ratios
complex
predicate:
e.g.,
p
is
a-multiple-of-^.
In
this
account,
the
relation
of
p
and
q
is
178
ALAN
C.
BOWEN
There
of the
a matter
of gram-
the first
objective
phenomenally, this
ratios. So
those
perceived
notes
(qua
series
of
consecutive
motions)
mentioned
in
sentence
[8]
to
180
ALAN
C.
BOWEN
argument as
should
allow
the
phrase,
Accordingly, let us
one
is
a
unlimited
interval
of
an
octave
and
p
2
- Ii^
other
words,
he
takes
it
even
if
the
interval
and his
(Nor,
given
that
[9]
the
claim
about
the
is about
superparticular notes be-
argument still needs an
182
ALAN
C.
BOWEN
(^4)
like
my
eliminative ontological
to the
concords
belong
to
concordant
numbers.
This
the
subsequent
theorems.
(ad
480-524)
the
Sectio
canonis
may
be
ill-founded,
so
may
this
consensus
about
its
philosophical
character.
There
are,
Archy-
tas
regarded as
a Pythagorean
Indeed, the story is
in
fied
with
ii
98,
107n40]
pitch on
to the
Pythagoreans. In
Yet
which
is
intel-
ligible
first
person
plural)
compare
with
what
of
which
agree
with
no
such
reduction
of
sound
as
heard
to
186
ALAN
C.
BOWEN
But
surely,
if it
to
be
adequate
does
pursue Barbera’s contention
on what
basis and
how are
reductive, eliminative analysis at
its core to the
at
on
that
one
would
expect
given
is
spelled
out
9
Aristoxenus’
Harmonics
and
Aristoxenus was
drew
heavily
reiterated
length,
the
methods
by
which
the
relationship,
one
in
to
our
understanding
of
Aristotle
own
treatises
offers
itself,
prima
facie,
as
an
example
of
the
sort
should
way of
be
Aristoxenus
his
own
making
but
are
in
fact
inherited
from
Aristotle.
2
territory
of
1951,
i
428.15-430.12,
435.15-436.13;
Bowen
1982],
other
the
correspondence
appropriate
Aristoxenus
unperceived
like,
which
4>naL9,
192
ANDREW
what
but
are
aspects of a
is
an
autonomous
form
inhering
in
certain
sequences
of
sounds
under
their
aspect
as
array of prin-
organisation in
will be
explored in
more detail
Aristoxenus’ harmonic
a
science
and
their
Aristoxenus
and
the overall match
shall
not
as
parts,
nevertheless
in
many
but there
ensure
that
they
cannot
originally
turity,
194
ANDREW
consists on
of
in
one
sense
or
another
inductive,
and
whose
stages
are
or
we
then
summarily
explanation
or
demon-
stration
in
terms
and
such.
Aristoxenus
and
constitutes a
of
under
the
explanations
of
why
post,
dpxaC
and
196
ANDREW
D.
BARKER
asserted
their
propositions
dvev
alTLa?
Kal
to
ydp
the
book
2
involves
types
broader
kind
rather
than
more
detailed
definitions
of
the
kind
itself),
the
terms
in
which
they
difficult to
dpxaC
include
Aristoxenus
and
not
fulfilled,
to
point
2,
but
concordant
intervals,
the
and
198
ANDREW
D.
BARKER
ticular examples of
melody drawn from
form
by
implicit
as
an
method by
way, the proposition
Aristoxenus
about
the
way
in
which
an
Aristotelian
a
tendentious
a
priori
dogma.
There
not.
2C3
ANDREW
D.
BARKER
written:
in so
analysing the
structure of
a system
mind,
not
Aristoxenus’
they
did
Aristoxenus
and
its
being
heard
as
melodic
or
as
being
of
some
principles
proper
to
subjects
in
another.
In
of relative
SeLKVuTar
Kal
202
reference to
ics.
The
reasoning
properties
in
things
4>i3aL9
them
and
why
I
instances
whose
perceptible
character
derives
conclusion
Aristoxenus
and
Aristotle
mentions
is
a
mathematical
one.
an example
from De
when
in terms of
numbers,
cm|i<j)a)-
204
ANDREW
D.
BARKER
any
passage).
19
fact
deter-
ratio
4:3,
and
so
on,
as
Science 205
two
a
concordant
limits
way correspond
mathematically significant:
musical experi-
ence, according
a
sequence
of
two
206
ANDREW
D.
BARKER
which
was wrong), since
between melodic systems,
no
doubt
be
of ratios. But if the rules of melodic progression turn
out
to
be
purely
accidental
perceptually
distinct
melodic
forms
we
mathematically
cirbi-
trary
grounds
of
their
coherence
and
order:
this
orderliness
will,
if
anything,
have
been
obscured
under
the
draw
attention.
Secondly,
however,
it
Aristoxenus
and
the
better
of
the
21
21
208
his
subject
those of
large a
perceived,
not even
pear).
Its
peaq
no
other
note
can
all
notes
13]:
but
they
do
not
depend
on
the
Archytan
theory
of
The
Aristoxenus
and
character
that
constitutes
its
8i
3
vapL
9
as
pair
that
pitches
210
ANDREW
D.
BARKER
insistence on the
of the
is Aristoxenus
melodic
8wapi9—
which
consists,
precisely,
by
such
argument
cannot
be
pursued
it may
relations of
it would be
why
constructed and
the intellect,
as accurate
plainly
committed
in
most
e
8
LavoCq
OecopoOpev
ras
between
two
placings
to
hear
at
which,
that
can
be
achieved
by
didvoLa:
to
say
that
quantitative
discriminations,
and
scientific
articulation
the
original
perception
of
a
magnitudes
as
such
is
no
part
(ouSev
eoTi
pepo?)
is
genera, nor,
Aristoxenus
and
notes as
the
of
behaviour
by
which
these
relations
are
ples
the
enharmonic
hvkvov:
and
tive
instance, 49.10-21].
relation
between
a
constituted
have
j
tried
to
to
persuade
us
that
that
can
be
intelligibly
accommodated
into
55.8-58.5: cf.
Aristoxenus
further, similar steps,
we can assign
an interval
that ‘calculative’
perception will
assess as
that
the
size
of
interval
relation,
but
a
given
concordant
relation
can
be
instantiated
in
an
interval
us
suppose
that it is true and that we can check its truth against
our
experience.
The
true,
it
seems
to
216
and
con-
functional
to
Aristoxenus
and
and
enharmonic
ones
end.
No
sense
can
be
made
of
this
task unless it is that of picking out the magnitude of
the sma,ll-
218
ANDREW
D.
BARKER
vary
from
at
once.
perceptions
of
intervallic
magnitudes
are
not
as
such
intrinsic
to
our
ex-
perience
from
remembered
perceptions
of
strings
of
peycOq.
Quantitative
hearing
as
yet
there
be so.
rules
governing
the
language
in
which
he
expresses
propositions
of
Aristoxenus
and
of the verb
extreme
caution.
Never-
theless,
into
Aristoxenus’
enterprise:
(Aristides
Quintilianus
helpfully
describes
KLi/qaLS
4
quence
requires
within
some
de-
terminate
range
in
this
dimension,
220
ANDREW
is
not,
Aristotelian
e.g.,
Meta,
conceived
Aristotle’s account
of relations between form and matter in Phys. ii 9. The
existence
of
suitable
matter
is
necessary
for
it.
But
what
contingent
is
as
potential,
conceived
materially,
is
potentially
an
living
organism,
independent
perceivers,
of natural necessity
necessity of logic,
constitution
we could decide to include in a definition of the
note
Xtxayog,
along
with
its
formal,
dynamic
properties,
an
account
classes of
relation
is
only
incidentally
related
to
impression
of a
incomplete
(where
or
distances
in
the
tottos
within
which
the
voice
can
places the
should be
difficulties.
injunction that it seemed to be. In the case of
domains,
what
something
to
form
of
gradually
become
discriminates
only
what
what
is
these
relations
are
is
224
the
superiority
the
first
decades
There
that
in
proceeding
discussion from
the
a-rrovSeLdCwv
TpoTTO?
is
Aristoxenus
and
be made comprehensible
20-21,
28-30].
It
was
melody.
When
fieXog:
he
argues
that
bad, can be
tainly
derived
from
Aristoxenus.
226
ANDREW
D.
BARKER
as
but
as
the
articulation
of
what
in
what
is
known.
course
my
own.
10
that
developed
Spherics
is
a
name
which
science
as
the
typical
of
228
J.
west
[1975,
301]
suggests
may
have
the length of
Greek
Pitane,
who
the
fourth
century
all
theorems
As
to
of
the
two
books
of
230
J.
there is
was
concern
with
eclipse-phenomena
that
j
Pythagoreans
be
the
elements
of
all
things
and
the
entire
heaven
to
had
been
cosmological-moral
tradition
leading up to works like Plato’s Republic and Timaeus. Indeed,
in
recent
studies,
Charles
Kahn
[1970,
the
construction
of
Greek
finally
the
alized
it
could
account
neither
and
Moon
will
think
the
slightest
deviation,
the following sec-
tion of the
astronomy
to
on
astronomy
genre
of
times
greater
than
when
was
the
for
a
few
That
Aristotle
[Meteor.
343b
1-7]
refers
neither
234
J.
L.
BERGGREN
with
a
terminus
ante
mathematical
two-
sphere
model
in
his
fairly
rapidly.
Autolycus,
one
must
agree
with
Neugebauer
that their
treatises were
all written
on
an
these
points
ouata
(substance)
as
well,
that
mentions by
Greek
would
conclude
what are
cosmos and occupies the place of the center in relation
to
the
cosmos)
puts
us
in
intends
us
to
Euclid’s,
science
whose
subject-matter
is
astronomical
phenomena.
These
one
rotation
(The parallel
they
abundant
used
to
and
may
case
with
the
whole
of
If in
a sphere
to
introduce
the
idea
of
given
day,
180°
of
the
ecliptic
must
rise,
one
may
use
9
and
10,
pro-
vide
qualitative
semicircles,H
while
X rises
explain
both
constant
the
in
period but
also by
lunar tablet
Euclid’s
Greek
Spherics
the
last
the
eastern
horizon
brings
P
to
the
subject
is
Autolycus’
nature of
this treatise,
so we
of
10 In a sphaera obliqua
a rotating
poles,
then
each
of
of theorems
that this
discussing
vari-
ous
There is
Greek
in
both
treatises
geometry
suggestion
[1886]
that
the
theorems
from
Theodosius’
Sphaerica
which
242
J.
definition
on
However,
it
is
instructive
to
compare
follows [cf.
Sphaer. i
as
It is significant, I think, that although Sphaer. iii prop.
1
back to
Sphaer.
i
props.
13
and
15
5
will,
when
the
seems
plain
put on
20
21
22
1
2
Sphaerica ii
Astronomy
245
horizon.
The
phrase
does
not
occur
in
going
from
greater
elevation
complication
in
the
it
seems
and,
more
to
the
18
246
Theodosius’ view
of construction-problems,
on
proof as
is required to
many great
from
now
on
that
A
and
B
are
not
diametrically
opposite.
Now
understood Sphaer. i
prop.
21,
which
requires
find-
ing
the
pole
Greek
Spherics
considerations
of
Given
that
Theodosius
methods
time
on rough
248
J.
L.
BERGGREN
themselves
of
spherics
produced
11
part these are
one
and ad-
250
there
are
temple
medicine
alone
necessarily
symmetrical,
respect.
We
On
the
was
always
those
group
itself.
Self-justification
was
the
order
The
Medical
Tex^
aspects
of
medical
Attacks
on
levelled
at
those
whom
principles
that
should
underlie
medical
practice,
and
of
medical
method
as
shared
concerns
the
in
its
practical
genuine
medicine
as
was directed
the author
epistemological
objections
raised
the
Texv'ai
authors
the
Art
has
simply
invented
conclusion
to
Art has
what
are
imagination,
it
in
particular
removal of the sufferings of the sick, together with the
alleviation
later chapters
be
infalli-
ble,
not
because
they
are
easy,
but
because
they
have
been
discovered’.
(Chapter
10
too
the
I
win,
tails
won the
mastery and
his
the
chapter
4,
The
Medical
for several
advance
the
criticisms
on
the
assump-
tion
disease for
error not
assertions,
the
plausibility
say
to
the
speaker
to
which
one
should
refer
to
and just a matter of chance [ch. 1: cf. ch.
12].
But
been
discovered
(cf.
But
regimens, he also
19
specifies
that
‘we
must
consider
the
causes
The
Medical
Tex^^
in
contribution
towards
the
definition
of
the
medical
9
begins
by
noting
that
substituting
weaker
for
stronger
I
think,
it
has
been
able
to
come
close
to
perfect
where
before
there
was
provide
258
tendency.
The
whole
topic
other
systems.
On
Ancient
the
the
Hip-
pocratic
Corpus,
to
illustrate
the
subject
of
explicit
debate,
maybe
not
anticipated
and that
w£LS
a test of the doctor’s skill. Some, whether or not aware of the
problems
that
remained
if
the
according to its
exact,
it
was
extent
were
al-
ready
dominating)
260
G.
E.
R.
LLOYD
of
12
Between
Data
in
off
decisively
from
the
study
of
coming
to
be
and
passing
away
in
general
[Me-
teor.
339a5-8,
390bl9-22;
du-
references
to
262
JAMES
G.
LENNOX
answer
the
‘what-is-it’
question
about
the
documents
Aristotle
refers
to
as
‘the
animal
histories’
or
sometimes
yevo<g
the taxonomic
certain features
of
cf. Keaney
move,
and
about
how
the
Hist.
an.
is
consistently
concerned
reflecting
on
scientific
research
and
explanation
done
or
in
about
given
part
on
to discover their
as
animals.
[Balme
1987b,
88]
The
reassessment:
like
Balme
[1987b,
80],
I
of
the
causes
264
JAMES
chapters
from
premises
which
are
is
An.
inquiry
predication
holds,
an
in
to
the
first
chapter,
Aristotle
states:
For
that
(otl
pev)
these
things
are
in
fact
thus
is
clear
from
our
inquiry
inquiries
(1)
listed
above.
If
[Balme
1972,
69;
too, must
generation occur in all the plants, it is appropriate to
distinguish
which
belong
use
of
an.
and
the
Hist,
266
so
constituted
Between
Data
and
itself,
which
explicitly
describe
which
in
the
nature
268
for
organizing
from what
selecting premises
from
270
JAMES
an
inquiry
into
the
causal
relationships
which
hold
among
the
a
particular
sort
of
toward
works
as
the
in
inqui

Bowen, Science and Philosophy in Classical Greece

Documents

Transcript of Bowen, Science and Philosophy in Classical Greece