Re: Tek Platonics

At the site:

http://sunsite.utk.edu/math_archives/.http/hypermail/historia/jul99/0063.html

I found a bit more stuff on "superparticularity" (quoted below):

1. SZABO'S ANALYSIS

The question was raised by Arpad Szabo in the II part of his "The beginnings
of Greek Mathematics" (D.Reidel, '74), where he sets out to show how the
pre-Eudoxian theory of proportions initially took place in the Pythagorean
theory of music.

He supported this thesis with a deep analysis of the Greek technical terms
of the theory (diasteeme, oroi, analogon, logos, etc.) and their recognition
in the supposedly Pythagorean experimental practice of a string stretched
across a ruler, the socalled "canon", divided in 12 parts. Here the
Pythagoreans found three main consonances: the 'octave' (diplasion, 12:6),
the 'fourth' (epitriton, 12:9 or 8:6) and the 'fifth' (emiolion, 12:8 or
9:6).

This musical theory was explicitly connected by Architas with the means:
"Now there are three means in music...": arithmetic a-b=b-c, geometric
a:b=b:c, harmonic or subcontrary a-b:a = b-c:c. (DK 47 B2)
However the first and third can be easily recognized in the fourth (such as
C-F) and the fifth (such as C-G) consonances, whereas the geometric mean
does not correspond to any consonance on the "canon", for it would yield
12:(6*radical(2)) or (6*radical(2)):6. With the word of Szabo (174): "an
octave cannot be divided into two equal subintervals by a number". Remark
than here 'equal' (in consonance) means 'proportional' (on the canon's
intervals), according to the logarithmic relationship between lengths and
tones on a string in our "equal temperament".

I want to remark however that the geometric mean is the natural
relationship between the different octaves and hence seems even a sort of
natural consonance, and that Aristotle (Rose, fragm. 43) recognized instead
that only the arithmetical and the harmonic means were related to musical
harmony.

Szabo did not ascribe to the Pythagorean theory of music "more than a start"
(171): the full development of the theory of proportions had to be found in
the "geometrical arithmetic of the Pythagoreans", where a crucial role was
played by the definition of "similar (omoioi) plane numbers" as "those which
have their sides proportional (analogon)" (Euclides' Elements , def VII,21).

Szabo moreover conjectures that "the concepts of the musical theory of
proportions were applied first of all in arithmetic... Furthermore the
application of this theory to geometrical arithmetic contributed towards an
understanding of the problem of geometric similarity, and this problem in
turn soon led to the problem of linear incommensurability" (173-4).
This displacement of the problem from music to geometry, to be ascribed in
my opinion probably to Archytas himself, was even the rationale of the name
of the 'geometrical' mean, because it could produce no musical consonances,
whereas had precise and easy geometrical instances.

I think that probably it is wrong to look for the 'first' proof of the
incommensurability, even because the numerical 'fact' of some
incommensurable results (for example side and diagonal of the square) was
already known to the Babylonians, whereas a rigorous proof required instead
the introduction of the "absurd" proof in an earlier visual and constructive
mathematics, so that the theorem had to be proved together with the
establishment of its method of proof.

However it is of great relevance the hypothesis of a 'musical' background
for the discovery. And actually a 'negative' proposition, asserting the
incommensurability in musical and arithmetic terms, can be found in the
prop.3 of the "Sectio Canonis", ascribed to Euclid (Appendix).

The same proposition can be found in Boethius' "De Institutione Musica"
iii,11 (DK 47 A19), ascribed to Archytas: "No mean proportional number can
ever be found between two numbers in a ratio superparticularis". 'Ratio
superparticularis' is the 'epimorion diasteema (logos)', i.e. the ratio
n+1:n or mn+m:nm, for n,m integers (Appendix).

In such proof some theorems in the arithmetic books of Euclid (VII, VIII and
IX usually partially ascribed to the Pythagoreans) are implicit, such as
VIII, 20: "If one mean proportional number fall between two numbers, the
numbers will be similar plane numbers".

A possible reconstruction of the missing steps could be found if we
cut-and-paste of the proofs of the propositions VIII.9, VIII.11 and VIII.20,
to prove something like "If one mean proportional number fall between two
numbers, the least numbers of those which have the same ratio of these two
numbers must be square numbers", and furthermore it is easy to show that two
consecutive numbers can not be both square, and this completes the proof.
A similar and detailed reconstruction of the proof is in Knorr's "The
evolution of the Euclidean Elements" (VII.1).
For n=1 in the ratio superparticularis the above results prove the
inexistence of an integer x such that 2m:x=x:m and then such that
2:x/m=x/m:1.

This 'negative' result in Archytas' form derived probably by some 'evident'
impossibility, which was credibly progressively substituted by a rigorous
'proof by absurd' according to the Euclidean style, such as the spurious
X.117 of the Elements (Appendix).

Szabo's conclusion is that "from a historical point of view the Greeks
originally thought of the problem of irrationality as belonging to the
theory of proportions. I hope that part II has shown how the theory of
proportions, whose initial development took place in the Pythagorean theory
of music, may in fact have led by way of arithmetic to the problem of
geometrical similarity and thence to problem of linear commensurability...We
have not yet shown how they reached the stage of being able to prove the
existence of linear incommensurability in a rigorous manner"