David Fowler
Date: Tue, 29 Dec 1998 01:27:32 -0500
From: Jim <kazorcht@CLARK.NET>
Subject: Re: Fractions in pythagorean algebra??
In Plato's Meno, the young slave boy is lead to discover the
square whose side is the square root of eight. He does this
by being told that one half of the unit square has an area of
one half and then by summing four whole squares plus eight
half squares to give a square whose area is eight and whose side
is the square root of 8.
This stands in direct contradiction to the assertion that ratios
were not added. Clearly Plato (and probably everyone else)
recognized that ratios could be added as numbers. The superior
mathematicians, whose work survived, may have chosen to always
consider fractions purely as ratios because there was some
benefit to doing so, and clearly there was and is. Couldn't
the hard line on ancient fractions be softened just a bit my the
realization that we have an incredibly limited sample of mathematics
in the ancient world. Most of Aristotle's works didn't survive.
What then might have happened to the more vulgar mathematicians who
concerned themselves with the merely mechanistic aspects of arithmetic
such as summing ratios? Consider whether your elementary school
math texts will survive over the next thousand years or so.
I was just thumbing through some books to see if I could come up
with more. I found, from Archimedes' "Measurement of a Circle":
Proposition 3.
"The circumference of any circle is greater than three times the
diameter and exceeds it by a quantity less than the seventh part of
the diameter but greater than ten seventy-first parts"
In English, anyway, it appears that Archimedes has no problem adding
a fraction to an integer, 3, to get an idea of the ratio of
circumference to diameter. This seems to me to be an indication that
he split numbers into integer and fractional parts and then presumably
back again to get an approximation of pi ( ratio), or if not, then
once he multiplied the the diameter by 3+fraction, he would be faced
with an "intractable" problem if the diameter was not an integer. And
since Archimedes was also an engineer, he probably didn't run into too
many diamaters that weren't fractions. Judging by his success as an
engineer and scientist, I would say that he solved these practical
problems in exactly the way many have been saying he didn't. Since I
can't read greek, I could be wrong on this. The editor was J.L.
Heiberg in the collection "Archimedes opera omnia cum commentariis
Eutocii" as translated by Ivor Thomas in case there is some error.
The comments:
These are good points that deserve a full and serious response.
* I would rather say that the slaveboy is adding numbers representing areas
here, which is in apparent direct contradiction of my general strict thesis
that Greek *mathematics* is non-arithmetised. There is a footnote about
this in the new Appendix to my book _The Mathematics of Plato's Academy_,
2nd ed, to appear very soon (sometime like last week, I hope):
"The only arithmetised passage I know, anywhere up to Archimedes and
beyond, is in Plato's Meno, 82c ff, where Socrates says to the slaveboy:
"Now if this side is two feet long, and this side the same, how many feet
will the whole be"; and the passage continues in this arithmetised vein for
the slaveboy's first two attempts. (I thank Wilbur Knorr for pointing this
out to me in January 1991 in a train somewhere between Verona and Venice; I
had used this passage for the introduction to the book without appreciating
this feature!) The switch to geometry is then indicated by Socrates when
he tells the slaveboy: "If you don't want to count it up, just show me on
the diagram" (84a). And I think one can explain this singular exception by
observing that the slaveboy is not a mathematician - that is the point of
the episode. Note that an arithmetisation of aspects of everyday life
occurs when a barter economy gives way to the use of money, which seems to
have happened in Greece by the 7th century BC, though taxation and other
documents are very often set out in terms of measures of wheat; see Section
7.3(c)."
I also add here that everyday Greek financial accounts, such as have
survived, are full of arithmetic of fractions, all of them expressed in
'parts', that is what are usually called 'Egyptian fractions' (but my point
here is that they are also Greek).
* Archimedes is here adding two *lines*, not numbers, and there is no
problem about that: Euclid is full of such operations. Also there is no
problem with the 'ten seventy-firsts': take a line, here the diameter,
split it into seventy-first parts (Elements VI 9 & 10), and then put
together ten of these (use Elements I 1-3 here,if you want to be strict).
One important difference between lines and numbers is the range of possible
arithmetical operations with them: in Euclidean terms, 'multiplying' two
lines gives a rectangle (II Def 1: Any rectangular parallelogram is
contained by the two straight lines containing the right angle), and this
kind of 'multiplication' very soon has dimensional restrictions.
((Descartes is different: using Euclidean ingredients, he defines a full
range of quasi-arithmetical operations on lines. In fact Euclid himself
deploys these same operations in Elements X, but with no aim of defining an
arithmetic with his lines.))
The text of Measurement of a Circle is corrupt, and the standard
translations have a very strong bias towards making the subsequent
arithmetic in it look like our kind of arithmetic with common fractions.
When you look closely and strip off the translator's additions and
clarifications, I think it become apparent that it also is carried out
using manipulations of parts.
There are a couple of long and detailed discussions of the treatise in my
book. For what it is worth, I think that Archimedes may have been trying to
work out the pattern of the 'Euclidean algorithm' (the 'anthyphairesis')
when applied to the perimeter and diameter of the circle, for which he gets
3-subtractions, 7-subtractions, 12-or-more-subtractions, and there is
corrupt further evidence elsewhere that he did a more involved evaluation.
And why? Again my suggestion is that he was exploring to see what one might
say about these two lines: Are they commensurable or, if not, does their
anthyphairesis exhibit any interesting features (because some pairs of
lines, like the sides of the n-fold and m-fold squares, do have a very very
interesting anthyphairesis). From the calculation as it survives, the only
thing that can be said is that, because of that '12-or-more', the ratio of
'three- then seven-subtractions, which translates to 22 to 7, will be a
good approximation, just what the proposition in effect enunciates.
(Explanation: the 12-or-more-subtractions -- in this context, 12 is quite a
big number -- shows that the remainder after the second subtraction is
small, so ignoring it will give a good approximation.)
David Fowler