[FOM] Historical queries

Wayne Aitken waitken at csusm.edu
Mon Nov 12 02:37:44 EST 2007

> For each of the following theorems Xi, I need to know:
> 1) who first assumed Xi (that is, wrote something which makes no sense
> unless they believed Xi to be true even if they did not state it)
> 2) who first explicitly stated Xi as a known result or conjecture
> 3) who first wrote a proof of Xi


> X3: Irrationality of all kth roots of integers that are not perfect kth
> powers, for all integers k>1 (or, again, if this was ever stated for
> some infinite subset, such as "the kth root of 2 is irrational for all
> k").

You will want to look at Proposition 8 of Book VIII of Euclid's Elements:
"If between two (natural) numbers there fall (natural) numbers in
continued proportion with them, then, however many numbers fall between
them in continued proportion, so many also fall in continued proportion
between the (natural) numbers which have the same ratios with the original

The statement X3 is an easy corollary of this proposition. Indeed, if the
kth root of N is the rational number a/b then
    b^k, b^{k-1} a, \ldots, b a^{k-1}, a^k
are in continued proportion (this is shown in the proof of Prop 2 of book
VIII), and so they satisfy the hypothesis of Proposition 8. Thus there
there are k-1 natural numbers, call them u, u', ..., between 1 and N such
    1, u, u', \ldots, N
are in continued proportion. Therefore N is the kth power of the integer u.

Euclid does not use rational numbers per se, so he would not have given
the exact statement of X3. Proposition VIII.8 is about as close as you can
get to X3 in the Euclidean milieu.

Thus an answer to 1,2,3 for both X2 and X3 is <= Euclid. Because of our
hazy knowledge of pre-Euclidean mathematics, this is about the best answer
you can expect.

Actually, there is an interesting passage in Plato's dialog Theaetetus
that suggests that Plato's friend Theaetetus was the first to address and
perhaps prove X2, and even X3 for the special case of k=3.
(Theaetetus 147 D - 148 B).

Wayne Aitken

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