Euclid's Elements - Book X, Prop. 117

[ken quirici]

Hi,

There's another controversy over in sci.logic about Euclid's Elements.
This time the controversy is over the above proposition, which one
person claims he saw in the original greek edition. However I can't
find it in the Heath translation, either the ClarkeU version or the
Perseus version. There, Book X only has 115 propositions.

And I can't read the original Greek in Perseus - which also,
unfortunately, doesn't list propositions in its table of contents, just
books.

The proposition is the proposition that the square root of 2 is
irrational. (and of course if any of you remember the original question
I posted about Euclid's Elements, it was over whether I.6 was a
reductio - same issue here).

So if anybody is so inclined, where is the proposition in the English
version of Euclid's Elements that the square root of 2 is irrational?
It might be hard to find because Euclid's terminology is kind of
geometry- based and not the same terminology mathematicians use today,

at least mostly.

Thanks in advance for any help.

Ken

[Petrushka]

Post by ken quirici
Hi,
There's another controversy over in sci.logic about Euclid's Elements.
This time the controversy is over the above proposition, which one
person claims he saw in the original greek edition. However I can't
find it in the Heath translation, either the ClarkeU version or the
Perseus version. There, Book X only has 115 propositions.
And I can't read the original Greek in Perseus - which also,
unfortunately, doesn't list propositions in its table of contents, just
books.
The proposition is the proposition that the square root of 2 is
irrational. (and of course if any of you remember the original question
I posted about Euclid's Elements, it was over whether I.6 was a
reductio - same issue here).
So if anybody is so inclined, where is the proposition in the English
version of Euclid's Elements that the square root of 2 is irrational?
It might be hard to find because Euclid's terminology is kind of
geometry- based and not the same terminology mathematicians use today,

The proof is an interpolation at the end of book 10, i.e. placed after
proposition 115. It is datable to the fourth century, though, as
Aristotle seems to make an allusion to the proof in Prior Analytics book
1, though I forget the reference.

The proof is often thought to originate among the Pythagoreans, though I
don't know of any evidence for that. But's it's definitely not by Euclid
himself, and I think it doubtful that he knew of it (surely he would
have put it into Elements book 10 himself if he had known it).

I can't provide any references for you to chase up off the top of my
head, alas. You might have to hunt down a printed edition (shock,
horror!). Good luck,
P.

Post by ken quirici
at least mostly.
Thanks in advance for any help.
Ken

[John W. Kennedy]

Post by ken quirici
Hi,
There's another controversy over in sci.logic about Euclid's Elements.
This time the controversy is over the above proposition, which one
person claims he saw in the original greek edition. However I can't
find it in the Heath translation, either the ClarkeU version or the
Perseus version. There, Book X only has 115 propositions.

"Original Greek" could mean ante-Theonine.

[ken quirici]

Post by John W. Kennedy
"Original Greek" could mean ante-Theonine.

Yes, I'm not sure what he meant by 'Original greek', or what
you mean by ante-Theonine?

I took him to mean in any standard classical-greek edition.
However he was emailing from Greece and consulting a Greek

Post by John W. Kennedy
Don't quote me, I asked today professor Michael Lambrou of the

University of

Post by John W. Kennedy
Crete, who specializes in Ancient Greek Mathematics and its history,

among
http://www.math.uoc.gr/dept/persons/lambrou.html

Post by John W. Kennedy
First, he said that Aristotle in his "Prior Analytics" discusses

*various*

Post by John W. Kennedy
forms of contradictions and as an example, he presents the proof that
sqrt(2) is not in Q. Indeed his proof in Euclid's Elements X, 117, is

by

Post by John W. Kennedy
contradiction.
We not only checked the above in the ancient text, but also performed

a

Post by John W. Kennedy
search for the words "oper atopon" (thus contradiction) in Euclid's

original

Post by John W. Kennedy
texts (he had it in his computer) and found _at least_ 27 occurences

of the

Post by John W. Kennedy
words, in Elements *only*.

Thanks.

Ken

[John W. Kennedy]

Post by ken quirici

Post by John W. Kennedy
"Original Greek" could mean ante-Theonine.

Yes, I'm not sure what he meant by 'Original greek', or what
you mean by ante-Theonine?
I took him to mean in any standard classical-greek edition.
However he was emailing from Greece and consulting a Greek

For most of history the /Elements/ meant the text produced in the 4th
century by Theon of Alexandria. In the 19th century, an ante-Theonine
ms. was discovered.

[William C Waterhouse]

Post by ken quirici
...
The proposition is the proposition that the square root of 2 is
irrational. (and of course if any of you remember the original question
I posted about Euclid's Elements, it was over whether I.6 was a
reductio - same issue here).
So if anybody is so inclined, where is the proposition in the English
version of Euclid's Elements that the square root of 2 is irrational?

This proof (and another that isn't really an independent proof)
is included in an appendix to Heiberg's edition of Euclid
(vol. 3, 408-413). There is general agreement that it wasn't in
Euclid to begin with but is ancient.

A member of the historia-matematica mailing list asked about it
a year ago, and I did a quick translation; here it is. The paragraph
breaks are mine.

------------------------------------------------------------------------------

IT IS SET FOR US TO PROVE THAT, IN A SQUARE FIGURE, THE
DIAMETER AND THE SIDE ARE INCOMMENSURABLE IN LENGTH.


Let ABCD be a square , with diameter AC. I say that CA, AB are
incommensurable in length.

For suppose, if possible, that they are commensurable. I say that
then the same number would be even and odd. For it is clear that
the square on AC is twice that on AB. As CA, AB are commensurable,
they have the ratio of a number to a number. Say CA is to AB as
EZ is to H, and let EZ and H be the smallest of those having that
same ratio.

Then EZ is not the unit. For if it is, and EZ is to H as AC to AB,
and AC is greater than AB, then EZ is greater than H, the unit
bigger than a number, which is absurd. Thus EZ is not the unit
and hence is a number.

Since CA is to AB as EZ is to H, then also the square on CA is
to the square on AB as the square on EZ is to the square on H.
But the square on CA is twice the square on AB, so the square on
EZ is twice the square on H. Hence the square on EZ is even.
Consequently, EZ itself is even; for if it were odd, its square
would also be odd, since when an odd number of odd summands are
combined, the total is odd. Thus EZ is even.

Split it into two equal parts at T. Since EZ, H are the smallest
with the same ratio, they are prime to each other. And EZ is
even, so H is odd. Indeed, if it were even, 2 would measure
EZ and H, since every even number has a half; this is impossible
for numbers prime to each other, and thus H is not even. Hence
it is odd.

Since EZ is twice ET, the square on EZ will be four times that
on ET. But the square on EZ is twice that on H, so the square
on H is twice the square on EZ. Hence the square on H is even.
By what was said earlier, H is even. But it is also odd; which is
impossible. Thus CA and AB are not commensurable in length, QED.

--------

IN ANOTHER WAY

Let A be for the diameter and B for the side. I say that A and B
are incommensurable in length. For is possible let the numbers
EZ be to H as A is to B; and let EZ and H be the smallest having
that same ratio. Thus EZ and H are prime to each other.

I say first that H cannot be the unit. For suppose it is. Since
A is to B as EZ is to H, then also the square on A is to the square
on B as the square on EZ is to the square on H. But the square on
A is twice the square on B, so the square on EZ is twice the square
on H. But H is the unit, and so 2 will be the square of EZ; which
is impossible. Thus H is not the unit and hence is a number.

Since the square on A is to the square on B as the square on EZ is
to the square on H, we know that the square on B is to the square
on A as the square on H is to the square on EZ. But the square on
B measures the square on A, so the square on H measures the square
on EZ. **Hence the side H measures EZ.** But H measures itself,
so H measures the numbers EZ and H, which are prime to each other.
That is impossible. Hence A and B are not commensurable in length.
Therefore they are incommensurable, QED.

-------------------------------------------------------------------------------

[Note that the second "proof" is not really a self-contained
argument; the sentence I have marked with **....** quotes
Euclid VIII.14, which uses VIII.7 ]


William C. Waterhouse
Penn State

[ken quirici]

Hi,

Post by ken quirici
Thanks in advance for any help.

Ken

And thanks after the fact for the help!

Ken