Proposition 7.4 of Euclid's Elements

Question

Here is an online version of Book 7 of Elements where the following definitions and proposition can be found: https://mathcs.clarku.edu/~djoyce/java/elements/bookVII/bookVII.html

Note: In his books on number theory, Euclid uses the word "measures" in place of "divides".

Definitions:

1. A unit is that by virtue of which each of the things that exist is called one.

2. A number is a multitude composed of units. (Thus, a number is a positive integer greater than one.)

3. A number is a part of a(nother) number, the less of the greater, when it measures the greater;

4. But (the lesser is) parts (of the greater) when it does not measure it.

Proposition 7.4 says:

Any number is either a part or parts of any (other) number, the less of the greater.

Euclid gives a somewhat long proof of this but isn't it obvious? If $a<b$, $a$ either measures $b$ or doesn't measure $b$. Thus, $a$ is either a part or parts of $b$.

What am I missing?

Answer 1

In Book VII you'll find

Definitions 3–5

3. A number is a part of a number, the less of the greater, when it measures the greater;

4. But parts when it does not measure it.

5. The greater number is a multiple of the less when it is measured by the less.

When you read these definitions it appears that Euclid's definition is an axiomatic statement:

$\quad$ IF $a \lt b$ THEN $[ \,a \text{ is a part of } b\,] \text{ xor } [\,a \text{ is parts of } b\,]$.

In the guide to the above definitions you'll find

There is one more difficulty with this definition. It seems obvious that when one number $a =mu$ is less than another $b =nu$, then in all cases $a$ would be parts of $b$, namely $a$ consists of $m \text{ one-}n^\text{th}$ parts of $b$. Yet, the proposition VII.4 has a proof to show that $a$ is either a part or parts of $b$. The reason is that the desired parts should be in lowest terms. For our example, where $a = 4u$ and $b = 6u$, it isn’t enough to say that $a$ is 4 one-sixth parts of $b$; what’s needed is that $a$ is $2$ one-third parts of $b$.

So I think Euclid felt the need for an $\text{XOR}$ algorithm that could decide the matter for any two distinct numbers. Lurking behind this is that he won't be defining rational numbers as an equivalence relation on a set - he needs to calculate a a canonical representation/idea/concept.

Euclid's logic of not regarding $1$ as a number also plays a part in this. My guess is that an attempt was being made to 'compartmentalize' concepts and to avoid trivial statements. For Euclid the number, say five, is a multitude of units. He doesn't want to also say that one is a part of five.

It is interesting how liberating the logic becomes, when, in modern mathematical treatments we don't ignore or shortchange the trivial numbers $1$ or $0$. Euclid would certainly be amazed to see how much things open up once we can all agree on the existence of the empty set.

Answer 2

Euclid's proof isn't much longer than that. He breaks things into three cases. Assuming that $a<b$ are positive integers:

• If $a\mid b$, then $a$ is a part of $b$ (e.g. $2$ is a part of $4$)
• If $(a,b)=1$, then $a$ are parts of $b$ (e.g. $3$ are parts of $5$)
• If $(a,b)>1$ but $a\not\mid b$, then $a$ are parts of $b$ (e.g. $4$ are parts of $6$)

Yes, today we would merge those last two cases without a second thought. But it seems from the commentary that there was a philosophical squabble at the time over whether $1$ was a proper number, so Euclid evidently felt it was worth a separate case.

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Your question asked "What am I missing?" TBH, I don't know. Was this it?