Commentary on Proposition VII.1

Back to Proposition VII.1.

[Folio 0303 | p. 297]

It is proper to remark here that the representation in Books VII. to IX. of numbers by straight lines is adopted by Heiberg from the MSS. The method of those editors who substitute points for lines is open to objection because it practically necessitates, in many cases, the use of specific numbers, which is contrary to Euclid’s manner.

“Let CD, measuring BF, leave FA less than itself.” This is a neat abbreviation for saying, measure along BA successive lengths equal to CD until a point F is reached such that the length FA remaining is less than CD; in other words, let BF be the largest exact multiple of CD contained in BA.

Euclid’s method in this proposition is an application to the particular case of prime numbers of the method of finding the greatest common measure of two numbers not prime to one another, which we shall find in the next proposition. With our notation, the method may be shown thus. Supposing the two numbers to be $a$, $b$, we have, say,

b) a ( p
           pb
           c) b ( q
              qc
              d) c ( r
                 rd
                 1
        

If now $a$, $b$ are not prime to one another, they must have a common measure $e$, where $e$ is some integer, not unity.

And since $e$ measures $a$, $b$, it measures $a - pb$, i.e. $c$.

Again, since $e$ measures $b$, $c$, it measures $b - qc$, i.e. $d$, and lastly, since $e$ measures $c$, $d$, it measures $c - rd$, i.e. $1$: which is impossible.

Therefore there is no integer, except unity, that measures $a$, $b$, which are accordingly prime to one another.

Observe that Euclid assumes as an axiom that, if $a$, $b$ are both divisible by $c$, so is $a - pb$. In the next proposition he assumes as an axiom that $c$ will in the case supposed divide $a + pb$.