Commentary on Proposition X.6

Back to Proposition X.6.

The same link to connect the proportion of numbers with the proportion of magnitudes as was necessary in the last proposition is necessary here. This being premised, the argument is as follows.

Suppose

$$a : b = m : n,$$

where $m, n$ are (integral) numbers.

Divide $a$ into $m$ parts, each equal to $c$, say, so that

$$a = mc.$$

Now take $d$ such that

$$d = nc.$$

Therefore we have

$$a : c = m : 1,$$

and

$$c : d = 1 : n,$$

so that, ex aequali,

$$ \begin{aligned} a : d &= m : n \\ &= a : b\text{, by hypothesis.} \end{aligned} $$

Therefore $b = d = nc$, so that $c$ measures $b$ $n$ times, and $a, b$ are commensurable.

The Porism is often used in the later propositions. It follows (1) that, if $a$ be a given straight line, and $m, n$ any numbers, a straight line $x$ can be found such that

$$a : x = m : n.$$

  1. We can find a straight line $y$ such that

$$a^2 : y^2 = m : n.$$

For we have only to take $y$, a mean proportional between $a$ and $x$, as previously found, in which case $a, y, x$ are in continued proportion and [V. Def. 9]

$$ \begin{aligned} a^2 : y^2 &= a : x \\ &= m : n. \end{aligned} $$