Commentary on Proposition VIII.4¶
Back to Proposition VIII.4.
The term “in continued proportion” is here not used in its proper sense, since a geometrical progression is not meant, but a series of terms each of which bears to the succeeding term a given, but not the same, ratio.
The proposition furnishes a good example of the cumbrousness of the Greek method of dealing with non-determinate numbers. The proof in fact is not easy to follow without the help of modern symbolical notation. If this be used, the reasoning can be made clear enough.
Euclid takes three given ratios and therefore requires to find four numbers. We will leave out the simpler particular case which he puts first, that namely in which E accidentally measures K, the multiple of D found in the first few lines; and we will reproduce the general case with three ratios.
Let the ratios in their lowest terms be
$$a : b,\quad c : d,\quad e : f$$
Take $l_1$, the L.C.M. of $b$, $c$, and suppose that
$$l_1 = mb = nc.$$
Form the numbers
$$\left. \begin{array}{r} ma,\, mb \\ = nc \end{array} \right\},\, nd.$$
These are in the ratios of $a$ to $b$ and of $c$ to $d$ respectively.
Next, let $l_2$ be the L.C.M. of $nd$, $e$, and let
$$l_2 = pnd = qe.$$
Now form the numbers
$$\left. \begin{array}{r} pma,\, pmb \\ = pnc \end{array} \right\},\, \left. \begin{array}{r} pnd \\ = qe \end{array} \right\},\, qf,$$
and these are the four numbers required.
If they are not the least in the given ratios, let
$$x,\, y,\, z,\, u$$
be less numbers in the given ratios.
Since $a : b$ is in its lowest terms, and
$$a : b = x : y,$$
$b$ measures $y$.
Similarly, since
$$c : d = y : z,$$
$c$ measures $y$.
Therefore $l_1$, the L.C.M. of $b$, $c$, measures $y$.
But
$$l_1 : nd\ [= c : d] = y : z.$$
Therefore $nd$ measures $z$.
And, since
$$e : f = z : u,$$
$e$ measures $z$.
Therefore $l_2$, the L.C.M. of $nd$, $e$, measures $z$: which is impossible, since $z < l_2$ or $pnd$.
The step (line 86) inferring that G : R = K : S is of course alternando from G : K [C : D] = R : S.
It will be observed that VIII. 4 corresponds to the portion of VI. 23 which shows how to compound two ratios between straight lines.