Let $AB$, / E B /, / C D /, / F D / be magnitudes proportional componendo, so that, as / A B / is to / E B /, so is / C D / to / F D /;
I say that they will also be proportional separando, that is, as / A E / is to / E B /, so is / C F / to / F D /.
For of / A E /, / E B /, / C F /, / F D / let equimultiples/ G H /, / H K /, / L M /, / M N / be taken, and of / E B /, / F D / other, chance, equimultiples, / K O /, / N P /.
Then, since / G H / is the same multiple of / A E / that / H K / is of / E B /,
[G.171]therefore / G H / is the same multiple of / A E / that / G K / is of / A B /.
But / G H / is the same multiple of / A E / that / L M / is of / C F /;
therefore / G K / is the same multiple of / A B / that / L M / is of / C F /.
Again, since / L M / is the same multiple of / C F / that / M N / is of / F D /,
[G.171]therefore / L M / is the same multiple of / C F / that / L N / is of / C D /.
But / L M / was the same multiple of / C F / that / G K / is of / A B /;
therefore / G K / is the same multiple of / A B / that / L N / is of / C D /.
Therefore / G K /, / L N / are equimultiples of / A B /, / C D /.
Again, since / H K / is the same multiple of / E B / that / M N / is of / F D /, and / K O / is also the same multiple of / E B / that / N P / is of / F D /,
therefore the sum / H O / is also the same multiple of / E B / that / M P / is of / F D /. [G.172]
And, since, as / A B / is to / E B /, so is / C D / to / F D /, and of / A B /, / C D / equimultiples / G K /, / L N / have been taken, and of / E B /, / F D / equimultiples / H O /, / M P /,
therefore, if / G K / is in excess of / H O /, / L N / is also in excess of / M P /, if equal, equal, and if less, less.
Let / G K / be in excess of / H O /; then, if / H K / be subtracted from each,
/ G H / is also in excess of / K O /.
But we saw that, if / G K / was in excess of / H O /, / L N / was also in excess of / M P /;
therefore / L N / is also in excess of / M P /,
and, if / M N / be subtracted from each,
/ L M / is also in excess of / N P /;
so that, if / G H / is in excess of / K O /, / L M / is also in excess of / N P /.
Similarly we can prove that, if / G H / be equal to / K O /, / L M / will also be equal to / N P /, and if less, less.
And / G H /, / L M / are equimultiples of / A E /, / C F /, while / K O /, / N P / are other, chance, equimultiples of / E B /, / F D /:
therefore, as / A E / is to / E B /, so is / C F / to / F D /.
