G.199 magnitudes proportional separando — Heath

V.18

If magnitudes be proportional separando, they will also be proportional componendo.

Let / A E /, / E B /, / C F /, / F D / be magnitudes proportional separando, so that, as / A E / is to / E B /, so is / C F / to / F D /;

I say that they will also be proportional componendo, that is, as / A B / is to / E B /, so is / C D / to / F D /.

For, if / C D / be not to / F D / as / A B / to / E B /, then, as / A B / is to / E B /, so will / C D / be either to some magnitude less than / F D / or to a greater.

First, let it be in that ratio to a less magnitude / G D /.

Then, since, as / A B / is to / E B /, so is / C D / to / G D /, they are magnitudes proportional componendo;

[G.198]so that they will also be proportional separando.

Therefore, as / A E / is to / E B /, so is / C G / to / G D /.

But also, by hypothesis,

as / A E / is to / E B /, so is / C F / to / F D /.

[G.189]Therefore also, as / C G / is to / G D /, so is / C F / to / F D /.

But the first / C G / is greater than the third / C F /;

therefore the second / G D / is also greater than the fourth / F D /. [G.192]

But it is also less: which is impossible.

Therefore, as / A B / is to / E B /, so is not / C D / to a less magnitude than / F D /.

Similarly we can prove that neither is it in that ratio to a greater;

it is therefore in that ratio to / F D / itself.

Therefore etc. Q. E. D.

Dependency Graph

Required for

G.207, G.236, G.238, G.239, G.240, G.241, G.243, G.407, G.408, G.413, G.414, G.419, G.420, G.437, G.438, G.443, G.444, G.448, G.449, G.451, G.458, G.459, G.464, G.465, G.477, G.478, G.479, G.480, G.481, G.482, G.486, G.487, G.491, G.492, G.494, G.495, G.496, G.557, G.563, G.565, G.570, G.571, G.572, G.573, G.589, G.597, G.601