Here's how Matsudaira handles the start of Book II of the Odyssey. First, for reference, Butler's version:

Now when the child of morning, rosy-fingered Dawn, appeared Telemachus rose ...

And here's Matsudaira's version:

朝のまだきに生れ指ばら色の曙の女神が姿を現わすと、オデュッセウスの寵愛の息子は床から身を起こし ...

So the word madaki is a time word sort of meaning "soon", almost "too soon." There is actually a word, asamadaki, which means "early morning when dawn is just about to break" (and it can be used with madaki, giving you asamadaki madaki: "too early in the early morning"). So literally the above means something like:

When, born as the morning broke, the rosy-fingered goddess of dawn showed her form, Odysseus's beloved son rose from his bed ...

(Note that Matsudaira is presumably restoring a roundabout way of saying "Telemachus" that Butler has simplified.)

Part of the reason that I decided to read this book in Japanese is that I read Japanese much closer than I do English. I'd never really even thought about the way that "morning" and "dawn" are related in the standard English version of the epithet, but all those Japanese books and poems about Heian nobles getting up to mischief at various named points on the evening → noon spectrum have primed me to pay closer attention to this stuff in Japanese. And so it pleases me to notice that to be the goddess of dawn (akebono), you have to get up (that is, be born) pretty early (asa no madaki).