In Tuck Everlasting, what secret about the Tucks is revealed in chapter five?

In chapter 5, Winnie runs away and meets Jesse Tuck in the woods.  He drinks from the spring, and she wants to drink, too, but he won't let her.  Then at the end of the chapter, Mae and Miles show up, and they decide they have to take her home with them because the worst possible outcome has arrived (someone has discovered the spring). 

Though nothing is stated outright in this chapter, it is revealed that the spring is significant to the Tucks, and they don't want anyone else to find out about it.  The reader may then infer the reason for this.

What is the amazing fact revealed about the Tucks in "Tuck Everlasting"?

In Chapter 7, the Tucks reveal to Winnie that they are immortal because they have drunk water from a spring in the forest that has stopped them from ever aging and dying. 87 years before Winnie met the Tucks, they had been pioneers coming through the woods looking for a place to settle. They stopped and drank from a spring, and soon found out, through a series of accidents, that nothing could kill them. Mae explains to Winnie, "The water -- it stops you right where you are. If you'd had a drink of it today, you'd stay a little girl forever. You'd never grow up, not ever" (Chapter 7). The Tucks are doomed to live forever, which is against the natural order of things. Winnie begins to understand, through her relationship with the Tucks, that immortality can be a sad thing, that the Tucks have to watch as the rest of the world grows and changes, but the Tucks stay stuck.