How long did Axel hold out before he finally revealed the answer to his uncle? What finally made him to do so?

Axel is very worried, especially when his uncle starts talking about how Axel will get some of the credit for their discoveries. Axel tries to dissuade Lidenbrock, suggesting that the document is a practical joke. To Axel's annoyance, Lidenbrock asks him to continue in a scholarly mode and to lay out his opinions.

Axel first asks about the place names, but his uncle is able to tell him what they refer to. (Snaefells, for instance, is the extinct crater that they must descend into.) Eventually Axel is worn down yet still maintains that it is unlikely that Saknussemm went down into the Earth and came back to tell about it. Such a journey would be impossible on account of the temperature of the center. The professor counters that no one actually knows what the temperature is and that it cannot be as hot as people say it is; if it were, the whole Earth would be compromised. Axel begins to agree with Lidenbrock, reluctantly. Even his ideas about internal heat fall away. In triumph, his uncle says that they must keep their find a secret from everyone.