How does act 1, scene 4 in Hamlet show reason under threat? (Reason is defined by the capacity of the mind to arrive at judgments based on evidence and sound logic.)

Act 1, scene 4, of Hamlet is when the ghost appears for the second time in the play. This is when Hamlet himself is present and is faced with the spirit of his dead father for the first time.
He immediately addresses the ghostly figure, trying to communicate with it. These actions can be interpreted as the first to indicate Hamlet’s descent into madness:

What may this mean,
That thou, dead corse, again in complete steel
Revisits thus the glimpses of the moon,
Making night hideous and we fools of nature,
So horridly to shake our disposition
With thoughts beyond the reaches of our souls?
Say why is this? Wherefore? What should we do? (1.4.54–61.)

He is wondering what meaning is behind the ghost’s sudden appearance and asks what he is to do. Once the ghost signals to Hamlet to follow it, Hamlet, curious as to where the spirit will lead him, announces to Horatio and Marcellus (friends who have accompanied him to see the ghost) that he has decided to follow it.
Even though both Horatio and Marcellus warn him and advise against him following the ghost, claiming its presence makes them feel uneasy, Hamlet has already decided to go after the spirit, choosing to ignore his friends’ warnings. He even threatens to kill them if they try to stop him.
That Hamlet impulsively and unthinkingly decides to follow the spirit, and that he immediately threatens his good friends with death, shows us how very unstable his mental state is. One could argue that it is apparent here that Hamlet is no longer on the verge of abandoning reason but has already dived deep into the sea of madness.