In "Dr. Heidegger's Experiment," if we examine the relationship between cause and effect, let's consider two questions about cause. First, what causes Dr. Heidegger's four companions to drink the water from the vase. This is quite obvious. They want to be young again. The lengthy first paragraph of the story tells the sad tale of each of them. Mr. Medbourne had been a successful businessman but now he is an old beggar. Colonel Killigrew fooled around when he was younger and now suffers from his irresponsible ways of his younger days. Mr. Gascoigne had been an evil politician and now, time having passed, he has been forgotten. Widow Wycherly had once been courted by each of these men, but now she is a withered old woman. Hawthorne concludes these introductions in saying that these four were "a little beside themselves" which means that they were almost out of their senses or out of their minds. In this case, they were beside themselves because of sadness and "woful recollections." So, they are perfect guinea pigs for Dr. Heidegger's experiment because they are so desperate to be young again.
And this is precisely why Heidegger chooses them in the first place. This answers the second question of cause. What causes him to choose them? They are desperate. Heidegger had also hoped to teach them a lesson, that time passes for a reason. Heidegger says that he will not take part in the experiment because he had trouble in growing old and doesn't want to go through it again. Not overtly expressed is the idea that a person has one chance in life to be young, old, and so on. It is therefore up to each of us to make the most of each time in our lives, even old age. Heidegger's four friends do not accept or understand any of these lessons. This is why (another cause) they make plans to seek the Fountain of Youth, even if that means drinking from it constantly.