In "Dr. Heidegger's experiment," what had been the relationship of the four people in the past?
The four guests who are invited to Dr. Heidegger's study to participate in one of his scientific experiments are described as "venerable friends" of the doctor, indicating their age and the kind of station that they possess. It is important to note in this tale how Hawthorne uses these characters in his allegory - each represents a kind of vice that when youth is restored to them they repeat and indulge in without much apparent thought. Therefore the descriptions of each of these characters that are given in the first paragraph are key to understanding the tale, as are their reactions to their new-found youth.
However, in the first paragraph, we are also given an indication of how these four people were involved with each other:
It is a circumstance worth mentioning, that each of these three old gentlemen, Mr. Medbourne, Colonel Killigrew, and Mr. Gascoigne, were early lovers of the Widow Wycherly, and had once been on the point of cutting each other's throats for her sake.
It is likewise important to note why Hawthorne adds this and how he uses it to support the theme of his story - that old age brings wisdom and reflection, whereas if we were to have our youth returned to us, we would probably fall into the same folly as before.