Plague is carried by rats and fleas, which are nonexistent in the cold temperatures. Therefore, it is ruled out. The bodies showed no signs of muscular atrophy, ruling out polio. None of the men had rabies symptoms so that too can be eliminated. The presence of lime juice and the absence of skin discoloration rules out scurvy.
Yellow fever is carried by mosquitoes in warm climates, so it is eliminated as a cause.
All five victims were bled and that is the only common factor. Therefore, phlebotomy is the cause of death. This would be apparent from excessive bleeding. The doctor is constantly bleeding the patients, this is fatal. Harper, who was one of the first to become sick but refused to be bled lived while others died.