The incident occurred on Rockett Street, in the northern part of Bastille Square, according to the France-3 television channel.
The scissor-wielding assailant struck a local postman in the throat and then attacked a passer-by and also wounded him.
Police officers arrived at the scene very quickly since their office is only a couple of hundred meters from the square. The law enforcement personnel had to taser the attacker before arresting him.
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Following the accident, Anne Hidalgo, the mayor of Paris, urged Interior Minister Gerard Collomb to boost police presence on the streets of the capital.
"The plea of the residents must be heard," she said on Twitter.
The mayor added that a medical and psychological center will be operating in the area where the attack took place.