Officers said the incidents happened in the early hours of Sunday, October 25, on two quiet streets in the Capital less than a mile away from each other.
Police said the first assault took place in Cranston Street, which joins the Canongate and East Market Street.
The second, police added, took place West College Street, which intersects Old College and the National Museum of Scotland.
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Officers have arrested and charged a 24-year-old man in connection with both attacks.
Detectives want to speak to anyone who was in the area between 6am and 9am and who may have any information that will assist their ongoing enquiries to come forward.
Police Scotland can be contacted on 101 quoting reference number 0905 of 25 October. Alternatively Crimestoppers can be contacted on 0800 555 111, where anonymity can be maintained.