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Adding flavoring to ice was popularized by Babur, a 12th century Muslim who founded the Mughal Empire of what is now India and surrounding countries. The ice in the summertime was brought down from the Himalaya mountains.
It may have been independently invented in several regions of the world. A legend says Roman Emperor Nero ate it in the first century.
Sherbet (Americans usually call it Italian ice) does not contain milk or cream. and it can mean a chilled drink flavored with fruit juice, spices or flower petals, or can mean a sweet powder for making such a drink, or can be a solid food. The word sorbet is of Persian origin,
In the United States, sherbert (originally spelled sherbet) is a chilled sweet dessert made from water, adding sugar, fruit flavoring and a small amount of milk or cream.
Photograph by Renee Comet of the United States Department of Health and Human Services, via Wikimedia Commons.