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An engine is something that produces some effect from a given input. The origin of engineering was the working of engines. There is an overlap in English between two meanings of the word "engineer": 'those who operate engines' and 'those who design and construct new items'.
In original usage, an engine was any sort of mechanical device. The term "gin" in cotton gin is a short form of this usage. Practically every device from the industrial revolution was referred to as an engine, and this is where the steam engine gained its name. This form of the term has recently come into use again in computer science, where terms like search engine, "3-D graphics rendering engine" and "text-to-speech engine" are common.
In more recent usage, the term is typically used to describe devices that perform mechanical work, follow-ons to the original steam engine. In most cases the work is supplied by exerting a torque, which is used to operate other machinery, generate electricity, pump water or compress gas.