so, three engined planes were A Thing for a while, but its mostly to do with legal restrictions why they were, and why they went away.
So, in general, designers want as few engines as they can get away with, as one big engine is lighter then two smaller engines for a given output, as the two engines need duplicate fuel lines, control runs, structural hardpoints, etc, that all increase the weight, and engines, being full of high precision engineering using exotic alloys, are very expensive. one big engine on the rear of the plane would be ideal for this.
however, theirs a bunch of factors that work against this: one factor is redundancy, in case of engine failure, another is the need to keep the main fuselage free for payload space, and so on. So, the next best set-up is two engines. three engines with the third in the rear fuselage are possible, and three engine designs are cheaper and slightly more fuel efficient, but the desire to maximise payload often pushes towards four wing mounted engines.
however, the early jet engines had a less than stellar reliability rating. Engine failures were not common, but not exactly unknown either. Like i said, high precision engineering.
Their was a standard applied by the FAA, and copied widely, that specified that no twin engine commercial plane could fly a route more than 120 minutes form the nearest airport that could land it. This 120 minute rule was so that in case of a engine failure, the plane would be close enough to a runway it could safely land. this placed limits on what routes a plane could fly, and specifically limited trans-oceanic flights.
three and four engine planes were not bound by the 120 minute rule, as it was felt not needed with the extra redundancy form three and four engines. These bIg, heavy planes affected airline route selection, as they could only run on specific, high volume routes and turn a profit. This, in turn, fed into the “hub and spoke” system of route selection, where air travel was routed though major airports to create the passenger numbers for 4 engine aircraft. for example, you’d fly form Manchester, England, to London Heathrow on a 2 engine, then over the Atlantic to JFK in New York on a 3 or 4 engine, then onwards from JFK to Baltimore on a 2 engine.
As technology improved and engine failures became less and less common, they were able to convince the FAA and equivalent bodies to allow type and airline specific relaxations for the 120 minute rule, called ETOPS (Extended-range Twin-engine Operations Performance Standards, aka “Engine Turns Or Passengers Swim”). This are a combination of plane performance standards and airline operating procedures that rate, for example, British Airways to fly the 787-200 direct form Manchester to Baltimore on a route that takes them 240 minutes flying time form the nearest suitable runway.
TLDR: early planes had reliability issues that mandated four engines for long haul, newer tech has allowed twin engine planes to take over a lot of these routes.
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