apostrophe - Etymology of "let us" and "let's"

The verb let means “allow”, “permit”, “not prevent or forbid”, “pass, go or come” and it's used with an object and the bare infinitive.

1. Are you going to let me drive or not?


2. Don't let him off the hook.


3. Before we let our children surf the Internet, they have to do their homework.



4. She lets the cats out before leaving.


5. He let us into the house.

The full form let us can be used similarly

6. Let us know as soon as possible.



7. Please, let us help you.


8. They will never let us forget.

None of the above can be contracted; however, when let us is used for making a suggestion; giving self-encouragement; expressing a consequence or plea, it is often contracted to let's

Let's go out
Let's have a party
Let's see what happens
Let's stand together in this emergency
Let's not forget those who sacrificed their lives

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Questions

• I believe that let + us is the only instance where this type of contraction occurs. Is there an explanation as to why verbs such as (i) give + us, (ii) get + us or (iii) take + us were not similarly contracted?




• When was the apostrophe used to replace the omitted letter "u" in let us?




• What is the origin of let's?