The term was introduced by American mathematician Benjamin Peirce in 1870[3][4] in the context of elements of algebras that remain invariant when raised to a positive integer power, and literally means "(the quality of having) the same power", from idem + potence (same + power).
^Original manuscript of 1870 lecture before National Academy of Sciences (Washington, DC, USA): Peirce, Benjamin (1870) "Linear associative algebra" From pages 16-17: "When an expression which is raised to the square or any higher power vanishes, it may be called nilpotent; but when raised to a square or higher power it gives itself as the result, it may be called idempotent.
The defining equation of nilpotent and idempotent expressions are respectively An = 0 and An = A; but with reference to idempotent expressions, it will always be assumed that they are of the form An = A unless it be otherwise distinctly stated."