“Invert, always invert.”
Carl Jacobi
“Invert, always invert” is widely linked to the 19th-century mathematician Carl Jacobi, but the attribution is not verified. The closest documented form is the German phrase “Man muss immer umkehren” — “one must always invert” — cited in 1916 by Edward B. Van Vleck, who said Jacobi was reported to have taught it to his students. No original source from Jacobi has been identified.
The idea remains useful. When a problem is difficult, reverse it. Work backwards, examine the opposite case, or ask what would make the situation fail. Inverting a question can reveal hidden assumptions and make a solution easier to see.
For a detailed source investigation, see Michael Molinsky, “Quotations in Context: Jacobi”, Mathematical Association of America.
This quote is commonly attributed to Carl Jacobi, but I have not been able to locate a primary source. Use with caution in academic or professional contexts.