Hi! Welcome to “Dusty Phrases.” You will find below an ancient phrase in one language or another, along with its English translation. You may also find the power to inspire your friends or provoke dread among your enemies.
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ὅπερ ἔδει δεῖξαι (ΟΕΔ)
hóper édei deîxai (OED)
The very thing it was required to have shown
This phrase relates to mathematical proofs and philosophical arguments. The phrase originates with Greek, but the common use today is from Latin: quod erat demonstrandum. From wiki:
Q.E.D. or QED is an initialism of the Latin phrase quod erat demonstrandum, meaning “which was to be demonstrated”. Literally it states “what was to be shown”. Traditionally, the abbreviation is placed at the end of mathematical proofs and philosophical arguments in print publications, to indicate that the proof or the argument is complete.
The phrase quod erat demonstrandum is a translation into Latin from the Greek ὅπερ ἔδει δεῖξαι (hoper edei deixai; abbreviated as ΟΕΔ). Translating from the Latin phrase into English yields “what was to be demonstrated”. However, translating the Greek phrase ὅπερ ἔδει δεῖξαι can produce a slightly different meaning. In particular, since the verb “δείκνυμι” also means to show or to prove, a different translation from the Greek phrase would read “The very thing it was required to have shown.”
The Latin phrase is attested in a 1501 Euclid translation of Giorgio Valla. Its abbreviation q.e.d. is used once in 1598 by Johannes Praetorius, more in 1643 by Anton Deusing, extensively in 1655 by Isaac Barrow in the form Q.E.D., and subsequently by many post-Renaissance mathematicians and philosophers.