Q.E.D. or QED is an initialism of the Latin phrase , 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. Etymology and early use The phrase quod erat demonstrandum is a translation into Latin from the Greek (; abbreviated as ΟΕΔ). Translating from the Latin phrase into English yields "what was to be demonstrated". However, translating the Greek phrase can produce a slightly different meaning. In particular, since the verb 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 Greek phrase was used by many early Greek mathematicians, including Euclid and Archimedes. 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. Modern philosophy During the European Renaissance, scholars often wrote in Latin, and phrases such as Q.E.D. were often used to conclude proofs. Perhaps the most famous use of Q.E.D. in a philosophical argument is found in the Ethics of Baruch Spinoza, published posthumously in 1677. Written in Latin, it is considered by many to be Spinoza's magnum opus. The style and system of the book are, as Spinoza says, "demonstrated in geometrical order", with axioms and definitions followed by propositions. For Spinoza, this is a considerable improvement over René Descartes's writing style in the Meditations, which follows the form of a diary. Difference from Q.E.F. There is another Latin phrase with a slightly different meaning, usually shortened similarly, but being less common in use. , originating from the Greek geometers' closing (), meaning "which had to be done". Because of the difference in meaning, the two phrases should not be confused. Euclid used the Greek original of Quod Erat Faciendum (Q.E.F.) to close propositions that were not proofs of theorems, but constructions of geometric objects. For example, Euclid's first proposition showing how to construct an equilateral triangle, given one side, is concluded this way. English equivalent There is no common formal English equivalent, although the end of a proof may be