Q.E.D.

This article is about Latin phrase Q.E.D. as used in proofs. For other uses, see QED.

Q.E.D. is an abbreviation of the Latin phrase quod erat demonstrandum, which literally means "which was to be demonstrated". The phrase is written in its abbreviated form at the end of a mathematical proof or philosophical argument to signify that the last statement deduced was the one to be demonstrated; the abbreviation thus signals the completion of the proof.

Etymology and early use

The phrase is a translation into Latin from Greek [ὅπερ ἔδει δεῖξαι] Error: {{Lang}}: text has italic markup (help) ([hoper edei deixai] Error: {{Lang}}: text has italic markup (help); abbreviated as ΟΕΔ), a phrase used by many early mathematicians, including Euclid and Archimedes. These mathematicians, in particular Euclid, are credited with founding axiomatic mathematics with its emphasis on establishing truths by logical deduction (rather than experimentation or assertion); their use of this phrase symbolizes this emphasis, as well as marking this important step in the development of mathematical philosophy.

Modern philosophy

Philippe van Lansberge's 1604 Triangulorum Geometræ used [quod erat demonstrandum] Error: {{Lang}}: text has italic markup (help) to conclude some proofs; others ended with phrases such as [figillatim deinceps demonstrabitur] Error: {{Lang}}: text has italic markup (help), [magnitudo demonstranda est] Error: {{Lang}}: text has italic markup (help), and other variants.

In 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 his magnum opus. The style and system of the book is, 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.

Q.E.F.

There is another Latin phrase with a slightly different meaning, and less common in usage. [Quod erat faciendum] Error: {{Lang}}: text has italic markup (help) is translated as "which was to have been done." This is usually shortened to Q.E.F. As with Q.E.D., Q.E.F. is a translation of the Greek geometers' closing [ὅπερ ἔδει ποιῆσαι] Error: {{Lang}}: text has italic markup (help) (hoper edei poiēsai). Euclid used this phrase to close propositions which were not proofs of theorems, but constructions. For example, Euclid's first proposition shows how to construct an equilateral triangle given one side.

Equivalents in other languages

Q.E.D. has acquired many translations in various foreign languages. In French, German, Italian, and Russian (with English, the main languages of Western mathematics) it is respectively C.Q.F.D., for [ce qu'il fallait démontrer] Error: {{Lang}}: text has italic markup (help) (or sometimes [ce qui finit la démonstration] Error: {{Lang}}: text has italic markup (help)), W.Z.B.W. for [was zu beweisen war] Error: {{Lang}}: text has italic markup (help), C.V.D. for [come volevasi dimostrare] Error: {{Lang}}: text has italic markup (help), and ч.т.д., for [что и требовалось доказать] Error: {{Lang}}: text has italic markup (help). There does not appear to be a common formal English equivalent, though the end of a proof may be announced with a simple statement such as "this completes the proof" or a similar locution. The equivalent in Ancient Greek was [ὅπερ ἔδει δεῖξαι] Error: {{Lang}}: text has italic markup (help). In modern Greek texts sometimes the initials ο.ε.δ. are used at the end of a mathematical proof.

Electronic forms

When typesetting was done by a compositor with letterpress printing, complex typography such as mathematics and foreign languages were called "penalty copy" (the author paid a "penalty" to have them typeset, as it was harder than plain text). With the advent of systems such as LaTeX, mathematicians found their options more open, so there are several symbolic alternatives in use, either in the input, the output, or both. Knuth, for example, developed TeX as a sideline, or perhaps diversion, to typeset Art of Computer Programming as he was upset with the standards of mathematical typography in journals in which he uses ■ (solid black square), also called by mathematicians tombstone or Halmos symbol (after Paul Halmos, who pioneered its use).

The tombstone is sometimes open: □ (hollow black square). Unicode explicitly provides the "End of Proof" character U+220E (∎), but also offers ▮ (U+25AE, black vertical rectangle) and ‣ (U+2023, triangular bullet) as alternatives. Some authors have adopted variants of this notation with other symbols, such as two forward slashes (//), or simply some vertical white space.

External links

• Earliest Known Uses of Some of the Words of Mathematics (Q)
• Robin Hartshorne note on QED