Q.E.D.

Q.E.D. is an initialism of the Latin phrase quod erat demonstrandum, which means "that which was to be demonstrated". The phrase is traditionally placed in its abbreviated form at the end of a mathematical proof or philosophical argument when that which was specified in the enunciation, and in the setting-out, has been exactly restated as the conclusion of the demonstration.^[1] The abbreviation thus signals the completion of the proof.

Etymology and early use

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 into English yields, "that which was to have been demonstrated"; however, translating the Greek phrase ὅπερ ἔδει δεῖξαι produces a slightly different meaning. A better translation from the Greek would read, "precisely what was required to be proved."^[2] The phrase was used by many early Greek mathematicians, including Euclid^[3] 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 Geometriæ used quod erat demonstrandum to conclude some proofs; others ended with phrases such as sigillatim deinceps demonstrabitur, magnitudo demonstranda est, and other variants.^[4]

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 Spinoza's 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.^[5]

Q.E.F.

There is another Latin phrase with a slightly different meaning, and less common in usage. Quod erat faciendum is translated as "which was to have been done". This is usually shortened to Q.E.F. The expression quod erat faciendum is a translation of the Greek geometers' closing ὅπερ ἔδει ποιῆσαι (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, including:

There is no common formal English equivalent, though the end of a proof may be announced with a simple statement such as "this completes the proof" or w⁵ which means which was what was wanted, or a similar locution. Most modern maths textbooks in English end proofs with a symbol, often a square. (See below.)

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)^[6]. 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. When creating TeX, Knuth provided the symbol ■ (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.

References

External links

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