quod erat demonstrandum

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 (hoper edei deixai; abbreviated as ΟΕΔ), a phrase used by many early mathematicians, including Euclid^[1] 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 to conclude some proofs; others ended with phrases such as figillatim deinceps demonstrabitur, magnitudo demonstranda est, and other variants.[2]

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 his magnum opus^{[citation needed]}. 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.^[3]

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. As with Q.E.D., Q.E.F. is a translation of the Greek geometers' closing ὅπερ ἔδει ποιῆσαι (hoper edei poiēsai). Euclid used this phrase to close propositions which were not precisely proofs, but rather exemplar constructions. The distinction between Q.E.D. and Q.E.F. is roughly equivalent to the distinction between a proof and an illustration of the proof.

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 (or sometimes ce qui finit la démonstration), W.Z.B.W. for was zu beweisen war, C.V.D. for come volevasi dimostrare, and ч.т.д., for что и требовалось доказать. 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. In modern Greek texts sometimes the initials ο.ε.δ. are used at the end of a mathematical proof.

RONGGOWARSITO : SERAT JOKO LODANG

SERAT JOKO LODANG
Megatruh

1. Mbok Parawan sangga wang duhkiteng kalbu
Jaka Lodang nabda malih
Nanging ana marmanipun
Ing waca kang wus pinesthi
Estinen murih kelakon

Mendengar segalanya itu Mbok Perawan merasa sedih. Kemudian Joko Lodang berkata lagi :
"Tetapi ketahuilah bahwa ada hukum sebab musabab, didalam ramalan yang sudah ditentukan haruslah diusahakan supaya segera dan dapat terjadi ".

2. Sangkalane maksih nunggal jamanipun
Neng sajroning madya akir
Wiku Sapta ngesthi Ratu
Adil parimarmeng dasih
Ing kono kersaning Manon

Jamannya masih sama pada akhir pertengahan jaman. Tahun Jawa 1877 (Wiku=7, Sapta=7, Ngesthi=8, Ratu=1). Bertepatan dengan tahun Masehi 1945. Akan ada keadilan antara sesama manusia. Itu sudah menjadi kehendak Tuhan.

3. Tinemune wong ngantuk anemu kethuk
Malenuk samargi-margi
Marmane bungah kang nemu
Marga jroning kethuk isi
Kencana sesotya abyor

Diwaktu itulah seolah-olah orang yang mengantuk mendapat kethuk (gong kecil)
yang berada banyak dijalan. Yang mendapat gembira hatinya sebab didalam benda tersebut
isinya tidak lain emas dan kencana.