QED

Here is a math trivia question for the day.

How and when did "QED" achieve its position as preeminent proof-ender?

I had always supposed that it must have been the standard ending for proofs in Latin, but a random survey of some early editions of Euclid's Elements proves otherwise.

In the authoritative edition of Commandino (1572), QED does not appear at all (as far as I looked). He prefers the impersonal "oportet" , "it is fitting or proper", coupled with the present or perfect infinitive, as in

• quod fecisse oportebat
  
• quod facere oportebat
  
• quod ostendere oportebat
  
• quod demonstrare oportuit
  
• quod demonstrasse oportuit.
  

Scheubel (1550) uses passive infinitives, as in

• quod demonstrari oportuit
  
• quod fieri oportuit.
  

Campanus (from the 1516 edition) uses most often

• quod est propositum.
  

The flamboyant Oronce Fine (1544) uses no less than 24 different expressions to end the proofs of the 48 propositions of Book I. Among these, our

• quod erat demonstrandum
  

with the gerundive expressing obligation or necessity, appears twice. Some of his other phrases are

• quod fuerat ostendendum
  
• quod faciendum susceperamus
  
• quod demonstrandum fuerat
  
• quod expediebat demonstrare
  
• quod demonstrare fuerat operaepretium
  
• quod faciendum proposueramus
  

and many others. Peletier (1557) uses some of the above, and a few others such as

• quod esse non potest
  
• quod erat constitutum
  
• sicque patet propositio.
  

By the time of Clavius (3rd ed. 1591),

• quod erat faciendum, and
  
• quod erat demonstrandum
  

have become predominant, still with other formulae mixed in, such as

• quod demonstrandum proponebatur.
  

Isaac Barrow (1678; first published 1655) who introduced many symbols into his proofs, also introduced abbreviations of proof endings after the first occurrence, and limits himself to these four:

• quod erat faciendum (subsequently Q.E.F. )
  
• quod erat demonstrandum (Q.E.D.)
  
• quod fieri nequit (Q.F.N.)
  
• quod est absurdum (Q.E.A.).
  

So is it to Barrow that we owe the use of the abbreviation QED? And how about reviving the latter two abbreviations, which seem to have fallen into disuse?

Baermann (1743) , who acknowledges his debt to Barrow, carries on with the systematic use of Q.E.F., Q.E.D., and Q.E.A.