Q.E.D.
B. T. Raven
esse velle.
Beer is living proof that God loves us and wants us to be happy.
Benjaminus Franklin
Johannes Patruus
Mike Lyle
The URL didn't work for me, but I hope the site includes the medievalism
"Goddesgode", for yeast. (I may be spelling it very wrongly, as it won't
come up on an OED search.)
--
Mike.
Johannes Patruus
For the spellings "goddis good" & "godesgood" see OED s.v. "God's good",
sense 3. (We of this Blighty'd Isle enjoy free online access to it!)
On retesting the link, it displayed instantly, and I reproduce its content
below.
Patruus
**************
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.
Robin Hartshorne
Berkeley
Ed Cryer
"Johannes Patruus" <inv...@invalid.invalid> wrote in message
news:6dn768F...@mid.individual.net...
Euclid used ὅπερ ἔδει δειξαι and ὅπερ ἔδει ποιησαι, which in Greek is
precisely the difference between "demonstrandum" and "faciendum".
The former was always used of proofs proper. The latter "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."
Baruch Spinoza, the philosopher, wrote his "Ethica Ordine Geometrico
Demonstrata" in 1677, and always ended his proofs with "Q.E.D".
Ed
John Briggs
> Mike Lyle wrote:
>> Johannes Patruus wrote:
>>> B. T. Raven wrote:
>>>> Cerevisia pro demonstatione vivida est Deum nos amare nosque
>>>> felices esse velle.
>>>>
>>>> Beer is living proof that God loves us and wants us to be happy.
>>>>
>>>> Benjaminus Franklin
>>> Q.E.D. variants - http://robin.hartshorne.net/QED.html
>>
>> The URL didn't work for me, but I hope the site includes the
>> medievalism "Goddesgode", for yeast. (I may be spelling it very
>> wrongly, as it won't come up on an OED search.)
>
> For the spellings "goddis good" & "godesgood" see OED s.v. "God's
> good", sense 3. (We of this Blighty'd Isle enjoy free online access
> to it!)
> On retesting the link, it displayed instantly, and I reproduce its
> content below.
Have a glance at the Wikipedia entry.
--
John Briggs
Mike Lyle
> Mike Lyle wrote:
>> Johannes Patruus wrote:
>>> B. T. Raven wrote:
>>>> Cerevisia pro demonstatione vivida est Deum nos amare nosque
>>>> felices esse velle.
>>>>
>>>> Beer is living proof that God loves us and wants us to be happy.
>>>>
>>>> Benjaminus Franklin
>>> Q.E.D. variants - http://robin.hartshorne.net/QED.html
>>
>> The URL didn't work for me, but I hope the site includes the
>> medievalism "Goddesgode", for yeast. (I may be spelling it very
>> wrongly, as it won't come up on an OED search.)
>
> For the spellings "goddis good" & "godesgood" see OED s.v. "God's
> good", sense 3. (We of this Blighty'd Isle enjoy free online access
> to it!)
Ah, thanks: I should have thought of looking at a modern spelling before
giving up. I never thought I'd get used to the online form, but, yes,
Glos Library's eleemosynary OED has almost completely displaced my old
compact codex version of the first ed.
--
Mike.