FOM: Euler's use of the Latin equivalent of the word ``arbitrary''

[Insall]

The following message is Ed Sandifer's reply to my initial enquiry on HM
about Euler's use of the word ``arbitrary''.

Matt

<Begin Quote>

Matt Insall asks about Euler's use of "arbitrary"

        Euler wrote mathematics mostly in Latin.  The Latin word he used
when
we mathematicians might use the word "arbitrary" was "quodlibet", which
translates best as "whatever."

        Latin has a richer arsenal of relative pronouns than does English.
We
have little choice beyond "this" and "that".  We used to have more, and our
seldom used terms "former" and "latter" are remnants of a relatively richer
population of pronouns.  (English hardly ever actually loses a word.)

        So, when you read mathematics in Latin, you read of "this curve",
"that
curve", "the curve before that one", "this very curve", "some curve",
"whatever
curve", all with their own relative pronouns.  They used relative pronouns a
lot, and it would have been awkward writing not to use them.  Though
logically,
expressions we would translate as "a curve" or "whatever curve" might be
equivalent, the second form would be better Latin.  And Euler's Latin was
very
good.

        Implicit in Matt's question is whether or not Euler used "quodlibet"
as
a surrogate quantifier.  The best answer would be, he did, but only in
retrospect.  Quantifiers wouldn't be formalized for at least another 100
years,
and it would be better to say that they clarified Euler's (and others') use
of
the terms rather than saying Euler used the terms instead of using
quantifiers.

        Another important aspect of Euler's nature is that he would change
the
way he did and said things, usually improving them.  Early, he spoke of
curves
and equations.  Later, he clarified his ideas and spoke instead of
functions.
Early, he used whatever symbol was convenient for the values we now denote
by
e, pi and i.  Later, he realized the value of consistency and he
standardized
his notation.

        Yet another thing to know about the times is that they had not yet
settled on definitions as precise as the ones we use today.  For example,
Euler
uses the word "algebraic" to mean "expressible by radicals" and also to mean
"expressible as the root of a polynomial" without worrying about whether the
two are the same.  He uses "transcendental" to mean "not algebraic" in one
case, and in another to mean "not algebraic or exponential" in another case.

        Euler, of course, would never have been able to speak of an
"arbitrary
group".  Though he did prove that the quadratic residues mod p form a
subgroup
of index 2 of the group of units mod p, he did it without having a notion of
a
"group."

        I hope this helps address some of your questions.

Ed Sandifer


<End Quote>