\chapter{The Greek alphabet}
\label{greek}
\index{Greek alphabet}
\index{Roman alphabet}

Mathematical experience finds the Roman alphabet to lack sufficient
symbols to write higher mathematics fluently.  Though not completely
solving the problem, the addition of the Greek alphabet helps.  See
Table~\ref{greek:288:greek}.%
\begin{table}
  \caption{The Roman and Greek alphabets.}
  \label{greek:288:greek}
  \bc
    \begin{tabular}{rlrlrlrl}
      \multicolumn{8}{c}{ROMAN}\\
      $Aa$ &Aa& $Gg$ &Gg& $Mm$ &Mm& $Tt$ &Tt \\
      $Bb$ &Bb& $Hh$ &Hh& $Nn$ &Nn& $Uu$ &Uu \\
      $Cc$ &Cc& $Ii$ &Ii& $Oo$ &Oo& $Vv$ &Vv \\
      $Dd$ &Dd& $Jj$ &Jj& $Pp$ &Pp& $Ww$ &Ww \\
      $Ee$ &Ee& $Kk$ &Kk& $Qq$ &Qq& $Xx$ &Xx \\
      $Ff$ &Ff& $L\ell$ &Ll& $Rr$ &Rr& $Yy$ &Yy \\
         &&    && $Ss$ &Ss& $Zz$ &Zz \\
      &&&&&&&\\
      \multicolumn{8}{c}{GREEK}\\
      A$\alpha$ &alpha& H$\eta$ &eta&
        N$\nu$ &nu& T$\tau$ &tau\\
      B$\beta$ &beta& $\Theta\theta$ &theta&
        $\Xi\xi$ &xi& $\Upsilon\upsilon$ &upsilon\\
      $\Gamma\gamma$ &gamma& I$\iota$ &iota&
        O$o$ &omicron& $\Phi\phi$ &phi\\
      $\Delta\delta$ &delta& K$\kappa$ &kappa&
        $\Pi\pi$ &pi& X$\chi$ &chi\\
      E$\epsilon$ &epsilon& $\Lambda\lambda$ &lambda&
        P$\rho$ &rho& $\Psi\psi$ &psi\\
      Z$\zeta$ &zeta& M$\mu$ &mu&
        $\Sigma\sigma$ &sigma& $\Omega\omega$ &omega\\
    \end{tabular}
  \ec
\end{table}

When first seen in English-language mathematical writing,
the Greek letters can seem to take on a
wise, mysterious aura.  Nevertheless, the Greek letters are just
letters.  We use them not because we want to be wise and mysterious%
\footnote{
  Well, you can use them to be wise and mysterious if you want to.  It's
  kind of fun, actually, when you're dealing with someone who doesn't
  understand math---if what you want is for him to go away and leave you
  alone.  Otherwise, we tend to use Roman and Greek letters in various
  conventional ways: Greek minuscules (lower-case letters) for angles;
  Roman capitals for matrices;~$e$ for the natural logarithmic base;~$f$
  and~$g$ for unspecified functions;~$i$, $j$, $k$, $m$, $n$, $M$
  and~$N$ for integers;~$P$ and~$Q$ for logical propositions and
  metasyntactic elements;
  %(the mathematical equivalents of \texttt{foo} and \texttt{bar});
  $t$, $T$ and~$\tau$ for time;~$d$, $\delta$ and~$\Delta$ for
  change;~$A$, $B$ and~$C$ for indeterminate coefficients; etc.
}
but rather because we simply do not have enough Roman letters.  An
equation like
\[
  \alpha^2 + \beta^2 = \gamma^2
\]
says no more than does an equation like
\[
  a^2 + b^2 = c^2,
\]
after all.  The letters are just different (though naturally one prefers
to use the letters one's audience expects when one can).

Applied as well as professional mathematicians tend to use Roman and
Greek letters in certain long-established conventional sets:~$abcd$;
$fgh$; $ijk\ell$; $mn$;
$pqr$; $st$; $uvw$; $xyz$.  For the Greek:~$\alpha\beta\gamma$;
$\delta\epsilon$; $\kappa\lambda\mu\nu\xi$; $\rho\sigma\tau$;
$\phi\chi\psi\omega$.
Greek letters are
frequently paired with their Roman congeners as appropriate: $a\alpha$;
$b\beta$; $cg\gamma$; $d\delta$; $e\epsilon$; $f\phi$; $k\kappa$;
$\ell\lambda$; $m\mu$; $n\nu$; $p\pi$; $r\rho$; $s\sigma$; $t\tau$;
$x\chi$; $z\zeta$.%
\footnote{
  The capital pair~$Y\Upsilon$ is occasionally seen but is awkward both
  because the Greek minuscule~$\upsilon$ is visually almost
  indistinguishable from the unrelated (or distantly related) Roman
  minuscule~$v$; and because the ancient Romans regarded the letter~$Y$
  not as a congener but as the Greek letter itself, seldom used but to
  spell Greek words in the Roman alphabet.  To use~$Y$ and~$\Upsilon$ as
  separate symbols is to display an indifference to, easily
  misinterpreted as an ignorance of, the Graeco-Roman sense of the
  thing---which is silly, arguably, if you think about it, since no one
  objects when you differentiate~$j$ from~$i$, or~$u$ and~$w$
  from~$v$---but, anyway, one is probably the wiser to tend to limit the
  mathematical use of the symbol~$\Upsilon$ to the very few instances in
  which established convention decrees it.  (In English particularly,
  there is also an old typographical ambiguity between~$Y$ and a
  Germanic, non-Roman letter named ``thorn'' which has practically
  vanished from English today, to the point that the typeface in which
  you are reading these words lacks a glyph for it---but which
  sufficiently literate writers are still expected to recognize on
  sight.  This is one more reason to tend to avoid~$\Upsilon$ when you
  can, a Greek letter that makes you look ignorant when you use it wrong
  and pretentious when you use it right.  You can't win.)

  The history of the alphabets is extremely interesting.  Unfortunately,
  a footnote in an appendix to a book on derivations of applied
  mathematics is probably not the right place for an essay on the topic,
  so we'll let the matter rest there.
}

Mathematicians usually avoid letters like the Greek capital~H (eta),
which looks just like the Roman capital~H, even though~H (eta) is an
entirely proper member of the Greek alphabet.  The Greek
minuscule~$\upsilon$ (upsilon) is avoided for like reason, for
mathematical symbols are useful only insofar as we can visually tell
them apart.  Interestingly, however, the Greek minuscules~$\nu$ (nu)
and~$\omega$ (omega) are often used in applied mathematics, so one needs
to learn to distinguish those ones from the Roman~$v$ and~$w$.

