% ----------------------------------------------------------------------

\chapter{Manuscript history}
\label{hist}
\index{Black, Thaddeus~H.\ (1967--)}

The book in its present form is based on various unpublished drafts and
notes of mine, plus a few of my wife Kristie's (n\'ee Hancock), going
back to 1983 when I was fifteen years of age.  What prompted the contest
I can no longer remember, but the notes began one day when I challenged
a high-school classmate to prove the quadratic formula.  The classmate
responded that he didn't need to prove the quadratic formula because the
proof was in the class' math textbook, then counterchallenged me to prove
the Pythagorean theorem.  Admittedly obnoxious (I was fifteen, after
all) but not to be outdone, I whipped out a pencil and paper on the spot
and started working.  But I found that I could not prove the theorem
that day.

The next day I did find a proof in the school's library,%
\footnote{
  A better proof is found in \S~\ref{alggeo:223} and the introduction to
  chapter~\ref{intro}.
}
writing it down, adding to it the proof of the quadratic formula plus
a rather inefficient proof of my own invention to the law of cosines.
Soon thereafter the school's chemistry instructor happened to mention
that the angle between the tetrahedrally arranged four carbon-hydrogen
bonds in a methane molecule was~$109^\circ$, so from a symmetry argument
I proved that result to myself, too, adding it to my little collection
of proofs.  That is how it started.%
\footnote{
  Fellow gear-heads who lived through that era at about the same age
  might want to date me against the disappearance of the slide rule.
  Answer: in my country, or at least at my high school, I was three
  years too young to use a slide rule.  The kids born in 1964 learned
  the slide rule; those born in 1965 did not.  I wasn't born till 1967,
  so for better or for worse I always had a pocket calculator in high
  school.  My family had an eight-bit computer at home, too, as we shall
  see.
}

The book actually has earlier roots than these.  In 1979, when I was
twelve years old, my father bought our family's first eight-bit
computer.  The computer's built-in \emph{BASIC} programming-language
interpreter exposed functions for calculating sines and cosines of
angles.  The interpreter's manual included a diagram much like
Fig.~\ref{trig:226:f1} showing what sines and cosines were, but it never
explained how the computer went about calculating such quantities.  This
bothered me at the time.  Many hours with a pencil I spent trying to
figure it out, yet the computer's trigonometric functions remained
mysterious to me.  When later in high school I learned of the use of the
Taylor series to calculate trigonometrics, into my growing collection of
proofs the series went.

Five years after the Pythagorean incident I was serving the U.S.\ Army
as an enlisted troop in the former West Germany.  Although those were
the last days of the Cold War, there was no shooting war at the time, so
the duty was peacetime duty.  My duty was in military signal
intelligence, frequently in the middle of the German night when there
often wasn't much to do.  The platoon sergeant wisely condoned neither
novels nor playing cards on duty, but he did let the troops read the
newspaper after midnight when things were quiet enough.  Sometimes I
used the time to study my German---the platoon sergeant allowed this,
too---but I owned a copy of Richard~P.\ Feynman's \emph{Lectures on
Physics}~\cite{Feynman} which sometimes I would read instead.

Late one night the battalion commander, a lieutenant colonel and West
Point graduate, inspected my platoon's duty post by surprise.  A
lieutenant colonel was a highly uncommon apparition at that hour of a
quiet night, so when that old man appeared suddenly with the sergeant
major, the company commander and the first sergeant in tow---the last
two just routed from their sleep, perhaps---surprise indeed it was.  The
colonel may possibly have caught some of my unlucky fellows playing
cards that night---I am not sure---but me, he caught with my boots
unpolished, reading the \emph{Lectures.}

I snapped to attention.  The colonel took a long look at my boots
without saying anything, as stormclouds gathered on the first sergeant's
brow at his left shoulder, and then asked me what I had been reading.

``Feynman's \emph{Lectures on Physics,} sir.''

``Why?''

``I am going to attend the university when my three-year enlistment is
up, sir.''

``I see.''  Maybe the old man was thinking that I would do better as a
scientist than as a soldier?  Maybe he was remembering when he had had
to read some of the \emph{Lectures} himself at West Point.  Or maybe it
was just the singularity of the sight in the man's eyes, as though he
were a medieval knight at bivouac who had caught one of the peasant
levies, thought to be illiterate, reading Cicero in the original Latin.
The truth of this, we shall never know.  What the old man actually said
was, ``Good work, son.  Keep it up.''

The stormclouds dissipated from the first sergeant's face.  No one ever
said anything to me about my boots (in fact as far as I remember, the
first sergeant---who saw me seldom in any case---never spoke to me
again).  The platoon sergeant thereafter explicitly permitted me to read
the \emph{Lectures} on duty after midnight on nights when there was
nothing else to do, so in the last several months of my military service
I did read a number of them.  It is fair to say that I also kept my
boots better polished.

In Volume I, Chapter 6, of the \emph{Lectures} there is a lovely
introduction to probability theory.  It discusses the classic problem of
the ``random walk'' in some detail and then states without proof that the
generalization of the random walk leads to the Gaussian distribution
(\S~\ref{prob:100}),
\[
  \Omega(x) = \frac{\exp(-x^2/2\sigma^2)}{\sigma\sqrt{2\pi}}.
\]
For the derivation of this remarkable theorem, I scanned the book in
vain.  One had no Internet access in those days, but besides a well-equipped
gym the Army post also had a tiny library, and in one yellowed
volume in the library---who knows how such a book got there?---I did
find a derivation of the $1/\sigma\sqrt{2\pi}$ factor.%
\footnote{
  The citation is now unfortunately long lost.
}
The exponential factor, the volume did not derive.  Several days later,
I chanced to find myself in Munich with an hour or two to spare, which I
spent in the university library seeking the missing part of the proof,
but lack of time and unfamiliarity with such a German site defeated me.
Back at the Army post, I had to sweat the proof out on my own over the
ensuing weeks.  Nevertheless, eventually I did obtain a proof which made
sense to me.  Writing the proof down carefully, I pulled the old
high-school math notes out of my military footlocker (for some reason I
had kept the notes and even brought them to Germany), dusted them off,
and added to them the new Gaussian proof.

That is how it has gone.  To the old notes, I have added new proofs from
time to time, and though I have somehow misplaced the original
high-school leaves I took to Germany with me the notes have nevertheless
grown with the passing years.  These years have brought me the
good things years can bring: marriage, family and career; a good life
gratefully lived, details of which interest me and mine but are mostly
unremarkable as seen from the outside.  A life however can take strange
turns, reprising earlier themes.  I had become an industrial building
construction engineer for a living
(and, appropriately enough, had most
lately added to the notes a mathematical justification of the standard
industrial building construction technique to measure the
resistance to ground of a new building's electrical grounding
system\footnote{%
  % diagn: check this new footnote
  The resistance-to-ground technique is too specialized to find place
  in this book.%
}),
when at a juncture between construction projects an unexpected
opportunity arose to pursue graduate work in engineering at Virginia Tech,
courtesy (indirectly, as it developed) of a research program not of the
United States Army as last time but this time of the United States Navy.
The Navy's research problem turned out to be in the highly mathematical
fields of theoretical and computational electromagnetics.  Such work
naturally brought a blizzard of new formulas, whose proofs I sought or
worked out and, either way, added to the notes---whence the manuscript
and, in due time, this book.

%I was delighted to
%discover that Eric W. Weisstein had compiled and published~\cite{EWW} a
%wide-ranging collection of mathematical results in a spirit not
%entirely dissimilar to that of my own notes.  A significant difference
%remained, however, between Weisstein's work and my own.  The difference
%was and is fourfold:
%\begin{enumerate}
%  \item Number theory, mathematical recreations and odd mathematical
%    \linebreak % b-a-d b-r-e-a-k
%    names interest Weisstein much more than they interest me; my own
%    tastes run toward math directly useful in known physical
%    applications.  The selection of topics in each body of work reflects
%    this difference.
%  \item Weisstein often includes results without proof.  This is fine,
%    but for my own part I happen to like proofs.
%  \item Weisstein lists results encyclopedically, alphabetically by
%    name.  I organize results more traditionally by topic, leaving
%    alphabetization to the book's index, that readers who wish to do so
%    can coherently read the book from front to back.%
%    \footnote{
%      There is an ironic personal story in this.  As children in the
%      1970s, my brother and I had a 1959 World Book encyclopedia in our
%      bedroom, about twenty volumes.  It was then a bit outdated (in
%      fact the world had changed tremendously in the fifteen or twenty
%      years following 1959, so the encyclopedia was more than a bit
%      outdated) but the two of us still used it sometimes.  Only years
%      later did I learn that my father, who in 1959 was fourteen years
%      old, had bought the encyclopedia with money he had earned
%      delivering newspapers daily before dawn, \emph{and then had read
%      the entire encyclopedia, front to back.} My father played
%      linebacker on the football team and worked a job after school,
%      too, so where he found the time or the inclination to read an
%      entire encyclopedia, I'll never know.  Nonetheless, it does prove
%      that even an encyclopedia can be read from front to back.
%    }
%  \item I have eventually developed an interest in the free-software
%    movement, joining it as a Debian Developer~\cite{Debian}; and by
%    these lights and by the standard of the Debian Free Software
%    Guidelines (DFSG)~\cite{DFSG}, Weisstein's work is not free.  No
%    objection to non-free work as such is raised here, but the book you
%    are reading \emph{is} free in the DFSG sense.
%\end{enumerate}
%A different mathematical reference, even better in some ways than
%Weisstein's and (if I understand correctly) indeed free in the DFSG
%sense, is emerging at the time of this writing in the on-line pages of
%the general-purpose encyclopedia Wikipedia~\cite{wikip}\@.  Although
%Wikipedia reads unevenly, remains generally un\-cit\-able,%
%\footnote{
%  Some ``Wikipedians'' do seem actively to be working on making
%  Wikipedia authoritatively citable.  The underlying philosophy and
%  basic plan of Wikipedia admittedly tend to thwart their efforts, but
%  their efforts nevertheless seem to continue to progress.  We shall
%  see.  Wikipedia is a remarkable, monumental creation.
%}
%forms no coherent whole, and seems to suffer a certain
%competition among some of its mathematical editors as to which of them
%can explain a thing most reconditely, it is laden with mathematical
%knowledge, including many proofs, which I have referred to more than a
%few times in the preparation of this text.

The book follows in the honorable tradition of Courant's and
Hilbert's
% bad break
1924 classic \emph{Methods of Mathematical
Physics}~\cite{Courant/Hilbert}---a
tradition subsequently developed by, among others,
Jeffreys and Jeffreys~\cite{Jeffreys/Jeffreys},
Arfken and Weber~\cite{Arfken/Weber},
and Weisstein%
\footnote{
  Weisstein lists results encyclopedically, alphabetically by
  name.  I organize results more traditionally by topic, leaving
  alphabetization to the book's index, that readers who wish to do so
  can coherently read the book from front to back.

  There is an ironic personal story in this.  As children in the
  1970s, my brother and I had a 1959 World Book encyclopedia in our
  bedroom, about twenty volumes.  The encyclopedia was then a bit
  outdated (in fact the world had changed tremendously during the
  fifteen or twenty years following 1959, so the book was more than a bit
  outdated) but the two of us still used it sometimes.  Only years
  later did I learn that my father, who in 1959 was fourteen years
  old, had bought the encyclopedia with money he had earned
  delivering newspapers daily before dawn, \emph{and then had read
  the entire encyclopedia, front to back.}  My father played
  linebacker on the football team and worked a job after school,
  too, so where he found the time or the inclination to read an
  entire encyclopedia, I'll never know.  Nonetheless, it does prove
  that even an encyclopedia can be read from front to back.
}~\cite{EWW}.
The present book's chief intended contribution to the tradition lies in
its applied-level derivations of the many results it presents.  Its
author always wanted to know why the Pythagorean theorem was so.  The
book is presented in this spirit.

%The book is also presented in the spirit of the free-software or
%open-source movement, which I have joined as a Debian
%Developer~\cite{Debian}.  I am an open-source contributor rather than an
%open-source partisan.  Not everything I write is free, but this
%particular work is sufficiently unusual in nature, is sufficiently
%distinctive in composition, and represents such a personal investment
%that one feels reluctant to risk entrusting it to a publisher who might
%pitch it to the wrong market and then, impatiently, let the book go out
%of print.  Publishers after all have their own problems.  Besides,
%desiring broad distribution even in the author's own country, one would
%rather not price the book at~\$~100 a copy in the United States and~\$~5
%a copy in China.  Open source thus makes
%sense for this book.  (Naturally, the publication of this or any book by
%open source subjects the book to the not unreasonable criticism that the
%book might never have been good enough to publish.  I do not believe
%that that is remotely the case here, but the reader, having been spared
%the book's presumed~\$~100 price, must be the judge of that.)

A book can follow convention or depart from it; yet, though occasional
departure might render a book original, frequent departure seldom
renders a book good.  Whether this particular book is original or
good, neither or both, is for the reader to tell, but in any case the
book does both follow and depart.  Convention is a peculiar thing: at
its best, it evolves or accumulates only gradually, patiently storing up
the long, hidden wisdom of generations past; yet herein arises the
ancient dilemma.  Convention, in all its richness, in all its
profundity, can, sometimes, stagnate at a local maximum, a hillock
whence higher ground is achievable not by gradual ascent but only by
descent first---or by a leap.  Descent risks a bog.  A leap risks a
fall.  One ought not run such risks without cause, even in such an
inherently unconservative discipline as mathematics.

Well, the book does risk.  It risks one leap at least: it employs
hexadecimal numerals.

This book is bound to lose at least a few readers for its unorthodox
use of hexadecimal notation (``The first primes are
$2,3,5,7,\mbox{0xB},\ldots$'').  Perhaps it will gain a few readers for
the same reason; time will tell.  I started keeping my own theoretical
math notes in hex a long time ago; at first to prove to myself that I
could do hexadecimal arithmetic routinely and accurately with a pencil,
later from aesthetic conviction that it was the right thing to do.  Like
other applied mathematicians, I've several own private notations, and in
general these are not permitted to burden the published text.  The hex
notation is not my own, though.  It existed before I arrived on the
scene and, since I know of no math book better positioned to risk its
use, I have with hesitation and no little trepidation resolved to let
this book use it.  Some readers will approve; some will tolerate;
undoubtedly some will do neither.  The views of the last group must be
respected, but in the meantime the book has a mission; and crass
popularity can be only one consideration, to be balanced against other
factors.  The book might gain even more readers, after all, had it no
formulas, and painted landscapes in place of geometric diagrams!  I like
landscapes, too, but anyway you can see where that line of logic leads.

More substantively: despite the book's title and despite the brief
philosophical discussion in its chapter~\ref{intro}, adverse criticism from
some quarters for lack of rigor is probably inevitable; nor is such
criticism necessarily improper from my point of view.  Still, serious
books by professional mathematicians tend to be \emph{for} professional
mathematicians, which is understandable but does not always help the
scientist or engineer who wants to use the math to model something.  The
ideal author of such a book as this would probably hold two doctorates:
one in mathematics and the other in engineering or the like.  The ideal
author lacking, I have written the book.

So here you have my old high-school notes, extended over
% diagn: how many years now?
thirty years---years that include professional engineering practice and
university study, research and teaching---now partly
typed and revised for the first time as a \LaTeX\ manuscript.  Where
this manuscript will go in the future is hard to guess.  Perhaps the
revision you are reading is the last.  Who can say?  The manuscript
met an uncommonly enthusiastic reception at Debconf~6~\cite{Debian}
May~2006 at Oaxtepec, Mexico---a reception that, as far as it goes,
augurs well for the book's future at least.  In the meantime, if
anyone should challenge you to prove the Pythagorean theorem on the
spot, why, whip this book out and turn to chapter~\ref{intro}.  That
should confound 'em.

\nopagebreak

\noindent\\
THB

