%% Thanks for your interest in A Gentle Introduction to TeX.  At the            
%% moment it is in draft form, i.e., subject to correction, but it              
%% is pretty close to the final copy.  Any comments on this manual              
%% would be appreciated: these may be typesetting, English, or                  
%% TeX criticisms.                                                              
%%                                                                              
%% This file is a complete TeX input file.  Just run it                         
%% through TeX and print out the resulting "DVI" file. If you are               
%% familiar with TeX, the macros at the top of the file have a few              
%% switches which you may want to set.  If you have problems or                 
%% can't run TeX at all, write to me and I'll send you a hard copy.             
%%                                                                              
%% You should feel free to photocopy and/or distribute this manual.             
%% My only request is that it remain in one piece and not be chopped            
%% up.  The only machine dependent section (#1.2) may need to be                
%% rewritten for your local site, of course.                                    
%%                                                                              
%%%                Michael Doob                                                 
%%                 Department of Mathematics                                    
%%                 The University of Manitoba                                   
%%                 Winnipeg, Manitoba  R3T 2N2                                  
%%                 Canada                                                       
%%                 mdoob@uofmcc                    (bitnet)                     
%%                 mdoob@ccu.umanitoba.ca          (internet)                   
%%                                                                              
%% Here is a character listing to check to be sure that no                      
%% unwanted translations took place within the bowels of the net.               
%% Upper case letters: ABCDEFGHIJKLMNOPQRSTUVWXYZ                               
%% Lower case letters: abcdefghijklmnopqrstuvwxyz                               
%% paretheses, brackets, braces: ()  []  {}                                     
%% Exclaim, at, sharp, dollar, percent: ! @ # $ %                               
%% Caret, ampersand, star, underscore, hyphen: ~ & * _ -                        
%% vertical bar, backslash, tilde, backprime, plus: ! \ ~ ` +                   
%% plus, equal, prime, quote, colon: + = ' " :                                  
%% less than, greater than, slash, question, comma: < > / ? ,                   
%% period, semicolon: . ;                                                       
%%                                                                              
%%                                                                              
                                                                                
                                                                                
                                                                                
                                                                                
%%%%%%%% Here are the fonts other than the sixteen defined in %%%%%%%%          
%%%%%%%%%%%%%        plain.tex that are used        %%%%%%%%%%%%%%%%%%          
                                                                                
       %%%% first, choose between amr or cmr fonts %%%                          
\newif \ifamrfonts                                                              
\amrfontsfalse  % use this line if you use the cmr fonts                        
%\amrfontstrue  % use this line if you use the old armfonts                     
                                                                                
\ifamrfonts \font\brm=amr10 scaled \magstep1                                    
      \else \font\brm=cmr10 scaled \magstep1 \fi                                
\ifamrfonts \font\halfrm=amr10 scaled \magstephalf                              
      \else \font\halfrm=cmr10 scaled \magstephalf \fi                          
\ifamrfonts \font\bbrm=amr10 scaled \magstep2                                   
      \else \font\bbrm=cmr10 scaled \magstep2 \fi                               
\ifamrfonts \font\bbbrm=amr10 scaled \magstep3                                  
      \else \font\bbbrm=cmr10 scaled \magstep3 \fi                              
\ifamrfonts \font\bbbbrm=amr10 scaled \magstep4                                 
      \else\font \bbbbrm=cmr10 scaled \magstep4 \fi                             
\ifamrfonts \font\bbbbbrm=amr10 scaled \magstep5                                
      \else \font\bbbbbrm=cmr10 scaled \magstep5 \fi                            
\ifamrfonts \font\sf = amssmc10                                                 
      \else \font\sf = cmss10 \fi                                               
\ifamrfonts \font\chapfont=ambx10 scaled \magstep2                              
      \else \font\chapfont=cmbx10 scaled \magstep2 \fi                          
\ifamrfonts \font\secfont=ambx10 scaled \magstep1                               
      \else \font\secfont=cmbx10 scaled \magstep1 \fi                           
\ifamrfonts \font\sc= amcsc10                                                   
      \else \font\sc= cmcsc10 \fi                                               
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%           
                                                                                
                                                                                
%%%%\vbadness=10000 %%%%%% I don't want to hear about underfull vboxes          
\raggedbottom                                                                   
\hsize=5.5in                                                                    
\vsize=7in                                                                      
\voffset=.75in                                                                  
\parskip=\baselineskip                                                          
\widowpenalty=1000 \clubpenalty=1000  %%% I hate widows and orphans! %%%        
                                                                                
%%%%%%%%% choosing between Canadian and American spellings %%%%%%%%%            
\newif \ifcanspell                                                              
\canspelltrue      %%%% Canadian spelling                                       
%\canspellfalse    %%% use this line for American spelling                      
                                                                                
\def\aesthetic{\ifcanspell \ae{}sthetic\else esthetic\fi}                       
\def\analogue{\ifcanspell analogue\else analog\fi}                              
     \let\analog=\analogue                                                      
\def\cancelled{\ifcanspell cancelled\else canceled\fi}                          
     \let\canceled=\cancelled                                                   
\def\centimetre{\ifcanspell centimetre\else centimeter\fi}                      
     \let\centimeter=\centimetre                                                
\def\centre{\ifcanspell centre\else center\fi}                                  
     \let\center=\centre                                                        
\def\centred{\ifcanspell centred\else centered\fi}                              
     \let\centered=\centred                                                     
\def\our{\ifcanspell our\else or\fi}                                            
\def\postcode{\ifcanspell postalcode\else zipcode\fi}                           
\def\province{\ifcanspell province\else state\fi}                               
\def\theatre{\ifcanspell theatre\else theater\fi}                               
     \let\theater=\theatre                                                      
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%           
                                                                                
                                                                                
%%%%%%%%%%%%%%%%% headline when in draft mode %%%%%%%%%%%%%%%%%                 
\newif \ifdraft                                                                 
\drafttrue                                                                      
\def\draftheadline{%                                                            
    \vbox {\hrule%                                                              
        \line {\strut \vrule \quad \tenrm DRAFT:                                
              \ifcanspell Canadian \else U.S. \fi spelling                      
              \hfil                                                             
              \ifnum \secnum > 0 Section \the\secnum: \fi \sectiontitle \quad   
              \vrule}%                                                          
        \hrule}%                                                                
}                                                                               
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
                                                                                
%%%%%%%%%%%%%%%%%% supress headline on title pages %%%%%%%%%%%%%%%%%%           
\newif \iftitlepage    \titlepagetrue                                           
\headline=                                                                      
  {\iftitlepage \hfil \global\titlepagefalse \else \draftheadline \fi}          
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%           
                                                                                
%%%%%%%%%%%%%%%%%%%% define contents and index files %%%%%%%%%%%%%%%%%%%%%%     
%%% Normally the contents and index are hard coded into the input file. %%%     
%% To generate new ones, use \writingcontentstrue and \writingindextrue. %%     
                                                                                
\newif \ifwritingcontents                                                       
\newif \ifwritingindex                                                          
\newwrite\contents         \newwrite\index                                      
                                                                                
\writingcontentsfalse      \writingindexfalse                                   
%%% \writingcontentstrue  %%% use this line to make the contents                
%%% \writingindextrue     %%% use this line to make the index                   
                                                                                
\ifwritingcontents \openout\contents=contents.tex     \fi                       
\ifwritingindex \openout\index=index.tex                                        
          \def\toindex#1{\immediate\write\index{#1 \the\pageno}}                
          \else \def\toindex#1{} \fi                                            
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
                                                                                
%%%%%%%%%%%%%%%%%%%% determine whether answers are printed %%%%%%%%%%%%%%%%     
\newif \ifwritinganswers                                                        
\writinganswersfalse        %%% use this line to suppress answer section        
\writinganswerstrue         %%% use this line to include answer section         
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
                                                                                
                                                                                
                                                                                
                                                                                
                                                                                
                                                                                
                                                                                
%%%%%%%%%%%%%%%% exercise, section, and subsection macros %%%%%%%%%%%%          
\newcount\exno                   %%%%%% counter for exercises %%%%%%%%          
\newcount\secnum \secnum=0       %%%% counter for section numbers %%%%          
\newcount\subsecnum                                                             
                                                                                
\def\section#1{                                                                 
            \vfill\eject %%%%% new section starts on a new page                 
            %%%\ifodd\pageno \else\ \vfill\eject \fi %start on an odd page      
            \advance\secnum by 1 \subsecnum=0 \exno=0                           
            \ifnum \secnum = 1 \pageno=1 \fi                                    
            \ifnum \secnum > 0                                                  
                 \leftline{\chapfont Section \the\secnum}                       
                 \vskip 3pt \fi                                                 
            \leftline{\chapfont #1}                                             
            \def\sectiontitle{#1}                                               
            \vskip\baselineskip                                                 
            \hrule                                                              
            \vskip 1cm                                                          
            \ifwritingcontents \write\contents{\string\line\string{#1           
                   \string\dotfill{}                                            
                   \ifnum \pageno < 0 \romannumeral-\pageno                     
                   \else \the\pageno \fi                                        
                   \string}}\fi                                                 
            \titlepagetrue}                                                     
                                                                                
\def\subsection#1{\advance\subsecnum by 1                                       
               \vskip 30pt                                                      
               \leftline{\secfont \the\secnum .\the\subsecnum\ #1}              
               \nobreak                                                         
               \ifwritingcontents                                               
                     \write\contents{\string\line\string{\string\qquad{}#1      
                     \string\dotfill{} \the\pageno\string}}\fi                  
                 }                                                              
                                                                                
\def\exercise{\advance \exno by 1                                               
         \vskip\baselineskip                                                    
         \noindent $\triangleright$ Exercise \the\secnum.\the\exno\quad         
         }                                                                      
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
                                                                                
                                                                                
%% definitions of control sequences for characters in the typewriter font %%%   
%%%% for short phrases this is easier than using a literal construction %%%%    
\def\\{\char92{}}          %%%%% backslash %%%%%                                
\def\lb{\char'173{}}       %%%%% left brace %%%%%                               
\def\rb{\char'175{}}       %%%%% right brace %%%%%                              
\def\sp{\char32{}}         %%%%% special space symbol %%%%%                     
                                                                                
\def\beginliteral{                                                              
\vskip\baselineskip                                                             
\begingroup                                                                     
\tt                                                                             
\obeylines                                                                      
%{\obeyspaces\global\let =\ }                                                   
\catcode`\@=0                                                                   
\parskip=0pt\parindent=0pt                                                      
\catcode`\$=12\catcode`\&=12\catcode`\~=12\catcode`\#=12                        
\catcode`\_=12\catcode`\~=12                                                    
\def\par{\leavevmode\endgraf}                                                   
\catcode`\{=12\catcode`\}=12\catcode`\%=12\catcode`\\=12                        
}                                                                               
                                                                                
\def\endliteral{\endgroup}                                                      
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%           
                                                                                
                                                                                
%%%%%%%%%%%%%% inhibit hyphenation of typewritter text %%%%%%%%%%%%%%           
\hyphenchar\tentt=-1                                                            
                                                                                
                                                                                
%%%%%%%%%%%% grouping to make input listing in typewriter type %%%%%%%          
\def\beginuser{\vskip\parskip                                                   
               \begingroup                                                      
               \tt \obeylines \parskip=0pt \parindent=0pt}                      
                                                                                
\def\enduser{\endgroup}                                                         
                                                                                
                                                                                
%%%%%%%%%%% macro to construct tables (easily) %%%%%%%%%%%%%%%%                 
%%% parameters: title goes between brackets, rest of the                        
%%%             paragraph is the table                                          
\def\maketable[#1]#2\par{                                                       
\setbox1=\vbox{#2}                                                              
$$\vbox{                                                                        
        \hbox to \wd1{\bf \hss #1 \hss}                                         
        \vskip 12pt                                                             
        \box1                                                                   
       }                                                                        
$$                                                                              
}                                                                               
%%%%%%%%%%%  end of macro to construct tables  %%%%%%%%%%%%%%%%                 
                                                                                
                                                                                
                                                                                
%%%%%%%%%%% macro to put a box around the text %%%%%%%%%%%%%%%%                 
\def\makebox#1#2#3% vsize, hsize, inserted text                                 
{\hbox{\vrule                                                                   
       \vbox to  #1{\hrule \vss                                                 
                   \hbox to #2{\hss#3\hss}\vss                                  
                   \hrule}\vrule}}                                              
\def\displaytext#1{$$\hbox{#1}$$}                                               
                                                                                
\def\LaTeX{{\rm L\kern-.36em\raise.3ex\hbox{\sc a}\kern-.15em                   
   T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}                              
                                                                                
\def\AMSTeX{{$\cal A$}\kern-.1667em\lower.5ex\hbox                              
 {$\cal M$}\kern-.125em{$\cal S$}-\TeX}                                         
                                                                                
                                                                                
%%%%%%%%%%%% macro to put TeX references in right margin %%%%%%%%               
\newdimen\theight                                                               
\def \TeXref#1{%                                                                
             \vadjust{\setbox0=\hbox{\sevenrm\quad\quad\TeX book: #1}%          
             \theight=\ht0                                                      
             \advance\theight by \dp0    \advance\theight by \lineskip          
             \kern -\theight \vbox to \theight{\rightline{\rlap{\box0}}%        
             \vss}%                                                             
             }}%                                                                
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%       
                                                                                
%%%%%%%%%%%% macro to write the date out the date and time %%%%%%%%%%%          
%%%%%%%%%%%%          TeXbook   p. 406 for date            %%%%%%%%%%%          
\def\today{\ifcase\month\or January\or February\or March\or April\or            
May\or June\or July\or August\or September\or October\or November\or            
December\fi \space\number\day, \number\year}                                    
                                                                                
\newcount\hour \newcount\minute                                                 
\hour=\time  \divide \hour by 60                                                
\minute=\time                                                                   
\loop  \ifnum \minute > 59 \advance \minute by -60 \repeat                      
\def\writetime{\ifnum \hour<13 \number\hour:%  % supresses leading 0's          
                      \ifnum \minute<10 0\fi%  % so add it it                   
                      \number\minute                                            
                      \ifnum \hour < 12 \ A.M. \else \ P.M. \fi                 
      \else \advance \hour by -12 \number\hour:%  % supresses leading 0's       
                      \ifnum \minute<10 0\fi%     % add it in                   
                      \number\minute \ P.M. \fi}                                
                                                                                
\def\datestamp{%                                                                
    \rightline{\sevenrm Run through \TeX{} on \today{} at \writetime}}          
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%           
                                                                                
\titlepagetrue                                                                  
{\nopagenumbers                                                                 
\def\sectiontitle{Introduction}                                                 
\topinsert \vskip 6 cm \endinsert                                               
\centerline{\chapfont A Gentle Introduction to \TeX{}}                          
\vskip 15 pt                                                                    
\centerline{\secfont A Manual for Self-study}                                   
\vskip 3cm                                                                      
\leftline{Michael Doob}                                                         
\leftline{Department of Mathematics}                                            
\leftline{The University of Manitoba}                                           
\leftline{Winnipeg, Manitoba, Canada R3T 2N2}                                   
\vskip\baselineskip                                                             
\leftline{MDOOB@UOFMCC.BITNET}                                                  
\leftline{MDOOB@CCU.UMANITOBA.CA}                                               
                                                                                
\vfill\eject                                                                    
\titlepagetrue.                                                                 
\vfill\eject                                                                    
}                                                                               
\secnum=-2 \pageno=-1                                                           
                                                                                
\section{Introduction}                                                          
                                                                                
First the bad news: \TeX{} is a large and complicated program that              
goes to extraordinary lengths to produce attractive typeset material.           
This very complication can cause unexpected things to happen at times.          
Now the good news: straightforward text is very easy to typeset using           
\TeX\null.  So it's possible to start with easier text and work up to           
more complicated situations.                                                    
                                                                                
The purpose of this manual is to start from the very beginning and              
to move towards these more complicated situations.  No previous knowledge       
of \TeX{} is assumed. By proceeding a section at a time, greater                
varieties of text can be produced.                                              
                                                                                
Here are a few suggestions: there are some exercises in each                    
section.  Be sure and do them! The only way to learn \TeX{} is by               
using it.  Better yet, experiment on your own; try to make some                 
variations on the exercises.  There is no way that you can damage               
the \TeX{} program with your experiments.  You can find an answer               
to most exercises by looking at the \TeX{} source file that was                 
used to produce this document.  You'll notice that there are                    
references in the right margin to {\bf The \TeX book}\null.  When you           
feel that you want more information on a topic, that's where to                 
look.                                                                           
                                                                                
Incidently, there are a few fibs that appear in this manual;                    
they are used to hide complications (I look at this as something                
like poetic license).  As you become more experienced at using                  
\TeX{}, you'll be able to find them.                                            
                                                                                
\TeX{} is a public domain program that is available for no license              
fee.  It was developed by Donald Knuth at Stanford University as                
a major project.                                                                
In the profit-oriented market place,                                            
the program would certainly cost many thousands of dollars.  The                
\TeX{} Users Group (TUG) is a nonprofit organization which distributes          
copies of \TeX{}, updates software, and gives information about                 
new developments in both hardware and software in its publications              
TUGboat and \TeX niques.  Joining this users group is inexpensive;              
please consider doing so.  The address is:                                      
\vskip\baselineskip                                                             
\centerline{\TeX{} Users Group}                                                 
\centerline{P.O. Box 9506}                                                      
\centerline{Providence, RI 02940}                                               
\centerline{U.S.A.}                                                             
                                                                                
This manual would not have come into being without the aid of others.           
In particular the proofreading and suggestions of the following                 
people have been invaluable:                                                    
Waleed A.~Al-Salam (University of Alberta),                                     
Nelson H.~F.~Beebe (University of Utah),                                        
Bart Childs (Texas A.~\&~M\null. University),                                   
Mary Coventry (University of Washington),                                       
Dimitrios Diamantaras (Temple University),                                      
Roberto Dominimanni (Naval Underwater Systems Center),                          
Victor Eijkhout (University of Nijmegen),                                       
Moshe Feder (St. Lawrence University),                                          
Jonas de Miranda Gomes                                                          
        (Instituto de Matematica Pura e Aplicada - Brazil),                     
Rob Gross (Boston College),                                                     
Klaus Hahn (University of Marburg),                                             
Anita Hoover (University of Delaware),                                          
J\"urgen Koslowski (Macalester College),                                        
John Lee (Northrop Corporation),                                                
Robert Messer (Albion College),                                                 
Emily H.~Moore (Grinnell College),                                              
Young Park (University of Maryland),                                            
Craig Platt (University of Manitoba),                                           
Kauko Saarinen (University of Jyv\"askyl\"a),                                   
Jim Wright (Iowa State University),                                             
and                                                                             
Dominik Wujastyk (Wellcome Institute for the History of Medicine).              
                                                                                
In addition several people have sent me parts or all of their local             
manuals.  I have received a few others on the rebound.                          
In particular Elizabeth Barnhart (TV Guide),                                    
Stephan v.~Bechtolsheim (Purdue University),                                    
Nelson H.~F.~Beebe (University of Utah)                                         
             and Leslie Lamport (Western Digital Corporation),                  
Marie McPartland-Conn and Laurie Mann (Stratus Computer),                       
Robert Messer (Albion College),                                                 
Noel Peterson (Library of Congress),                                            
Craig Platt (University of Manitoba),                                           
Alan Spragens (Stanford Linear Accelerator Center, now of Apple Computers),     
Christina Thiele (Carleton University),                                         
and Daniel M.~Zirin (California Institute of Technology)                        
have written various types of lecture notes that have been most helpful.        
                                                                                
\vfill \eject                                                                   
\section{Contents}                                                              
                                                                                
\line{Introduction \dotfill{} i}                                                
\line{Contents \dotfill{} iii}                                                  
\line{1. Getting Started \dotfill{} 1}                                          
\line{\qquad{}1.1 What \TeX{} is and what \TeX{} isn't \dotfill{} 1}            
\line{\qquad{}1.2 From \TeX{} file to readable output,                          
                                        the big set up \dotfill{} 2}            
\line{\qquad{}1.3 Let's do it! \dotfill{} 4}                                    
\line{\qquad{}1.4 \TeX{} has everything under control \dotfill{} 7}             
\line{\qquad{}1.5 What \TeX{} won't do \dotfill{} 8}                            
\line{2. All characters great and small \dotfill{} 9}                           
\line{\qquad{}2.1 Some characters are more special than others \dotfill{} 9}    
\line{\qquad{}2.2 Typesetting with an accent \dotfill{} 10}                     
\line{\qquad{}2.3 Dots, dashes, quotes,                                         
       $\textfont0=\tenbf \mathinner {\ldotp \ldotp \ldotp }$ \dotfill{} 13}    
\line{\qquad{}2.4 Different fonts \dotfill{} 14}                                
\line{3. The shape of things to come \dotfill{} 18}                             
\line{\qquad{}3.1 Units, units, units \dotfill{} 18}                            
\line{\qquad{}3.2 Page shape \dotfill{} 19}                                     
\line{\qquad{}3.3 Paragraph shape \dotfill{} 21}                                
\line{\qquad{}3.4 Line shape \dotfill{} 25}                                     
\line{\qquad{}3.5 Footnotes \dotfill{} 26}                                      
\line{\qquad{}3.6 Headlines and Footlines \dotfill{} 27}                        
\line{\qquad{}3.7 Overfull and underfull boxes \dotfill{} 28}                   
\line{4. $\Bigl\{$Groups, $\bigl\{$Groups,                                      
        $\{$and More Groups$\}\bigr\}\Bigr\}$ \dotfill{} 30}                    
\line{5. No math anxiety here! \dotfill{} 32}                                   
\line{\qquad{}5.1 Lots of new symbols \dotfill{} 32}                            
\line{\qquad{}5.2 Fractions \dotfill{} 36}                                      
\line{\qquad{}5.3 Subscripts and superscripts \dotfill{} 37}                    
\line{\qquad{}5.4 Roots, square and otherwise \dotfill{} 38}                    
\line{\qquad{}5.5 Lines, above and below \dotfill{} 38}                         
\line{\qquad{}5.6 Delimiters large and small \dotfill{} 39}                     
\line{\qquad{}5.7 Those special functions \dotfill{} 40}                        
\line{\qquad{}5.8 Hear ye, hear ye! \dotfill{} 41}                              
\line{\qquad{}5.9 Matrices \dotfill{} 42}                                       
\line{\qquad{}5.10 Displayed equations \dotfill{} 44}                           
\line{6. All in a row \dotfill{} 47}                                            
\line{\qquad{}6.1 Picking up the tab \dotfill{} 47}                             
\line{\qquad{}6.2 Horizontal alignment with more sophisticated patterns         
      \dotfill{} 50}                                                            
\line{7. Rolling your own \dotfill{} 54}                                        
\line{\qquad{}7.1 The long and short of it \dotfill{} 55}                       
\line{\qquad{}7.2 Filling in with parameters \dotfill{} 56}                     
\line{\qquad{}7.3 By any other name \dotfill{} 59}                              
\line{8. To err is human \dotfill{} 60}                                         
\line{\qquad{}8.1 The forgotten bye \dotfill{} 60}                              
\line{\qquad{}8.2 The misspelled or unknown control sequence \dotfill{} 60}     
\line{\qquad{}8.3 The misnamed font \dotfill{} 62}                              
\line{\qquad{}8.4 Mismatched mathematics \dotfill{} 62}                         
\line{\qquad{}8.5 Mismatched braces \dotfill{} 63}                              
\line{9. Digging a little deeper \dotfill{} 66}                                 
\line{\qquad{}9.1 Big files, little files \dotfill{} 66}                        
\line{\qquad{}9.2 Larger macro packages \dotfill{} 67}                          
\line{\qquad{}9.3 Horizontal and vertical lines \dotfill{} 68}                  
\line{\qquad{}9.4 Boxes within boxes \dotfill{} 70}                             
\line{10. Control word list \dotfill{} 74}                                      
\ifwritinganswers                                                               
    \line{11. I get by with a little help\dotfill{} 77}                         
\fi                                                                             
                                                                                
                                                                                
                                                                                
\section{Getting Started}                                                       
                                                                                
\subsection{What \TeX{} is and what \TeX{} isn't}                               
                                                                                
First of all, let's see what steps are necessary to produce a                   
document using \TeX\null.  The first step is to type the file                   
that \TeX{} reads. This is usually called the \TeX{} file or the                
input file, and it can be created using a simple text editor (in                
fact, if you're using a fancy word processor, you have to be sure               
that your file is saved in ASCII or nondocument mode without any                
special control characters). The \TeX{} program then reads your                 
input file and produces what is called a DVI file (DVI stands for               
DeVice Independent). This file is not readable, at least not by                 
humans. The DVI file is then read by another program (called a                  
device driver) that produces the output that is readable by                     
humans. \TeXref{23} Why the extra file? The same DVI file can be                
read by different device drivers to produce output on a dot                     
matrix printer, a laser printer, a screen viewer, or a                          
phototypesetter. Once you have produced a DVI file that gives                   
the right output on, say, a screen viewer, you can be assured                   
that you will get exactly the same output on a laser printer                    
without running the \TeX{} program again.                                       
                                                                                
The process may be thought of as proceeding in the following way:               
$$                                                                              
\lower .3cm \makebox{1cm}{2.5cm}{\TeX{} input file}                             
{\hbox{\quad \TeX{} program \quad} \atop \longrightarrow}                       
\lower .3cm \makebox{1cm}{2.5cm}{DVI file}                                      
{\hbox{\quad device driver\quad} \atop \longrightarrow}                         
\lower .3cm \makebox{1cm}{2.7cm}{readable output}                               
$$                                                                              
                                                                                
This means that we don't see our output in its final form when it               
is being typed at the terminal. But in this case a little                       
patience is amply rewarded, for a large number of symbols not                   
available in most word processing programs become available. In                 
addition, the typesetting is done with more precision, and the                  
input files are easily sent between different computers by                      
electronic mail or on a magnetic medium.                                        
                                                                                
Our focus will be on the first step, that is, creating the \TeX{}               
input file and then running the \TeX{} program to produce                       
appropriate results.  There are two ways of running the \TeX{}                  
program; it can be run in batch mode or interactively.  In                      
batch mode you submit your \TeX{} input file to your computer;                  
it then runs the \TeX{} program without further intervention and                
gives you the result when it is finished.  In interactive mode                  
the program can stop and get further input from the user, that                  
is, the user can interact with the program.  Using \TeX{}                       
interactively allows some errors to be corrected by the user,                   
while the \TeX{} program makes the corrections in batch mode as                 
best it can. Interactive is the preferred mode, of course. All                  
personal computer and many mainframe implementations are                        
interactive.  On some mainframes, however, the only practical                   
method of running \TeX{} is in batch mode.                                      
                                                                                
                                                                                
\subsection{From \TeX{} file to readable output, the big set up}                
                                                                                
{\parskip = 0pt \noindent                                                       
[Note from MD:                                                                  
{\sl This is the only system dependent section in the                           
manual. No reference is made to it outside of the section itself.               
The following local information should be included:                             
\item{$\bullet$} What initial steps, if any, should be taken by the reader      
to permit the running of \TeX{} and your local device driver(s).                
\item{$\bullet$} How to run \TeX\null.                                          
\item{$\bullet$} How to read the log file.                                      
\item{$\bullet$} How to preview and/or print the dvi file.                      
                                                                                
The following sample is applicable here at the University of                    
Manitoba.  We use a locally written editor (MANTES) on an                       
Amdahl running MVS; I'm fairly certain that it's the worst possible case.}]     
}                                                                               
                                                                                
                                                                                
In this section we'll see how to run \TeX{} at the University of                
Manitoba.  It is assumed that the reader is familiar with MANTES                
and can create text files using it.                                             
                                                                                
First, there are several things that must be done {\bf one time                 
only}. To start you must do the following (you type in the                      
material that looks like typewriter type):                                      
\item{(1)} allocate the files that \TeX{} will use by typing the                
following lines (while in MANTES):                                              
\itemitem{}  C: {\tt alloc da=source.tex format=vb,256,6144}                    
\itemitem{}  C: {\tt alloc da=dvi format=fb,1024,6144}                          
                                                                                
\item{(2)} Create a file called RUNTEX in your MANTES aggregate                 
containing the following JCL:                                                   
\par\noindent\vbox{                                                             
\beginuser                                                                      
//        JOB  ,'RUN TEX'                                                       
//        EXEC TEXC                                                             
//SRC     DD DSN=<userid>.SOURCE.TEX,DISP=SHR                                   
//DVI     DD DSN=<userid>.DVI,DISP=OLD                                          
\enduser                                                                        
}                                                                               
                                                                                
The name {\tt <userid>} is replaced by your own user id, of                     
course. The use of upper case and the spaces must be followed                   
exactly.                                                                        
                                                                                
\item{(3)} Create a file called PRINTTEX in your MANTES aggregate               
containing the following JCL:                                                   
\par\noindent\vbox{                                                             
\beginuser                                                                      
//        JOB  ,'PRINT TEX'                                                     
//        EXEC TEXP                                                             
//DVIFILE DD DSN=<userid>.DVI,DISP=SHR                                          
\enduser                                                                        
}                                                                               
                                                                                
Once you have completed these three steps, you are ready to run a               
\TeX{} job. The files you have created will allow you to produce                
about ten pages of ordinary text.                                               
                                                                                
Here are the steps you use {\bf each time you run a job}.                       
                                                                                
\item{(1)} create a MANTES file containing your \TeX{} input.                   
\item{(2)} save and submit your file using the commands                         
\itemitem{} C: {\tt save f/l to da=source.tex noseq}                            
\itemitem{} C: {\tt submit runtex}                                              
\item{(3)} when you get a message saying that your job is finished,             
enter the command                                                               
\itemitem{} C: {\tt out <jobname>; list ttyout}                                 
                                                                                
In this command, {\tt <jobname>} is replaced by your user id with               
a dollar sign appended.  This file listing will tell you of any                 
errors that might have occured.  It is an abbreviated version of                
what is called the ``log file''; we will use the term ``log                     
file'' to refer to the ttyout file produced by \TeX\null.                       
                                                                                
If you want, you can check on the status of your job while it is                
executing by using the command                                                  
\itemitem{} C: {\tt q <jobname>}                                                
                                                                                
When you are finished looking at the log file, the command {\tt                 
end scratch} will throw away the log file while the command {\tt                
end release} will send the log file to the printer, and it can                  
then be picked up with your \TeX{} output.                                      
                                                                                
\item{(4)} when your output from RUNTEX program is error free, \TeX{}           
will have created a legal DVI file.  To print it, use the                       
command                                                                         
\itemitem{} C: {\tt submit printtex nohold}                                     
                                                                                
As in (3), you can check on the status of your job while it is                  
executing.                                                                      
\item{(5)} Pick up your output at the I/O window, sixth floor                   
Engineering building.  It usually takes about twenty minutes for the            
output to be ready.  Ask for it by ``{\tt <jobname>}''.                         
                                                                                
The files created are large enough for running \TeX{} jobs of about             
10 pages.  A job of that size will take about one second of CPU time            
to run through \TeX\null.  It will take about 15 seconds of CPU time to         
print 10 pages on the Xerox 8600 using the current device driver.               
                                                                                
\subsection{Let's do it!}                                                       
                                                                                
So, from our viewpoint, the name of the game is to create the \TeX{}            
file that produces the right readable output. What does                         
a \TeX{} file look like? It consists of characters from an                      
ordinary terminal, that is, upper and lower case letters,                       
numbers, and the usual punctuation and accent characters (these                 
are the usual ASCII characters). Text, for the most part, is just               
typed normally. Special instructions usually start with one of a                
few special symbols such as {\tt \#} or {\tt \&} (these will be                 
described in more detail later). Here is an example of a \TeX{}                 
input file:                                                                     
\vskip\parskip                                                                  
                                                                                
\beginuser                                                                      
Here is my first \\TeX\\ sentence.                                              
\\bye                                                                           
\enduser                                                                        
\toindex{TeX{} } \toindex{ }                                                    
                                                                                
                                                                                
First note that the characters in this example look like                        
typewriter type. We use these characters with all examples that                 
are meant to be typed from the computer terminal. Second, note                  
that the backslash appears three times in the text. We'll soon                  
see that this is one of the special symbols mentioned                           
previously, and it is used very frequently when making \TeX{}                   
documents. Make a file containing this example. Use the \TeX{}                  
program to make a DVI file and a device driver to see what you                  
have produced. If all goes well, you'll have a single page with                 
the following single sentence:                                                  
                                                                                
Here is my first \TeX\ sentence.                                                
                                                                                
  There will also be a page number at the bottom of the page. If                
you've gotten this far, congratulations! Once you can produce one               
\TeX{} document, it's just a matter of time before you can do                   
more complicated ones. Now let's compare what we typed in with                  
what we got out. The straightforward words were just typed in                   
normally, and \TeX{} set them in ordinary type. But the word ``\TeX'',          
which can't be typed in on a terminal because the letters                       
aren't on the same line, is entered by using a word starting with               
a backslash, and \TeX{} made the proper interpretation. Most                    
symbols that are not ordinary letters, numbers, or punctuation                  
are typeset by entering a word starting with a backslash. If we                 
look a little closer, we'll note that the word ``first'' is also                
changed.                                                                        
\TeXref{4}                                                                      
The first two letters have been joined together and there isn't                 
a separate dot over the letter ``i''. This is standard                          
typesetting practice: certain letter combinations are joined up                 
to form what are called {\sl ligatures}. There is actually a good               
\aesthetic{} reason for this. Compare the first two letters of                  
``first'' and ``{f}irst'' to see the difference. We note that                   
{\tt \\bye} appears in the input file with no corresponding word                
in the final copy.                                                              
\toindex{bye}                                                                   
This a typesetting instruction that tells \TeX{} that the input                 
is finished. We'll learn about lots of different typesetting                    
instructions as we go along.                                                    
                                                                                
Let's look at the log file that was created when we ran \TeX                    
\null. It may vary slightly at your site, but should look                       
something like this:                                                            
                                                                                
\noindent\vbox{                                                                 
\beginuser                                                                      
   1.                                                                           
   2.  This is TeX, MVS Version 2.9 (no format preloaded)                       
   3.   ** File  PLAIN.FMT <-- DD=FMTLIB MEM=PLAIN                              
   4.                                                                           
   5.   ** File  SRC.TEX <-- DD=SRC                                             
   6.  (src.tex [1]                                                             
   7.  Output written on DVI (1 page, 256 bytes).                               
   8.  Transcript written on TEXLOG.                                            
\enduser                                                                        
}                                                                               
                                                                                
This is the file that will contain any error messages. On line 6,               
{\tt (src.tex\ } indicated that \TeX{} has started reading that                 
file. The appearance of {\tt [1]} indicates that page 1 has been                
processed. If there were errors on page 1, they would be listed                 
at that point.                                                                  
                                                                                
\vskip .5cm                                                                     
\exercise Add a second sentence to your original \TeX{} file to get:            
\beginuser                                                                      
Here is my first \\TeX\\ sentence.                                              
I was the one who typeset it!                                                   
\\bye                                                                           
\enduser                                                                        
                                                                                
Use \TeX{} and look at your output. Is the second sentence on a                 
new line?                                                                       
                                                                                
                                                                                
\exercise Now add this line to the beginning of your file:                      
\beginuser                                                                      
\\nopagenumbers                                                                 
\enduser                                                                        
\noindent                                                                       
Guess what will happen when you run the new file through                        
\TeX\null.  Now try it and see what happens.                                    
\toindex{nopagenumbers}                                                         
                                                                                
\exercise Add three or four more sentences to your file.                        
Use letters, numbers, periods, commas, question marks,                          
and exclamation points, but don't use any other symbols.                        
                                                                                
                                                                                
\exercise Leave a blank line and add some more sentences.                       
You can now get new paragraphs.                                                 
                                                                                
We have now seen a major principle concerning the preparation of                
\TeX{} input files. The placement of the text on your computer                  
terminal does not necessarily correspond to the placement of the                
text on your output. You can not, for example, add space between                
words in your output by adding spaces in your input file. Several               
consecutive spaces and one space will produce exactly the same                  
output. As would be expected, a word at the end of one line will                
be separated from the first word of the following line. In fact,                
sometimes when working on a file that will be heavily edited, it                
is convenient to start each sentence on a separate line. Spaces                 
at the beginning of a line, however, are always ignored.                        
                                                                                
\exercise Add the following sentence as a new paragraph, and then               
typeset it:                                                                     
\beginuser                                                                      
Congratulations! You received a grade of 100\% on your latest                   
examination.                                                                    
\enduser                                                                        
\noindent                                                                       
The {\tt\%} sign is used for comments in your input file.                       
Everything on a line following this symbol is ignored. Notice                   
that even the space that normally separates the last word on one                
line from the first word on the next line is lost. Now put a                    
backslash in front of the {\tt\%} sign to correct the                           
sentence. \toindex{\$} \toindex{\%}                                             
                                                                                
\exercise Add the following sentence as a new paragraph:                        
\beginuser                                                                      
You owe me \$10.00 and it's about time you sent it to me!                       
\enduser                                                                        
\noindent                                                                       
This will produce an error in your log file (if your                            
implementation of \TeX{} is interactive, that is, sends you                     
messages and waits for answers, just hit the carriage return or                 
enter key when you get the error message). You will get output,                 
but not what you might expect. Look at the log file and see                     
where the error messages are listed. Don't worry about the                      
actual messages. We'll have a lot more to say about errors                      
(including this one) later. Now fix the error by putting a                      
backslash in front of the {\tt \$} sign, and typeset the result                 
(there are a small number of characters like the per cent and                   
dollar signs that \TeX{} uses for its own purposes. A table of                  
these characters will be provided shortly).                                     
                                                                                
\subsection{\TeX{} has everything under control}                                
                                                                                
We have seen that the backslash has a special role. Any word                    
starting with a backslash will be given a special interpretation                
when \TeX{} reads it from your input file. Such a word is called                
a {\sl control sequence}. There are, in fact, two types of                      
control sequences: a {\sl control word\/} is a backslash followed               
by letters only (for example, {\tt \\TeX}) and a {\sl control                   
symbol\/} is a backslash followed by a single nonletter (for                    
example, {\tt \\\$})\null. Since a space is a nonletter, a                      
backslash followed by a space is a legitimate control symbol.                   
\TeXref{7--8}                                                                   
When we want to emphasize that a space is present, we will use a                
special symbol {\tt\sp} to indicate the space, as in the control                
symbol {\tt\\\sp}.  This convention is used in {\bf The \TeX                    
book} as well as in this manual.                                                
                                                                                
When \TeX{} is reading your input file and comes to a backslash                 
followed by a letter, it knows that a control word is being                     
read. So it continues reading the name of the control word until                
a nonletter is read. So if your file contains                                   
                                                                                
\displaytext{\tt I like \\TeX!}                                                 
                                                                                
\noindent                                                                       
the control word {\tt \\TeX} is terminated by the exclamation                   
point. But this presents a problem if you want to have a space                  
after a control word. If you have, for example, the sentence                    
                                                                                
\displaytext{\tt I like \\TeX and use it all the time.}                         
                                                                                
\noindent                                                                       
in your input file, the control word {\tt\\TeX} is terminated by                
the space (which is, of course, a nonletter). But then you won't                
have a space between the words ``\TeX{}'' and ``and''; inserting                
more spaces won't help, since \TeX{} doesn't distinguish between                
one space and several consecutive spaces. But if you put the                    
control symbol {\tt\\\sp} after a control word, you will both                   
terminate the control word and insert a space. It's really easy                 
to forget to put in something like {\tt\\\sp} after a control                   
word. I promise you that you will do it at least once while                     
you're learning to use \TeX\null.                                               
                                                                                
\exercise Make an input file that will produce the following                    
paragraph:                                                                      
                                                                                
I like \TeX! Once you get the hang of it, \TeX{} is really easy                 
to use. You just have to master the \TeX nical aspects.                         
                                                                                
Most control words are named so as to give a hint of their uses.                
You can use {\tt \\par} to make a new paragraph, for example,                   
instead of putting in a blank line.                                             
\toindex{par}                                                                   
                                                                                
\subsection{What \TeX{} won't do}                                               
                                                                                
\TeX{} is very good at setting type, but there are things that \TeX{}           
can't do well.  One is making figures.  Space can be left                       
to insert figures, but there are no graphic procedures built into               
the language (at present).  Some implementations allow graphic                  
instructions to be inserted using the {\tt \\special} control                   
word but these are exceptional and definitely site dependent.                   
                                                                                
\TeX{} sets type in horizontal lines.  Text that has a curve for                
a baseline or whose type is perspective (gradually increasing or                
decreasing) is not handled well by \TeX\null.                                   
                                                                                
We have seen that there is an ``edit, \TeX{}, driver'' cycle that               
is necessary for each different copy of output.  This is true                   
even when the output device is a terminal.  In particular, it's                 
not possible to type the input file and see the results on the                  
screen immediately without going through the full cycle.  Some                  
implementations have both text and graphics displays with                       
reasonably quick turnaround (a few seconds for a single page); as               
hardware becomes less expensive and processors become faster, we                
may see improvement.                                                            
                                                                                
                                                                                
                                                                                
                                                                                
                                                                                
                                                                                
\section{All characters great and small}                                        
                                                                                
\subsection{Some characters are more special than others}                       
                                                                                
We saw in the last section that most text is entered at the                     
terminal as sentences of ordinary words just as when typing with                
a typewriter. But we also saw that, in particular, the backslash                
could be used for at least two different purposes. It can be                    
used for symbols (or combinations of symbols) that don't appear                 
on the keyboard such as typing {\tt \\TeX} to get \TeX\null. It                 
can also be used to give \TeX{} special instructions such as                    
typing {\tt \\bye} to indicate the end of the input file. In                    
general, a word starting with a backslash will be interpreted by                
\TeX{} as one requiring special attention. There are several                    
hundred words that \TeX{} knows, and you can define more                        
yourself, and so the backslash is very important. We'll spend a                 
lot of time as we proceed learning some of these words;                         
fortunately we'll only need to use a small number of them most of               
the time.                                                                       
                                                                                
There are ten characters which, like the backslash, are used                    
by \TeX{} for special purposes, and we now want to give the                     
complete list.                                                                  
\TeXref{37--38}                                                                 
What if we want to use a sentence with one of these special                     
characters in it? With this in mind we'll ask the following                     
questions:                                                                      
\item{(1)} What are the different special characters?                           
\item{(2)} How do we use a special character if we really want to               
typeset it in our text?                                                         
                                                                                
Here is a table of each special character, its purpose, and the                 
method of typesetting the special character itself if you need                  
it:                                                                             
                                                                                
                                                                                
\maketable [Characters requiring special input]                                 
\halign{                                                                        
   \strut \hfil#\hfil & \quad # \hfil & \quad\tt # \hfil\cr                     
   \bf Character & \bf Purpose & \bf Input for literal output \cr               
   \noalign{\hrule} \noalign{\smallskip}                                        
   $\backslash$   & Special symbols and instructions & \$\\backslash\$   \cr    
   $\{$ & Open group                       & \$\\\lb\$         \cr              
   $\}$ & Close group                      & \$\\\rb\$         \cr              
   \%   & Comments                         & \\\%                   \cr         
   \&   & Tabs and table alignments        & \\\&                   \cr         
   \~{} & Unbreakable space                & \\\~{}\lb\rb \cr                   
   \$   & Starting or ending math text     & \\\$                   \cr         
   \~{} & Math superscripts                & \\\~{}\lb\rb \cr                   
   \_{} & Math subscripts                  & \\\_{}\lb\rb \cr                   
   \#   & Defining replacement symbols     & \\\#                   \cr         
       }                                                                        
                                                                                
\toindex{\lb}                                                                   
\toindex{\rb}                                                                   
\toindex{\%}                                                                    
\toindex{\&}                                                                    
\toindex{\~{}}                                                                  
\toindex{\$}                                                                    
\toindex{\~{}}                                                                  
\toindex{\_{}}                                                                  
\toindex{\#}                                                                    
                                                                                
\subsection{Typesetting with an accent}                                         
                                                                                
Now we're going to start using some of \TeX{}'s goodies! So far                 
we've just been using \TeX{} to make our output look attractive,                
but now we'll start to do things that are difficult or impossible               
on the typewriter. In particular, we're going to look at accents                
now. How do you produce accents when the symbol doesn't appear on               
the keyboard? Just as with the symbol \TeX{}, it is necessary to                
enter a word starting with a backslash. For the word ``premi\`ere'',            
as a first example, you need to type in {\tt premi\\\`{}ere}                    
(you may have to hunt around to find the ``back prime''                         
or ``grave'' symbol {\tt \`{}} on your keyboard, but it's there                 
somewhere\footnote{$~1$}{If you have a very old or obscure                      
keyboard and the back prime is {\it really\/} not there, you can                
use {\tt \\lq\lb\rb} instead.  Similarly {\tt \\rq\lb\rb} can be                
used for the symbol {\tt \'{}}. You can think of the symbols as                 
being abbreviations for ``left quote'' and ``right quote.'' In                  
addition, {\tt \\lq\\lq\lb\rb} and {\tt \\rq\\rq\lb\rb} give the                
usual quotation marks.}). In general, to put an accent on a                     
letter, the appropriate control sequence precedes the letter.                   
                                                                                
Here are some examples:                                                         
$$\vbox{                                                                        
                                                                                
\halign{                                                                        
\strut \hfil\tt # & \quad # \hfil\cr                                            
\bf \TeX{} input & \bf \TeX{} output \cr                                        
\noalign{\hrule} \noalign{\smallskip}                                           
\\\`{}a la mode          & \`a la mode \cr                                      
r\\\'{}esum\\\'{}e          & r\'esum\'e  \cr                                   
soup\\c{\sp}con   & soup\c con  \cr                                             
No\\"el               & No\"el      \cr                                         
na\\"\\i{\sp}ve    & na\"\i ve   \cr                                            
}                                                                               
}$$                                                                             
                                                                                
                                                                                
We see several principles illustrated by these examples. Most                   
accents are produced by using a control symbol with a similar                   
shape. A few of them are produced by control words containing a                 
single letter. Some care must be used in this case, for a space                 
must be used to terminate to the control word. If you have {\tt                 
soup\\ccon} in your file, for example, \TeX{} will look for the                 
control word {\tt\\ccon}.                                                       
\TeXref{52--53}                                                                 
                                                                                
Notice that there is a control word {\tt\\i} also. This produces                
the letter ``i'' without the dot over it; this allows an accent                 
to be put over the lower part of the letter. There is an                        
analogous control word {\tt\\j} which produces a dotless ``j''                  
for accenting purposes.                                                         
                                                                                
\maketable [Accents that may be immediately followed by a letter]               
\halign{                                                                        
   \strut \hfil # & \quad \hfil \tt # \hfil & \hfil \quad # \hfil\cr            
   \bf Name & \bf \TeX{} input & \bf \TeX{} output \cr                          
   \noalign{\hrule} \noalign{\smallskip}                                        
   grave           & \\\`{}o & \`o \cr                                          
   acute           & \\\'{}o & \'o \cr                                          
   circumflex      & \\\~{}o & \~o \cr                                          
   umlaut/dieresis/tr\'emat & \\"{}o    & \"o \cr                               
   tilde           & \\\~{}o & \~o \cr                                          
   macron          & \\={}o    & \=o \cr                                        
   dot             & \\\char'056{}o    & \.o \cr                                
       }                                                                        
                                                                                
\toindex{`}                                                                     
\toindex{'}                                                                     
\toindex{\~{}}                                                                  
\toindex{"}                                                                     
\toindex{i}                                                                     
\toindex{j}                                                                     
\toindex{=}                                                                     
\toindex{.}                                                                     
                                                                                
\maketable [Accents requiring an intervening space]                             
\halign{                                                                        
   \strut \hfil # \hfil & \quad \hfil \tt # \hfil & \hfil \quad # \hfil\cr      
   \bf Name & \bf \TeX{} input & \bf \TeX{} output \cr                          
   \noalign{\hrule} \noalign{\smallskip}                                        
   cedilla         & \\c o   & \c{o} \cr                                        
   underdot        & \\d o   & \d{o} \cr                                        
   underbar        & \\b o   & \b{o} \cr                                        
   h\'a\v{c}ek     & \\v o   & \v{o} \cr                                        
   breve           & \\u o   & \u{o} \cr                                        
   tie             & \\t \lb oo\rb  & \t{oo} \cr                                
   Hungarian umlaut& \\H o   & \H{o} \cr                                        
      }                                                                         
                                                                                
\toindex{c}                                                                     
\toindex{d}                                                                     
\toindex{b}                                                                     
\toindex{v}                                                                     
\toindex{u}                                                                     
\toindex{t}                                                                     
\toindex{H}                                                                     
                                                                                
\TeX{} also allows some letters from languages other than English to            
be typeset.                                                                     
                                                                                
\maketable [A\kern-1pt vailable foreign language symbols]                       
\halign{                                                                        
   \strut \hfil # \hfil & \quad \hfil \tt # \hfil & \hfil \quad # \hfil\cr      
   \bf Example& \bf \TeX{} input & \bf \TeX{} output \cr                        
   \noalign{\hrule} \noalign{\smallskip}                                        
   \AE gean, \ae sthetics     & \\AE, \\ae    & \AE, \ae \cr                    
   \OE uvres, hors d'\oe uvre & \\OE \\oe     & \OE, \oe \cr                    
   \AA ngstrom                & \\AA, \\aa    & \AA, \aa \cr                    
   \O re, K\o benhavn         & \\O, \\o      & \O, \o   \cr                    
   \L odz, \l\'odka           & \\L, \\l      & \L, \l   \cr                    
   Nu\ss                      & \\ss          & \ss      \cr                    
   !`Si!                      & {!}{\`{}}     & !`       \cr                    
   ?`Si?                      & {?}{\`{}}     & ?`       \cr                    
                             & \lb\\it\\\$\rb & {\it\$}  \cr                    
      }                                                                         
                                                                                
\toindex{AE}                                                                    
\toindex{ae}                                                                    
\toindex{OE}                                                                    
\toindex{oe}                                                                    
\toindex{AA}                                                                    
\toindex{aa}                                                                    
\toindex{O}                                                                     
\toindex{o}                                                                     
\toindex{L}                                                                     
\toindex{l}                                                                     
\toindex{ss}                                                                    
                                                                                
Typeset the sentence in each of the following exercises:                        
                                                                                
\exercise Does \AE schylus understand \OE dipus?                                
                                                                                
\exercise The smallest internal unit of \TeX{} is about 53.63\AA.               
                                                                                
\exercise They took some honey and plenty of money wrapped up in a              
{\it \$}5 note.                                                                 
                                                                                
\exercise \'El\`eves, refusez vos le\c cons! Jetez vos cha\~\i nes!             
                                                                                
\exercise Za\v sto tako polako pijete \v caj?                                   
                                                                                
\exercise Mein Tee ist hei\ss.                                                  
                                                                                
\exercise Peut-\~etre qu'il pr\'ef\`ere le caf\'e glac\'e.                      
                                                                                
\exercise ?`Por qu\'e no bebes vino blanco? !`Porque est\'a avinagrado!         
                                                                                
\exercise M\'\i\'\j n idee\"en worden niet be\"\i nvloed.                       
                                                                                
\exercise Can you take a ferry from \"Oland to \AA land?                        
                                                                                
\exercise T\"urk\c ce konu\c san ye\u genler nasillar?                          
                                                                                
                                                                                
\def\bdots{$\textfont0=\tenbf \ldots$} %%% boldface version of \ldots           
                                                                                
\subsection{Dots, dashes, quotes, \bdots}                                       
                                                                                
Typing has always been a compromise: the small number of keys                   
(compared to the number of typeset symbols available) has forced                
some changes on the typist.  When preparing material using \TeX{},              
there is no need to be so restricted. In this section we'll look                
at some differences between typing and using \TeX\null.                         
                                                                                
There are four types of dashes that are used.  The hyphen is used               
for combining words into one unit as with mother-in-law.                        
\TeXref{3--5}                                                                   
The en-dash is used to indicate a sequence of page numbers, years               
or such things. The em-dash is a grammatical symbol.  The minus                 
sign is used for negative numbers. Here they are with their uses:               
                                                                                
\maketable [Different types of dashes]                                          
\halign{                                                                        
   \strut \hfil # \hfil & \quad \hfil \tt # \hfil & \hfil \quad # \hfil         
                        & \hfil # \hfil\cr                                      
   \bf Name& \bf \TeX{} input & \bf \TeX{} output & \bf Example\cr              
   \noalign{\hrule} \noalign{\smallskip}                                        
   hyphen     & -         & -   & The space is 3-dimensional. \cr               
   en-dash    & {-}{-}    & --  & Read pages 3--4. \cr                          
   em-dash    & {-}{-}{-} & --- & I saw them---there were 3 men alive. \cr      
   minus sign & \$-\$     & $-$ & The temperature dropped to $-$3 degrees. \cr  
      }                                                                         
                                                                                
\exercise I entered the room and---horrors---I saw both my                      
father-in-law and my mother-in-law.                                             
                                                                                
\exercise The winter of 1484--1485 was one of discontent.                       
                                                                                
                                                                                
                                                                                
Another difference between typing and using \TeX{} is the use of                
quotation marks.  Opening and closing quotation marks are the                   
same on a typewriter.  They are produced in \TeX{} by using the                 
apostrophe or prime key {\tt \'{}} and backprime key {\tt \`{}}\null.           
An open quote is produced by {\tt {\`{}}{\`{}}} and the                         
close quote by {\tt {\'{}}{\'{}}}\null.\TeXref{3} Similarly the                 
opening single quote is produced by {\tt {\`{}}} and the closing                
single quote by {\tt {\'{}}}\null. Notice that there is no need                 
to use the usual typing quotation mark (it normally gives a close               
quote, but don't count on it).                                                  
                                                                                
\exercise His ``thoughtfulness'' was impressive.                                
                                                                                
\exercise Frank wondered, ``Is this a girl that can't say `No!'?''              
                                                                                
Sometimes ellipses (three dots) are used to indicate a passage                  
of time or missing material.  Typing three periods will give                    
three dots very close together. The dots with proper spacing will               
be produced by having {\tt \\dots} in your input file.                          
\TeXref{173}                                                                    
\toindex{dots}                                                                  
                                                                                
\exercise He thought, ``\dots and this goes on forever, perhaps to the          
last recorded syllable.''                                                       
                                                                                
Another problem with dots is that the period after an                           
abbreviation normally has less space after it than does the                     
period at the end of a sentence. There are two ways to solve this               
problem: the dot can be followed by either {\tt\\\sp} or {\tt\~{}}              
to change the spacing.                                                          
\TeXref{91--92}                                                                 
The second alternative will give an unbreakable space; when such                
a space is between two words, these words must typeset on the                   
same line. The input {\tt Prof.\~{}Knuth} would cause those two                 
words to be typeset on one line.  This is desirable for names                   
like Vancouver, B.~C. and Mr.~Jones so that ``Mr.'' and ``Jones''               
do not end up on separate lines. Notice that no backslash is used               
with the unbreakable space.                                                     
                                                                                
\exercise Have you seen Ms.~Jones?                                              
                                                                                
\exercise Prof.~Smith and Dr.~Gold flew from                                    
\ifcanspell                                                                     
Halifax N.~S. to Montr\'eal, P.~Q. via Moncton, N.~B.                           
\else                                                                           
L.~A. to Washington, D.~C., via Fargo, N.~D.                                    
\fi                                                                             
                                                                                
                                                                                
\subsection{Different fonts}                                                    
                                                                                
The most obvious difference between typewritten text and \TeX{}                 
output is undoubtedly the different fonts or types of symbols                   
used. When \TeX{} starts up it has sixteen fonts available.  Some               
of these are used for technical purposes, but even so there are                 
several different typefaces available as can be seen in the                     
following table. A complete list of the sixteen fonts is given in               
Appendix F of {\bf The \TeX{}book}.                                             
\TeXref{427--432}                                                               
Most of them are used automatically; a mathematical subscript,                  
for example, is put in smaller type by \TeX{} with no special                   
user intervention.                                                              
                                                                                
To change from the usual (roman) typeface to italic, the control                
word {\tt \\it} is used.  To change back to the usual roman                     
typeface, use {\tt \\rm}. For example, you could have the                       
following sentence: {\tt I started with roman type, \\it switched               
to italic type, \\rm and returned to roman type}. This would give               
the following: I started with roman type, \it switched to italic                
type, \rm and returned to roman type.                                           
                                                                                
Typefaces other than italic are also available. Those given below               
are the ones most commonly used.  They are available immediately                
when \TeX{} starts. A little later we'll see how to use other                   
typefaces that aren't available when \TeX{} starts.                             
                                                                                
\maketable [Font Samples]                                                       
\halign{                                                                        
   \strut \hfil # \hfil & \quad \hfil \tt # \hfil & \hfil \quad # \hfil\cr      
   \bf Font name & \bf \TeX{} switch sequence & \bf Sample of typeface \cr      
   \noalign{\hrule} \noalign{\smallskip}                                        
   Roman       & \\rm  & \rm This is roman type. \cr                            
   Boldface    & \\bf  & \bf This is boldface type. \cr                         
   Italic      & \\it  & \it This is italic type. \cr                           
   Slanted     & \\sl  & \sl This is slanted type. \cr                          
   Typewriter  & \\tt  & \tt This is typewriter type. \cr                       
   Math symbol${}~1$ & \\cal & $\cal SOME\hbox{\quad}SCRIPT                     
                                     \hbox{\quad}LETTERS$\rm. \cr               
      }                                                                         
                                                                                
\footnote{}{\llap{$~1$} This example is cheating since you need                 
to know a little about mathematics input to use it, but I wanted                
to include it anyway.  You'll be able to use these letters after                
you study the use of mathematical symbols.}                                     
\nobreak                                                                        
\toindex{it}                                                                    
\toindex{rm}                                                                    
\toindex{bf}                                                                    
\toindex{sl}                                                                    
\toindex{tt}                                                                    
                                                                                
The slanted and italic fonts seem quite similar at first.  It is                
easy to see the difference between the letter ``a'' in each                     
sample. When changing from a slanted or italic font to a roman                  
font, the last letter in the first font will lean towards the                   
first letter of the roman font; this looks cramped, and to                      
compensate there is a little extra space that can be added called               
the {\sl italic correction}. This is done using the control                     
symbol {\tt \\/}.                                                               
\toindex{/}                                                                     
In the following sentence, there is no italic correction after                  
the first sequence of italic letters but there is after the                     
second sequence; see the difference: {\it If} the italic                        
correction is not used the letters are too close together, but                  
{\it if\/} the correction is used, the spacing is better.  There                
is no need for the italic correction when the italic characters                 
are followed by a comma or a period.                                            
                                                                                
                                                                                
It is possible to use fonts other than those initially defined in               
\TeX{} when desired (assuming that they are available on                        
your computer system, of course).  Different sizes can be used                  
with the aid of the control word {\tt \\magstep}. To define your                
new font you will have to know its name on your computer system.                
For example, the roman typeface is called ``cmr10'' on most                     
systems.  If you use {\tt \\font\\bigrm = cmr10 scaled \\magstep                
1}, you can then use {\tt\\bigrm} in exactly the same way as you                
use {\tt\\it} or {\tt\\rm}.                                                     
\TeXref{13--17}                                                                 
\toindex{magstep}                                                               
\toindex{font}                                                                  
\toindex{scaled}                                                                
                                                                                
Switching via {\tt\\bigrm} will give roman type that is larger                  
than the usual by 20\%\null.  {\tt \\font\\bigbigrm = cmr10                     
scaled \\magstep 2} will define a font that is about 44\% larger                
than the usual roman typeface.  The sizes {\tt \\magstep~0} to                  
{\tt \\magstep~5} are available.  On most computers {\tt \\magstephalf}         
is also available; this represents an enlargement of                            
about 9.5\%\null. Here are some samples at various sizes:                       
                                                                                
\maketable [Different Font Magnifications]                                      
\halign{                                                                        
   \strut \hfil\tt # \hfil & \quad # \hfil \cr                                  
   \bf Magnification & \bf Sample \cr                                           
   \noalign{\hrule} \noalign{\smallskip}                                        
   \\magstep0     & \rm  Sample text at magstep 0. \cr                          
   \\magstephalf  & \halfrm  Sample text at magstephalf. \cr                    
   \\magstep 1     & \brm  Sample text at magstep 1. \cr                        
   \\magstep 2     & \bbrm  Sample text at magstep 2. \cr                       
   \\magstep 3     & \bbbrm  Sample text at magstep 3. \cr                      
   \\magstep 4     & \bbbbrm  Sample text at magstep 4. \cr                     
   \\magstep 5     & \bbbbbrm  Sample text at magstep 5. \cr                    
      }                                                                         
                                                                                
It's also possible to use completely new typefaces.  These                      
are dependent on system availability, of course.  Many systems                  
have a file called CMSS10 which is a font that contains sans                    
serif letters.  Using {\tt \\font\\sf = cmss10} will allow the                  
control word {\tt\\sf} to be used in the same manner as {\tt\\bf}.              
Having made this definition, the input                                          
\displaytext{\tt \\sf Here is a sample of our new Sans Serif font.}             
will produce                                                                    
\displaytext{\sf Here is a sample of our new Sans Serif font.}                  
                                                                                
\exercise What problem might have arisen if we had used {\tt \\ss}              
instead of {\tt \\sf} to turn on the Sans Serif font? Hint: if                  
the answer doesn't occur to you at first, think about the German                
alphabet and it will finally come to you.                                       
                                                                                
\exercise Typeset a paragraph of magnified sans serif text.                     
                                                                                
The extra fonts available may vary from site to site.  The ones                 
in the following table are available at most places.                            
                                                                                
\maketable [External Fonts Names]                                               
\halign{                                                                        
\strut \tt # \hfil & \quad \tt # \hfil & \quad \tt # \hfil                      
                   & \quad \tt # \hfil & \quad \tt # \hfil                      
                   & \quad \tt # \hfil  \cr                                     
\noalign{\hrule} \noalign{\smallskip}                                           
CMBSY10   &CMBXSL10  &CMBXTI10  &CMBX10    &CMBX12    &CMBX5      \cr           
CMBX6     &CMBX7     &CMBX8     &CMBX9     &CMB10     &CMCSC10    \cr           
CMDUNH10  &CMEX10    &CMFF10    &CMFIB8    &CMFI10    &CMITT10    \cr           
CMMIB10   &CMMI10    &CMMI12    &CMMI5     &CMMI6     &CMMI7      \cr           
CMMI8     &CMMI9     &CMR10     &CMR12     &CMR17     &CMR5       \cr           
CMR6      &CMR7      &CMR8      &CMR9      &CMSLTT10  &CMSL10     \cr           
CMSL12    &CMSL8     &CMSL9     &CMSSBX10  &CMSSDC10  &CMSSI10    \cr           
CMSSI12   &CMSSI17   &CMSSI8    &CMSSI9    &CMSSQI8   &CMSSQ8     \cr           
CMSS10    &CMSS12    &CMSS17    &CMSS8     &CMSS9     &CMSY10     \cr           
CMSY5     &CMSY6     &CMSY7     &CMSY8     &CMSY9     &CMTCSC10   \cr           
CMTEX10   &CMTEX8    &CMTEX9    &CMTI10    &CMTI12    &CMTI7      \cr           
CMTI8     &CMTI9     &CMTT10    &CMTT12    &CMTT8     &CMTT9      \cr           
CMU10     &CMVTT10                                                \cr           
}                                                                               
                                                                                
Here's a little background about these names.  The first two                    
letters {\tt CM} stands for Computer Modern, the name given to                  
this font family by the designer. The number at the end is the                  
point size: 10 point type is the normal size, 7 point is the                    
normal size for subscripts, and 5 point is the normal size for                  
subscripts of subscripts, 12 point is 20\% larger than 10 point,                
etc. If the letters {\tt CM} are followed by {\tt B}, it is a                   
boldface type.  Similarly {\tt R} indicates roman, {\tt I} is for               
italics, {\tt CSC} is for small caps, {\tt SL} is for slanted,                  
{\tt SS} is for sans serif, and {\tt TT} is for typewriter type,                
                                                                                
\exercise Find the fonts available on your system, and print out                
all the letters and numbers in several of them.                                 
                                                                                
\exercise The font {\tt CMR12} is 20\% larger than {\tt CMR10}.                 
Also {\tt \\magstep~1} magnifies the type by 20\%.  Take some                   
text and print it using {\tt CMR12} then using {\tt CMR10} scaled               
by {\tt \\magstep~1}. The results are quite different!                          
                                                                                
                                                                                
                                                                                
                                                                                
\section{The shape of things to come}                                           
                                                                                
In this section we want to see how to make text have different                  
shapes or sizes.  One may use \TeX{} with the default sizes, of                 
course, just as we have been doing so far.  We'll now become more               
creative with our output.  In discussing the size of various                    
parts of a page of text, there are several units of measure we                  
can use.                                                                        
                                                                                
\subsection{Units, units, units}                                                
                                                                                
\TeX{} can measure length using many types of units.  The most                  
common are the inch, the \centimeter, the point, the pica. The                  
abbreviations for these units are {\tt in}, {\tt cm}, {\tt pt},                 
and {\tt pc} respectively. The point is defined by the equation 1               
inch = 72.27 points, and the pica by 1 pica = 12 points. Thus a                 
point is rather small and is about the size of a decimal point.                 
Here is a sample to give an idea of the comparative sizes:                      
\def\pip{\vrule height 4 true pt }                                              
\TeXref{57}                                                                     
                                                                                
$$ \hbox{1 inch: } \hbox to 1 true in{\pip\hrulefill\pip}$$                     
$$ \hbox{1 \centimeter: }\hbox to 1 true cm{\pip\hrulefill\pip}$$               
$$ \hbox{20 points: }\hbox to 20 true pt{\pip\hrulefill\pip}$$                  
$$ \hbox{1 pica: } \hbox to 1 true pc{\pip\hrulefill\pip}$$                     
                                                                                
Thus points are used to make fine changes; a pica is about the                  
distance between the baselines of two consecutive lines of                      
(unmagnified) normal text.  \TeX{} is quite exact about                         
dimensions; internally its smallest unit is less than one                       
four-millionth of an inch.  Hence it is the resolution of the                   
output device that will determine the accuracy of the output.                   
                                                                                
There are two other units that are sometimes useful that have                   
different sizes in different fonts. The {\tt ex} is about the                   
height of a small ``x'' and the {\tt em} is a little smaller than               
the width of a capital ``M''\null.                                              
\TeXref{60}                                                                     
                                                                                
The shape of the output is generally determined by control                      
words. There are many such words; these allow very fine control                 
of the resulting text.  But most of the time only a small number                
of them are necessary.                                                          
                                                                                
\subsection{Page shape}                                                         
                                                                                
The text on a page consists of three basic parts.  There is the                 
headline above the main text; this often contains a chapter                     
title, section title, or a page number, and may differ on odd and               
even pages. Below this is the main text, which includes any                     
footnotes. And finally there is the footline that might contain a               
page number.                                                                    
                                                                                
In the examples we have seen so far, the headline has been blank.               
The footline has contained either a \centred{} number or, if {\tt               
\\nopagenumbers} had been used, has also been blank.  We'll have                
more to say about headlines and footlines later in this section.                
For the moment let's concentrate on the main text.                              
                                                                                
To finish one page and start a new one, {\tt \\vfill \\eject} can               
be used.  The control word {\tt \\eject} forces the completion of               
the current page, while {\tt \\vfill} causes any left over                      
vertical space to gathered at the bottom of the page (you might                 
try leaving out the {\tt \\vfill} as an experiment to see how the               
vertical \analog{} of justified horizontal lines of text works).                
\toindex{vfill}                                                                 
\toindex{eject}                                                                 
                                                                                
                                                                                
{\hsize=4in                                                                     
The current horizontal width of the text on the page is described               
by the control word {\tt \\hsize}.  It can be changed, say to 4                 
inches, by {\tt \\hsize = 4 in\ } at any desired time using                     
methods to be described in the next few sections. The value of                  
{\tt \\hsize} in effect when the paragraph is completed                         
determines the width of the paragraph. As can be seen in this                   
paragraph, the text width can be changed (in this case to 4                     
inches) for a single paragraph.  Because it represents the                      
current value of the horizontal width, expressions such as {\tt                 
 \\hsize = 0.75\\hsize} can make a new value that is relative                   
to the old one. \par}                                                           
\toindex{hsize}                                                                 
                                                                                
The vertical \analog{} of {\tt \\hsize} is {\tt \\vsize}, which                 
is the current height of the main text.  It can be reset, just                  
like {\tt \\hsize}.  Thus, {\tt \\vsize = 8 in\ } can be used to                
change the vertical height of the main text. Note that {\tt \\vsize}            
is the size of the text exclusive of any headline or                            
footline material.                                                              
\toindex{vsize}                                                                 
                                                                                
Text can also be shifted across the page.  The upper left corner                
of the text is one inch down and one inch in from the upper left                
corner of the page.  The control words {\tt \\hoffset} and {\tt \\voffset}      
are used to shift the text horizontally and                                     
vertically.  Thus {\tt \\hoffset = .75 in} and {\tt \\voffset =                 
-.5 in} would shift the text an additional .75 inches to the                    
right and .5 inches up. Most of the time you'll need to set {\tt                
\\hoffset}, {\tt \\voffset}, and {\tt \\vsize} at the beginning                 
of your document only.                                                          
\TeXref{251}                                                                    
\toindex{hoffset}                                                               
\toindex{voffset}                                                               
                                                                                
\maketable [Control words for page sizes]                                       
\halign{                                                                        
   \strut \hfil # & \quad \hfil \tt # \hfil & \hfil \quad # \hfil\cr            
   \bf Name & \bf \TeX{} Control Word & \bf \TeX{} default value (inches)\cr    
   \noalign{\hrule} \noalign{\smallskip}                                        
   horizontal width  & \\hsize   & 6.5 \cr                                      
   vertical width    & \\vsize   & 8.9 \cr                                      
   horizontal offset${}~1$  & \\hoffset & 0 \cr                                 
   vertical offset${}~1$   & \\voffset & 0 \cr                                  
      }                                                                         
                                                                                
\footnote{}{\llap{$~1$}The text is initially 1 inch down and 1                  
inch in from upper left corner of the page.}                                    
\nobreak                                                                        
                                                                                
\exercise Enter a paragraph of text that is a few lines long.                   
Take several copies of your paragraph and put {\tt \\hsize = 5                  
in\ } before one and {\tt \\hsize = 10 cm\ } in front of a second               
one.  Try a few other values for {\tt \\hsize}.                                 
                                                                                
\exercise Put {\tt \\hoffset = .5 in\ } and {\tt \\voffset = 1                  
in\ } before the first paragraph of your previous exercise.                     
                                                                                
\exercise Take your previous exercise and put {\tt \\vsize = 2                  
in\ } before the first paragraph.                                               
                                                                                
In the previous section we saw that it is possible to use fonts                 
of larger size by using the {\tt \\magstep} control word.  It's                 
also possible to magnify the entire document at once. If {\tt                   
\\magnification = \\magstep 1} appears at the top of your document,             
then all of your output is magnified by 20\%\null.  The other                   
values of {\tt \\magstep} can be used, too. {\bf It should be                   
emphasized that {\tt \\magnification} can only be used before                   
even a single character is typeset.}  This magnification creates                
a problem in units; if the document is to be magnified by 20\%                  
and {\tt \\hsize = 5 in} appears in the \TeX{} input file, should               
the final output have an {\tt \\hsize} of 5 inches or should it                 
be magnified by 20\% to 6 inches? Unless told otherwise, all the                
dimensions will be magnified so that, in this case, {\tt \\hsize}               
would be 6 inches in the final output.  This means that all the                 
dimensions will increase uniformly when {\tt \\magnification} is                
used.  But for some special situations it might not be desirable                
for this to happen; for example, you might want to leave a space                
of exactly 3 inches to insert a figure.  In this case any unit                  
can be modified by {\tt true} so that {\tt \\hsize = 5 true in}                 
will always set the line width to 5 inches, regardless of                       
magnification. \TeXref{59--60} \toindex{magnification}                          
                                                                                
\exercise Put {\tt \\magnification = \\magstep 1} as the first                  
line in one of your input files and see how the output changes.                 
                                                                                
\subsection{Paragraph shape}                                                    
                                                                                
When \TeX{} is reading the text to be typeset from your input                   
file, it reads in a paragraph at a time and then typesets it.                   
This means that there is a lot of control over paragraph shapes,                
but also that some care must be taken.  We've already seen that                 
{\tt \\hsize} can be used to control the width of a single                      
paragraph.  But suppose the following appeared as part of your                  
input file:                                                                     
                                                                                
\beginuser                                                                      
\\hsize = 5 in                                                                  
Four score and seven years $\ldots$                                             
$\vdots$                                                                        
$\ldots$ from this earth.                                                       
\\hsize = 6.5 in                                                                
                                                                                
\enduser                                                                        
                                                                                
What is the width of the paragraph?  The {\tt \\hsize} was                      
changed at the start of the paragraph and again at the end.                     
Since the paragraph was not completed (by putting in a blank line               
or {\tt \\par}) until after the second change, the paragraph                    
would be typeset with a width of 6.5 inches. However, if a blank                
line were inserted before {\tt \\hsize = 6.5 in}, then the                      
paragraph would be typeset with a width of 5 inches.  So, in                    
general, when a paragraph is set, the values of the parameters                  
that are in effect when the paragraph is completed are the ones                 
that are used.                                                                  
                                                                                
Here is a table with some paragraph parameters:                                 
                                                                                
\maketable [Some paragraph shape parameters]                                    
\halign{                                                                        
   \strut \hfil # & \quad \hfil \tt # \hfil & \hfil \quad # \hfil\cr            
   \bf Function & \bf \TeX{} Control Word & \bf \TeX{} default\cr               
   \noalign{\hrule} \noalign{\smallskip}                                        
   width & \\hsize & 6.5 inches \cr                                             
   indentation on first line & \\parindent & 20 points\cr                       
   distance between lines   & \\baselineskip & 12 points\cr                     
   distance between paragraphs & \\parskip & 0 points \cr                       
      }                                                                         
                                                                                
\toindex{parindent}                                                             
\toindex{parskip}                                                               
\toindex{baselineskip}                                                          
                                                                                
The control word {\tt \\noindent} may be used at the beginning of               
a paragraph to avoid the automatic indentation on the first line.               
This control word will only affect the paragraph being set when                 
it is invoked (of course {\tt \\parindent = 0 pt} will cause no                 
indentation for all paragraphs).                                                
\toindex{noindent}                                                              
                                                                                
{\narrower                                                                      
A more flexible way to control the width of a paragraph is to use               
{\tt \\rightskip} and {\tt \\leftskip}.  Setting {\tt \\leftskip                
= 20 pt} causes the left margin of the paragraph to be moved in                 
an extra twenty points. Negative values may be assigned to {\tt                 
\\leftskip} to move the left margin out. The control word {\tt                  
\\rightskip} does the same to the right side of the paragraph.                  
The single control word {\tt \\narrower} is the equivalent of                   
setting both {\tt \\leftskip} and {\tt \\rightskip} equal to {\tt               
\\parindent}.  This is quite useful for long quotations, and                    
this paragraph is an example of its use. As with {\tt \\hsize},                 
the value of {\tt \\leftskip} and {\tt \\rightskip} in effect                   
when the paragraph is completed is the one which will apply to                  
the whole paragraph.                                                            
\TeXref{100}                                                                    
                                                                                
}                                                                               
\toindex{leftskip}                                                              
\toindex{rightskip}                                                             
\toindex{narrower}                                                              
                                                                                
\exercise Make two paragraphs with the following specifications:                
the left margin of both paragraphs is indented by 1.5 inches,                   
the right margin of the first paragraph is indented by 0.75                     
inches, and the right margin of the second paragraph is indented                
by 1.75 inches.                                                                 
                                                                                
\def\hangparagraph{ Lines can be made with different lengths                    
within one paragraph by using {\tt \\hangindent} and {\tt                       
\\hangafter}.  The amount of the indentation is determined by value             
of {\tt \\hangindent}. If {\tt \\hangindent} is positive, the                   
indentation is made from the left, and if it is negative it is                  
made from the right. The lines on which the indentation occurs is               
controlled by {\tt \\hangafter}.  If {\tt \\hangafter} is                       
positive then it determines the number of lines at full width                   
before the indentation starts.  Thus if {\tt \\hangindent = 1.75                
in} and {\tt \\hangafter = 6}, then the first six lines will be                 
at full width and the rest will be indented by 1.75 inches from                 
the left. On the other hand if {\tt \\hangindent = \hbox{-1.75}                 
in} and {\tt \\hangafter = -6}, then the first six lines will be                
indented by 1.75 inches from the right and the rest will be at                  
full width. \TeX{} resets to the default values {\tt \\hangindent               
= 0 pt} and {\tt \\hangafter = 1} after each paragraph. These                   
control words are useful for paragraphs with ``hanging indents''                
and for flowing a paragraph around space reserved for a figure.                 
\TeXref{355}                                                                    
The control word {\tt \\hang} at the beginning of the                           
paragraph will cause the first line to be of full width ({\tt                   
\\hangafter=1}) and the rest of the paragraph to be indented by                 
the current value of {\tt \\parindent}. But you do have to use                  
{\tt \\noindent} if you want the first line to extend all the way               
to the left margin.                                                             
\TeXref{102}                                                                    
\par}                                                                           
                                                                                
\hangafter=6 \hangindent=1.75in                                                 
\hangparagraph                                                                  
\toindex{hangindent}                                                            
\toindex{hangafter}                                                             
\toindex{hang}                                                                  
                                                                                
Here is the previous paragraph repeated with {\tt \\hangafter = -6}             
and {\tt \\hangindent = -1.75}.                                                 
                                                                                
\hangafter=-6 \hangindent=-1.75in \hangparagraph                                
                                                                                
                                                                                
The control word {\tt \\parshape} can be used to make paragraphs with           
a greater variety of shapes.                                                    
\TeXref{101}                                                                    
\toindex{parshape}                                                              
                                                                                
Another useful control word for setting paragraphs is {\tt \\item}.             
It can be used to make various types of itemized lists,                         
and is invoked using the pattern {\tt \\item\lb$\ldots$\rb}\null.               
This causes the next  paragraph to be formed with every line is                 
indented by {\tt \\parindent} and, in addition, the first line                  
labeled on the left by whatever is between the braces.  It is                   
usually used with {\tt \\parskip = 0 pt}, since that control word               
determines the vertical space between the different items. The control          
word {\tt \\itemitem} is the same as {\tt \\item} except that the               
indentation is twice as far, that is, twice the value of                        
{\tt \\parindent}.                                                              
\TeXref{102}                                                                    
Here is an example:                                                             
\toindex{item}                                                                  
\toindex{itemitem}                                                              
                                                                                
\beginliteral                                                                   
\parskip = 0pt \parindent = 30 pt                                               
\noindent                                                                       
Answer all the following questions:                                             
\item{(1)} What is question 1?                                                  
\item{(2)} What is question 2?                                                  
\item{(3)} What is question 3?                                                  
\itemitem{(3a)} What is question 3a?                                            
\itemitem{(3b)} What is question 3b?                                            
@endliteral                                                                     
                                                                                
                                                                                
\noindent                                                                       
will produce                                                                    
\vskip\baselineskip                                                             
                                                                                
{\parskip = 0pt \parindent = 30 pt                                              
\noindent                                                                       
Answer all the following questions:                                             
\item{(1)} What is question 1?                                                  
\item{(2)} What is question 2?                                                  
\item{(3)} What is question 3?                                                  
\itemitem{(3a)} What is question 3a?                                            
\itemitem{(3b)} What is question 3b?                                            
                                                                                
}                                                                               
                                                                                
\exercise Make a paragraph several lines long and use it with                   
the {\tt \\item} control word to see the ``hanging indent.'' Now                
take the same paragraph and use it with different values of {\tt                
\\parindent} and {\tt \\hsize}.                                                 
                                                                                
Now let's see how to put space between paragraphs.  The control                 
word {\tt \\parskip} is used to determine how much space is                     
normally left between paragraphs. So if you put {\tt \\parskip =                
12 pt} at the beginning of your \TeX{} source file, there will be               
12 points between paragraphs unless other instructions are                      
given. The control word {\tt \\vskip} can be used to insert extra               
vertical space between paragraphs.  If {\tt \\vskip 1 in} or {\tt               
\\vskip 20 pt} appears between two paragraphs, then the extra                   
space is inserted.                                                              
                                                                                
There are a couple of peculiarities of {\tt \\vskip} that seem                  
quite strange at first.  If you have {\tt \\vskip 3 in} and the                 
skip starts two inches from the bottom of the page, the rest of                 
the page is skipped, but the extra one inch is {\it not\/}                      
skipped at the top of the next page.  In other words, {\tt \\vskip}             
{\bf will not insert space across page boundaries}.  In                         
fact, {\tt \\vskip~1~in} will have no effect at all if it happens               
to appear at the top of a page! For many types of vertical                      
spacing this is quite appropriate.  The space before a section                  
heading, for example, should not continue across page boundaries.               
                                                                                
A similar phenominon occurs at the beginning of your document.                  
If, for example, you want a title page with the title about half                
way down the page, you can not insert the space at the top of the               
page using {\tt \\vskip}.                                                       
                                                                                
But what if you really want some blank space at the top of a                    
page? You could start the page with {\tt\\\sp} but this in effect               
typesets a one line paragraph containing a blank. So while                      
nothing is typeset, the extra space due to the values of {\tt                   
\\baselineskip} and {\tt \\parskip} will add extra space. An easier             
method is to use {\tt \\vglue} instead of {\tt \\vskip} to get                  
the desired result.  Thus {\tt \\vglue 1 in\ }will leave one inch               
of blank space at the top of the page.\TeXref{352}                              
\toindex{vglue}                                                                 
                                                                                
                                                                                
We can note in passing that there is another more general method                
to put material at the top of a page using the control words {\tt               
\\topinsert} and {\tt \\endinsert}. If {\tt \\topinsert $\ldots$                
\\endinsert} is used within a page, the material between                        
{\tt \\topinsert} and {\tt \\endinsert} will appear at the top                  
of the page, if possible. For example:                                          
                                                                                
\beginuser                                                                      
\\topinsert                                                                     
\\vskip 1 in                                                                    
\\centerline\lb Figure 1\rb                                                     
\\endinsert                                                                     
\enduser                                                                        
\noindent                                                                       
is useful for inserting figures of a prescribed size.                           
\TeXref{115}                                                                    
\toindex{topinsert}                                                             
\toindex{endinsert}                                                             
                                                                                
There are also some special control words for making small                      
vertical skips. They are {\tt \\smallskip}, {\tt \\medskip}, and                
{\tt \\bigskip}. Here is the size of each one:                                  
                                                                                
\def\hrl{\hrule width .5 in}                                                    
                                                                                
\centerline{{\tt \\smallskip}: \vbox{\hrl \smallskip \hrl} \quad                
            {\tt \\medskip}: \vbox{\hrl \medskip \hrl} \quad                    
            {\tt \\bigskip}: \vbox{\hrl \bigskip \hrl}                          
            }                                                                   
\toindex{smallskip}                                                             
\toindex{medskip}                                                               
\toindex{bigskip}                                                               
                                                                                
\subsection{Line shape}                                                         
                                                                                
For most material \TeX{} does a good job of breaking up a                       
paragraph into lines.  But sometimes it's necessary to add                      
further instructions. It's possible to force a new line by                      
inserting {\tt \\hfill \\break} in your input file. It's also                   
possible to  put some text on a line by itself using the control                
word {\tt \\line\lb $\ldots$\rb}; then the material between the                 
braces will be restricted to one line (although the result will                 
be spread out over the whole line and may be horrible). The                     
control words {\tt \\leftline\lb $\ldots$\rb}, {\tt \\rightline\lb              
$\ldots$\rb}, and {\tt \\centerline\lb $\ldots$\rb} will,                       
respectively, set the material between the braces on a single                   
line on the left margin, on the right margin, or \centred.  Thus                
\toindex{hfill}                                                                 
\toindex{break}                                                                 
\toindex{centerline}                                                            
\toindex{leftline}                                                              
\toindex{rightline}                                                             
\toindex{line}                                                                  
                                                                                
                                                                                
\beginliteral                                                                   
\leftline{I'm over on the left.}                                                
\centerline{I'm in the @centre.}                                                
\rightline{I'm on the right.}                                                   
\line{I just seem to be spread out all over the place.}                         
@endliteral                                                                     
                                                                                
\noindent                                                                       
produces the four lines:                                                        
\vskip\baselineskip                                                             
                                                                                
                                                                                
{                                                                               
\hbadness = 10000                                                               
\leftline{I'm over on the left.}                                                
\centerline{I'm in the \center.}                                                
\rightline{I'm on the right.}                                                   
\line{I just seem to be spread out all over the place.}                         
                                                                                
Other types of spacing can be obtained by using the control word {\tt           
\\hfil}.  This causes all the extra space in the line to be                     
accumulated at the position where {\tt \\hfil} appears.                         
Thus if we alter our last example to                                            
{\tt \\line\lb I just seem to be spread out \\hfil all over the place.\rb}      
we will get \hfil\break                                                         
\medskip                                                                        
\line{I just seem to be spread out \hfil all over the place.}                   
\medskip                                                                        
If we have more than one {\tt \\hfil}, they will divide any extra               
space among themselves equally.  Hence                                          
{\tt \\line\lb left text \\hfil \centre{} text\\hfil right text.\rb}            
will produce                                                                    
\medskip                                                                        
\line{left text \hfil \centre{} text \hfil right text.}                         
\toindex{hfil}                                                                  
}                                                                               
                                                                                
\exercise Typeset the following line:                                           
                                                                                
\line{left end \hfil left tackle \hfil left guard \hfil \centre{} \hfil         
right guard \hfil right tackle \hfil right end}                                 
                                                                                
\exercise Typeset the following line with twice as much space between           
``left'' and ``right-\centre{}'' as between ``right-\centre{}'' and             
``right'':                                                                      
                                                                                
\line{left \hfil \hfil right-\centre{} \hfil right}                             
                                                                                
It's possible to move horizontally using {\tt \\hskip} in a                     
manner analogous with {\tt \\vskip}.                                            
\toindex{hskip}                                                                 
                                                                                
\exercise What happens with the following input:\hfil\break                     
{\tt \\line\lb\\hskip 1 in ONE \\hfil TWO \\hfil THREE\rb}                      
                                                                                
The right justification can be canceled by using the                            
control word {\tt \\raggedright}.                                               
\toindex{raggedright}                                                           
                                                                                
\subsection{Footnotes}                                                          
                                                                                
The general pattern to make footnotes using \TeX{} is                           
{\tt \\footnote\lb$\ldots$\rb\lb$\ldots$\rb}\null.                              
                                                                                
The footnote mark goes in between the first set of braces.  Some                
available marks are {\tt \\dag (\dag)}, {\tt \\ddag (\ddag)}, {\tt \\S          
(\S)}, and {\tt \\P (\P)}\null.  The text of the footnote                       
goes between the second set of braces. The use of numbers as                    
marks is a little less straightforward. The footnote%                           
\footnote{${}~{21}$}{This is the footnote at the bottom of the page.}           
at the bottom of the page was produced by using {\tt \\footnote\lb\$\lb         
\rb\~{}\lb{21}\rb\$\rb\lb This is the footnote at the                           
bottom of the page.\rb}  after the word ``footnote'' in the                     
text. This construction is somewhat complicated; we'll see why                  
it's this way when we know a little more about typesetting                      
mathematics. But for the moment we can look at it as a way of                   
getting the job done. You may want to use {\tt \\rm} within the                 
footnote to ensure that the right font appears.                                 
\TeXref{117}                                                                    
\toindex{footnote}                                                              
\toindex{ddag}                                                                  
\toindex{S}                                                                     
\toindex{P}                                                                     
                                                                                
\exercise Make up a page with a relatively long footnote spanning               
several lines.                                                                  
                                                                                
\exercise Make up a page with two different footnotes on it.                    
                                                                                
\subsection{Headlines and Footlines}                                            
                                                                                
\headline={\hfil \tenrm Page \the\pageno} %example for this section             
                                                                                
The lines for title and page numbers that go above and below the                
main text are produced by using {\tt \\headline=\lb$\dots$\rb}                  
and {\tt \\footline=\lb$\dots$\rb}\null.                                        
\TeXref{252--253}                                                               
                                                                                
The principle is the same as using the control word {\tt \\line\lb              
$\dots$\rb} within the usual text on the page.  A helpful                       
control word is {\tt \\pageno} which represents the current page                
number. Thus {\tt \\headline=\lb\\hfil \\tenrm Page \\the\\pageno\rb}           
would cause the page number to appear in the upper                              
right corner preceded by the word ``Page'' (now look at the upper               
right corner of this page). It is safer to explicitly name the                  
font that you want to use (in this case {\tt \\tenrm} to use the                
10 point roman font), since you are never guaranteed that a                     
particular font will be in use when the headline or footline is                 
set. The control word {\tt \\the} takes the internal value of the               
next word if it is an appropriate control word and prints it as                 
text. You can also use the control word {\tt \\folio} instead of                
{\tt \\the \\pageno}. The difference is that {\tt \\folio} will                 
give roman numerals when {\tt \\pageno} is negative.                            
\toindex{headline}                                                              
\toindex{footline}                                                              
\toindex{pageno}                                                                
\toindex{the}                                                                   
\toindex{folio}                                                                 
                                                                                
You can also assign values to {\tt \\pageno} if you want your                   
document to use a different sequence of page numbers from the                   
usual.  Roman numerals can be produced by using negative numbers;               
{\tt \\pageno=-1} at the beginning of a document will cause page                
numbers to be in roman numerals.                                                
\TeXref{252}                                                                    
                                                                                
Different headlines can be produced for even and odd pages in                   
the following manner:                                                           
                                                                                
{\tt                                                                            
\\headline=\lb\\ifodd \\pageno \lb$\dots$\rb \\else \lb$\dots$\rb               
\\fi\rb}                                                                        
                                                                                
\noindent                                                                       
where the material between the first set of braces is for the                   
right-hand pages and the material between the second set of                     
braces is for the left-hand pages.                                              
                                                                                
\exercise Change the footline so that the page number is \centred{}             
with a en-dash on either side.                                                  
                                                                                
\subsection{Overfull and underfull boxes}                                       
                                                                                
One of the most frustrating experiences for the new \TeX{} user                 
is the occurrence of overfull and underfull boxes.  They are duly               
noted in the log file and are also written to the screen when \TeX{}            
is being run interactively.  Overfull boxes are also marked                     
on the output by a slug (a large filled-in black rectangle that                 
looks like this: \vrule width \overfullrule) in the right                       
margin. Various obscene terms are applied to this slug.  It                     
appears even though there is nothing wrong with the \TeX{} input.               
So why is the slug there and what can be done about it?                         
                                                                                
A good way to visualize the way \TeX{} organizes a page is to                   
think of the printed material as being put into boxes. There are                
two types of boxes: {\sl hboxes\/} and {\sl vboxes}.  Most of the               
time these correspond to the organization of horizontal text                    
into lines and vertical paragraphs into pages.  In particular, it               
is the spacing of the words in a hbox corresponding to a line of                
text that causes the slug to appear.                                            
                                                                                
Recall that \TeX{} reads in the complete paragraph before                       
deciding how to break it up into lines.  This is better than                    
working a line at a time, since a slight improvement in one line                
might cause catastrophic changes farther down in the paragraph.                 
When words are gathered together to form a line, space is added                 
between the words to justify the right margin. Very large spaces                
between words is obviously undesirable; the badness of the line                 
is a measure of how badly the words are spaced.  An underfull                   
hbox means that there is too much space between words.  More                    
specifically, the badness of any line is a number between 0                     
(perfect) and 10000 (horrible).  There is a parameter called {\tt               
\\hbadness} whose default value is 1000\null.  Any line whose                   
badness is greater than {\tt \\hbadness} is reported as an                      
underfull hbox.  If the value of {\tt \\hbadness} is increased,                 
then fewer underfull hboxes will be reported. In fact {\tt                      
\\hbadness = 10000} will suppress the reporting of underfull                    
hboxes altogether.                                                              
\toindex{hbaddness}                                                             
                                                                                
%%%%%%%%%%%%%% restore original headline %%%%%%%%%%%%%%%%%%%%%%%%               
\headline={\iftitlepage \hfil \global\titlepagefalse                            
                                      \else \draftheadline \fi}                 
                                                                                
In a similar way, sometimes \TeX{} allows a line to be slightly                 
longer than {\tt \\hsize} in order to achieve a more balanced                   
appearance. The control word {\tt \\tolerance} determines when                  
this happens.  If the badness of a line is greater than {\tt                    
\\tolerance}, \TeX{} will make the line longer by adding a new word             
at the end of the line, even though it might exceed the value of                
{\tt \\hsize}. A line which is only slightly longer is set                      
without being reported. The control word {\tt \\hfuzz} determines               
how much excess is allowed; the default is {\tt \\hfuzz                         
=~0.1~pt}.  A line that is more than slightly longer than {\tt \\hsize}         
is obviously a serious problem; \TeX{} puts a slug in                           
the margin to make its point forcefully. It's possible to avoid                 
overfull hboxes altogether by increasing the value of {\tt \\tolerance}.        
With {\tt \\tolerance = 10000} there will never be                              
overfull boxes or slugs. The default is {\tt \\tolerance = 200}\null.           
\TeXref{29}                                                                     
\toindex{hfuzz}                                                                 
\toindex{tolerance}                                                             
                                                                                
The width of the slug is determined by the control word {\tt                    
\\overfullrule}.  Including {\tt \\overfullrule = 0 pt} in your                 
file will delete any further  printing of the slugs.  The                       
overfull boxes will still be there, of course, and they will                    
probably be harder to spot.                                                     
\toindex{overfullrule}                                                          
                                                                                
So we see why overfull boxes and underfull boxes are reported; we               
can also change the reporting by changing the values of {\tt                    
\\badness}, {\tt \\hfuzz}, and {\tt \\tolerance}.  In addition, a               
small value of {\tt \\hsize} obviously makes it more difficult to               
set lines and causes more overfull and underfull hboxes to be                   
reported. These are warnings that you ignore at your own peril!                 
                                                                                
Adding new possibilities of hyphenation will sometimes eliminate                
an overfull box. \TeX{} has automatic hyphenation and usually                   
finds good places to hyphenate a word; however, it's possible to                
add hyphenations to let lines break at new places.  For example,                
the automatic hyphenation will never put a hyphen in the word                   
``database''.  If you type in {\tt data\\-base}, it will allow a                
hyphen to be inserted after the second letter ``a'' in the word.                
More generally, if you put {\tt \\hyphenation\lb data-base\rb} in               
the beginning of your input file, then all occurrences of the                   
word ``database'' will allow hyphenation after the letter ``a''.                
\TeXref{28}                                                                     
The log file will show the possible hyphenations on the line                    
containing the overfull or underfull box. Sometimes the best                    
solution to an overfull or underfull hbox is a little judicious                 
editing of the original document.                                               
\toindex{hyphenation}                                                           
                                                                                
Our discussion has involved the setting of type into lines, that is,            
the horizontal page structure.  There are several vertical \analog{}s.          
Overfull and underfull hboxes indicate how well words are gathered              
into lines.  Similarly, overfull and underfull vboxes are reported              
when paragraphs are gathered to form pages.  A large table that                 
can't be broken in the middle, for example, can produce an underfull            
vbox that is reported in the log file when the page being typeset is            
completed.  The control word {\tt \\vbadness} works for the vertical            
placement of text in the same way as {\tt \\hbadness} works for                 
horizontal text.                                                                
\toindex{vbadness}                                                              
                                                                                
\exercise Take a few paragraphs and print them using various (small)            
values of {\tt \\hsize} to see what kind of overfull boxes result.              
Repeat with various values of {\tt \\hbadness}, {\tt \\hfuzz}, and              
{\tt \\tolerance}.                                                              
                                                                                
                                                                                
                                                                                
                                                                                
                                                                                
                                                                                
\section{$\Bigl\{$Groups, $\bigl\{$Groups,                                      
        $\{$and More Groups$\}\bigr\}\Bigr\}$}                                  
                                                                                
The concept of gathering text into groups allows \TeX{} input                   
files to be greatly simplified.  A new group is started by the                  
character {\tt\lb} and terminated by the character {\tt \rb}\null.              
Changes made within a group will lose their effect when the                     
group terminates.  So, for example if {\tt \lb \\bf three                       
boldface words\rb} appears in your text, the opening brace starts               
the group, the {\tt \\bf} control word changes to a boldface                    
font, and the closing brace finishes up the group.  Upon                        
completion of the group the font being used is the one in effect                
before the group started.  This is the (easier) way of having a                 
few words in a different font.  It's also possible to have groups               
nested within groups.                                                           
                                                                                
                                                                                
As another example, size changes can be made in the text that are               
only temporary.  For example                                                    
\beginuser                                                                      
\lb                                                                             
\\hsize = 4 in                                                                  
\\parindent = 0 pt                                                              
\\leftskip = 1 in                                                               
will produce a paragraph that is four                                           
$\vdots$                                                                        
(this is an easy mistake to make).                                              
\\par                                                                           
\rb                                                                             
\enduser                                                                        
                                                                                
{\hsize = 4 in                                                                  
\parindent = 0 pt                                                               
\leftskip = 1 in                                                                
will produce a paragraph that is four inches wide with the text                 
offset into the paragraph by one inch regardless of the settings                
in effect before the start of the group.  This paragraph is set                 
with those values. After the end of the group, the old settings                 
are in effect again. Note that it is necessary to include                       
{\tt \\par} or to use a blank line before the closing brace to end              
the paragraph, since otherwise the group will end and \TeX{} will               
go back to the old parameters before the paragraph is actually                  
typeset (this is an easy mistake to make). \par}                                
                                                                                
When a control word (like {\tt \\centerline}) acts on text                      
following it in braces, that text is implicitly in a group. Thus                
{\tt \\centerline\lb\\bf A bold title\rb} will produce a \centred{}             
boldface line, and the text following that line will be in                      
whatever font was in effect before the {\tt \\centerline} was                   
invoked.                                                                        
                                                                                
The empty group {\tt\lb\rb} is useful.  One use allows accents to be            
typeset with no accompanying letter.  For example, {\tt \\\~{}\lb\rb}           
will print a tilde with no letter under it.  It can also be used to             
stop \TeX{} from eating up consecutive spaces. Hence {\tt I use                 
\\TeX\lb\rb{} all the time}  will leave a space after ``\TeX'' in the           
output. This is an alternative to using {\tt \\\sp} as we did in                
Section~1.                                                                      
\TeXref{19--21}                                                                 
                                                                                
\exercise Change the dimensions of one paragraph on a page using the            
grouping idea.                                                                  
                                                                                
\exercise Mathematicians sometime use the word ``i{f}f'' as an                  
abbreviation for ``if and only if''\null.  In this case it looks                
better if the first and second letter ``f'' are {\sl not\/}                     
joined as a ligature.  How do you do this (there are several                    
solutions)?                                                                     
                                                                                
It's really easy to forget to match braces properly.  The effect                
can be dramatic; if you get output that suddenly changes to an                  
italic font for the rest of the document, a mismatched brace is                 
probably the cause.  If you have an extra {\tt \lb} \TeX{} will                 
give a message in the log file: {\tt (\\end occurred inside a                   
group at level 1)}. An extra {\tt\rb} will result in the message                
\hbox{\tt! Too many \rb's.}                                                     
                                                                                
Here's a little hint to help you keep track of the braces in more               
complicated groups: put the opening brace on a line by itself and               
do the same for the closing brace.  If there are braces nested                  
within the original ones, put them on separate lines also, but                  
indent them a few spaces.  The text within the nested braces can                
also be indented since \TeX{} ignores all spaces at the beginning               
of a line. The matching braces will then stand out when you look                
at your \TeX{} source file.  In fact, if your editor is smart                   
enough, you can create the two lines with the braces first and                  
then insert the appropriate material within them with automatic                 
indenting.                                                                      
                                                                                
                                                                                
                                                                                
\section{No math anxiety here!}                                                 
                                                                                
\TeX{} is at its best when typesetting mathematics.  The                        
conventions for doing this are many and complex, and the ability                
of \TeX{} to take them into account makes the production of high                
quality, attractive mathematical output possible.  If you plan to               
produce papers with mathematical symbols in them, this section                  
will give you all the basics necessary for creating beautiful                   
output in almost all circumstances; \TeX{} may be used without                  
any mathematics, of course, and if this is your goal, then the                  
following two subsections are probably sufficient for your needs.               
                                                                                
\subsection{Lots of new symbols}                                                
                                                                                
Mathematical text is inserted into normal text in two possible                  
ways: it can be {\sl in-line\/}, that is, as part of the usual                  
lines of text.  It can also be {\sl displayed}, that is, \centred{}             
in a gap between the usual text.  The results in the spacing and                
placement of symbols can be quite different in each case.  The                  
in-line equation $\sum_{k=1}~{\infty} {1\over k~2} = {\pi~2\over6}$             
is easily seen to be different from the same equation when                      
displayed:                                                                      
$$\sum_{k=1}~{\infty} {1\over k~2} = {\pi~2\over6}.$$                           
                                                                                
Since the spacing and the fonts used for mathematics are quite                  
different from those of ordinary text, \TeX{} needs to be told                  
when mathematics rather than normal text is being typeset.  This                
is done using the {\tt\$} symbol.  More specifically, mathematics               
is typeset in-line by surrounding the symbols to be entered by                  
single dollar signs: {\tt \$$\ldots$\$}, and is typeset displayed               
by surrounding the symbols to be entered by double dollar signs:                
{\tt \$\$$\ldots$\$\$}\null.  Hence {\tt \$x = y+1\$} will give                 
$x=y+1$ in-line while {\tt \$\$x = y+1.\$\$} will display                       
$$x=y+1.$$                                                                      
                                                                                
The spacing for both in-line and displayed mathematics is                       
completely controlled by \TeX\null.  Adding spaces to your input                
has no effect at all.  What if you need a space or some text in                 
the middle of some mathematics?  You can insert text by inserting               
it into an hbox (don't worry about the definition of an hbox for                
now): {\tt \\hbox\lb$\ldots$\rb}\null.  This is particularly                    
useful for displayed mathematics.  Hence ``$x=y+1 \hbox{ whenever               
} y=x-1$'' can be typeset using {\tt \$x=y+1 \\hbox\lb\ whenever                
\rb y=x-1\$}. Note the spaces on either side of the word within                 
the braces. Usually it isn't necessary to insert spaces into                    
mathematical text, but here are the control sequences that do the               
job.                                                                            
\TeXref{167}                                                                    
                                                                                
\maketable [Adding space to mathematical text]                                  
\halign{                                                                        
\strut \hfil # & \quad \hfil\tt# \hfil \quad                                    
   & \hbox to 2cm{\hrulefill\vrule height 8pt#\vrule height 8pt\hrulefill} \cr  
   Name & \rm Control Sequence & \hfil{}$\gets$Size$\to$\cr                     
   \noalign{\hrule} \noalign{\smallskip}                                        
   Double quad         & \\qquad &\qquad \cr                                    
   Quad                & \\quad  &\quad \cr                                     
   Space               & \\\sp   &\ \cr                                         
   Thick space         & \\;     &$\;$\cr                                       
   Medium space        & \\>     &$\>$\cr                                       
   Thin space          & \\,     &$\,$\cr                                       
   Negative thin space & \ \\!\    &$\!$\cr                                     
       }                                                                        
                                                                                
\toindex{quad}                                                                  
\toindex{qquad}                                                                 
\toindex{\sp}                                                                   
\toindex{;}                                                                     
\toindex{>}                                                                     
\toindex{,}                                                                     
\toindex{!}                                                                     
                                                                                
If you look closely at the negative thin space, you'll notice                   
that, unlike the other entries, the two arms overlap.  This is                  
because the negative space is one in the opposite direction, that               
is, while the other control sequences increase the amount of                    
space between two symbols being typeset, the negative thin space                
decreases the space between them, even if it causes them to                     
overlap.                                                                        
                                                                                
\exercise Typeset the following: $C(n,r) = n!/(r!\,(n-r)!)$\null. Note the      
spacing in the denominator.                                                     
                                                                                
You shouldn't have any blank lines between the dollar signs                     
delimiting the mathematical text. \TeX{} assumes that all the                   
mathematical text being typeset is in one paragraph, and a blank                
line starts a new paragraph; consequently, this will generate an                
error message. This turns out to be useful, for one of the                      
easiest errors to make is to forget to put in the trailing dollar               
sign(s) after the mathematical input (I promise that you'll do it               
at least once while learning \TeX{}); if \TeX{} allowed more than               
one paragraph to be between the dollars signs, then one omitted                 
trailing dollar sign might cause the rest of the document to be                 
typeset as mathematics.                                                         
                                                                                
Most mathematical text is entered in exactly the same way for                   
in-line typesetting as for displayed typesetting (except for the                
surrounding dollar signs, of course). The exceptions, such as                   
aligning multiline displays and placing equation numbers at the                 
left or right margin will be discussed in the last part of this                 
section.                                                                        
                                                                                
Many new symbols can appear when typesetting mathematics.  Most                 
of the ones that actually appear on the keyboard can be used                    
directly. The symbols {\tt + - / * = ' ! < > ( )} are all entered               
directly. Here they are as mathematics: $+ - / * =\> '\> ! <\>                  
>\> ( \> )$.                                                                    
                                                                                
\exercise Typeset the equation $a+b=c-d=xy=w/z$ as in-line and                  
displayed mathematical text.                                                    
                                                                                
\exercise Typeset the equation $(fg)' = f'g + fg'$ as in-line and               
displayed mathematical text.                                                    
                                                                                
Many other symbols, as you would expect, are predefined control                 
words.  All Greek letters are available.  Here is a table of them:              
\TeXref{434}                                                                    
                                                                                
\maketable [Greek letters]                                                      
\halign{                                                                        
\strut \hfil$#$ & \quad \tt# \hfil \qquad &\hfil$#$ & \quad \tt# \hfil \qquad   
      &\hfil$#$ & \quad \tt# \hfil \qquad &\hfil$#$ & \quad \tt# \hfil \cr      
\noalign{\hrule} \noalign{\smallskip}                                           
\alpha & \\alpha &\beta & \\beta &\gamma & \\gamma &\delta & \\delta \cr        
\epsilon & \\epsilon & \varepsilon & \\varepsilon & \zeta & \\zeta &            
\eta & \\eta \cr                                                                
\theta & \\theta & \vartheta & \\vartheta & \iota & \\iota & \kappa &           
\\kappa \cr                                                                     
\lambda & \\lambda & \mu & \\mu & \nu & \\nu & \xi & \\xi \cr                   
o & o & \pi & \\pi & \rho & \\rho & \varrho & \\varrho \cr                      
\sigma & \\sigma & \varsigma & \\varsigma & \tau & \\tau & \upsilon &           
\\upsilon \cr                                                                   
\phi & \\phi & \varphi & \\varphi & \chi & \\chi & \psi & \\psi \cr             
\omega & \\omega & \Gamma & \\Gamma & \Delta & \\Delta & \Theta &               
\\Theta \cr                                                                     
\Lambda & \\Lambda & \Xi & \\Xi & \Pi & \\Pi & \Sigma & \\Sigma \cr             
\Upsilon & \\Upsilon & \Phi & \\Phi & \Psi & \\Psi & \Omega & \\Omega \cr       
       }                                                                        
                                                                                
\toindex{alpha}                                                                 
\toindex{beta}                                                                  
\toindex{gamma}                                                                 
\toindex{delta}                                                                 
\toindex{epsilon}                                                               
\toindex{varepsilon}                                                            
\toindex{zeta}                                                                  
\toindex{eta}                                                                   
\toindex{theta}                                                                 
\toindex{vartheta}                                                              
\toindex{iota}                                                                  
\toindex{kappa}                                                                 
\toindex{lambda}                                                                
\toindex{mu}                                                                    
\toindex{nu}                                                                    
\toindex{xi}                                                                    
\toindex{pi}                                                                    
\toindex{rho}                                                                   
\toindex{varrho}                                                                
\toindex{sigma}                                                                 
\toindex{varsigma}                                                              
\toindex{tau}                                                                   
\toindex{upsilon}                                                               
\toindex{phi}                                                                   
\toindex{varphi}                                                                
\toindex{chi}                                                                   
\toindex{psi}                                                                   
\toindex{omega}                                                                 
\toindex{Gamma}                                                                 
\toindex{Delta}                                                                 
\toindex{Theta}                                                                 
\toindex{Lambda}                                                                
\toindex{Xi}                                                                    
\toindex{Pi}                                                                    
\toindex{Sigma}                                                                 
\toindex{Upsilon}                                                               
\toindex{Phi}                                                                   
\toindex{Psi}                                                                   
\toindex{Omega}                                                                 
                                                                                
                                                                                
\exercise Typeset $\alpha\beta=\gamma+\delta$ as in-line and displayed          
mathematical text.                                                              
                                                                                
\exercise Typeset $\Gamma(n) = (n-1)!$ as in-line and displayed                 
mathematical text.                                                              
                                                                                
Sometimes accents are put above or below symbols.  The control                  
words used for accents in mathematics are different from those                  
used for normal text.   The normal text control words may not be                
used for mathematics and vice-versa.                                            
\TeXref{135--136}                                                               
                                                                                
\maketable [Mathematical accents]                                               
\halign{                                                                        
   \strut \hfil$#$ & \quad \tt# \hfil \qquad &\hfil$#$ &                        
      \quad \tt# \hfil \qquad &\hfil$#$ & \quad \tt# \hfil \qquad  \cr          
   \noalign{\hrule} \noalign{\smallskip}                                        
   \hat o   & \\hat o   & \check o & \\check o  & \tilde o & \\tilde o \cr      
   \acute o & \\acute o & \grave o & \\grave o  & \dot o & \\dot o \cr          
   \ddot o  & \\ddot o  &\breve o  & \\breve o  & \bar o & \\bar o \cr          
   \vec o   & \\vec o   & \widehat {abc} & \\widehat \lb abc\rb                 
                    & \widetilde {abc} & \\widetilde \lb abc\rb\cr              
       }                                                                        
                                                                                
\toindex{hat}                                                                   
\toindex{check}                                                                 
\toindex{tilde}                                                                 
\toindex{acute}                                                                 
\toindex{grave}                                                                 
\toindex{dot}                                                                   
\toindex{ddot}                                                                  
\toindex{breve}                                                                 
\toindex{bar}                                                                   
\toindex{vec}                                                                   
\toindex{widehat}                                                               
\toindex{widetilde}                                                             
                                                                                
                                                                                
                                                                                
Binary operators combine two mathematical objects to get another                
object.  Ordinary addition and multiplication, for example,                     
combine two numbers to get another number, and so they are binary               
operators. When a binary operator such as $+$ or $\times$ is                    
typeset, a little extra space is put around it. Here is a list of               
some of the available binary operators:                                         
\TeXref{436}                                                                    
                                                                                
\maketable [Binary operators]                                                   
\halign{                                                                        
\strut \hfil$#$ & \quad \tt# \hfil \qquad &\hfil$#$ & \quad \tt# \hfil \qquad   
      &\hfil$#$ & \quad \tt# \hfil \qquad &\hfil$#$ & \quad \tt# \hfil \cr      
\noalign{\hrule} \noalign{\smallskip}                                           
\cdot & \\cdot &\times & \\times &\ast & \\ast &\star & \\star \cr              
\circ & \\circ & \bullet & \\bullet & \div & \\div & \diamond & \\diamond \cr   
\cap & \\cap & \cup & \\cup & \vee & \\vee & \wedge & \\wedge \cr               
\oplus & \\oplus &\ominus & \\ominus & \otimes &\\otimes &\odot &\\odot \cr     
       }                                                                        
                                                                                
\toindex{cdot}                                                                  
\toindex{times}                                                                 
\toindex{ast}                                                                   
\toindex{star}                                                                  
\toindex{circ}                                                                  
\toindex{bullet}                                                                
\toindex{div}                                                                   
\toindex{diamond}                                                               
\toindex{cap}                                                                   
\toindex{cup}                                                                   
\toindex{vee}                                                                   
\toindex{wedge}                                                                 
\toindex{oplus}                                                                 
\toindex{ominus}                                                                
\toindex{otimes}                                                                
\toindex{odot}                                                                  
                                                                                
Ellipses are commonly used with binary operators.  The control                  
word {\tt \\cdots} will raise the dots so that they are level                   
with the binary operator. Thus {\tt \$a + \\cdots + z\$} will                   
produce $a + \cdots + z$.  The control word {\tt \\ldots} will                  
put the dots on the baseline, and so {\tt \$1\\ldots n\$}                       
produces $1\ldots n$.                                                           
\toindex{cdots}                                                                 
\toindex{ldots}                                                                 
                                                                                
\exercise Typeset: $x\wedge (y\vee z) = (x\wedge y) \vee (x\wedge z)$.          
                                                                                
\exercise Typeset: $2+4+6+\cdots +2n = n(n+1)$.                                 
                                                                                
A relation indicates a property of two mathematical objects.  We                
already know how to show two objects equal, or how to show one                  
number less than or greater than another number (since these are                
symbols on most terminal keyboards).  To negate a relation, the                 
control word {\tt \\not} is put in front of the relation.  Here                 
are some relations:                                                             
\TeXref{436}                                                                    
\toindex{not}                                                                   
                                                                                
\maketable [Relations ]                                                         
\halign{                                                                        
\strut \hfil$#$ & \quad \tt# \hfil \qquad &\hfil$#$ & \quad \tt# \hfil \qquad   
      &\hfil$#$ & \quad \tt# \hfil \qquad &\hfil$#$ & \quad \tt# \hfil \cr      
\noalign{\hrule} \noalign{\smallskip}                                           
\leq  & \\leq  &\not\leq & \\not \\leq                                          
       & \geq & \\geq & \not\geq & \\not \\geq \cr                              
\equiv & \\equiv & \not\equiv & \\not \\equiv                                   
       & \sim & \\sim & \not\sim & \\not \\sim \cr                              
\simeq & \\simeq & \not\simeq & \\not \\simeq                                   
       & \approx & \\approx & \not\approx & \\not \\approx \cr                  
\subset & \\subset & \subseteq & \\subseteq                                     
        & \supset & \\supset & \supseteq & \\supseteq \cr                       
\in & \\in & \ni & \\ni & \parallel & \\parallel & \perp & \\perp \cr           
       }                                                                        
                                                                                
\toindex{leq}                                                                   
\toindex{geq}                                                                   
\toindex{equiv}                                                                 
\toindex{sim}                                                                   
\toindex{simeq}                                                                 
\toindex{approx}                                                                
\toindex{subset}                                                                
\toindex{subseteq}                                                              
\toindex{supset}                                                                
\toindex{supseteq}                                                              
\toindex{in}                                                                    
\toindex{ni}                                                                    
\toindex{parallel}                                                              
\toindex{perp}                                                                  
                                                                                
                                                                                
\exercise                                                                       
Typeset: $\vec x\cdot \vec y  = 0$ if and only if $\vec x \perp \vec y$.        
                                                                                
\exercise                                                                       
Typeset: $\vec x\cdot \vec y \not= 0$ if and only if $\vec x \not\perp \vec y$. 
                                                                                
Here are some other available mathematical symbols:\TeXref{435--438}            
                                                                                
\maketable [Miscellaneous symbols ]                                             
\halign{                                                                        
\strut \hfil$#$ & \quad \tt# \hfil \qquad &\hfil$#$ & \quad \tt# \hfil \qquad   
      &\hfil$#$ & \quad \tt# \hfil \qquad &\hfil$#$ & \quad \tt# \hfil \cr      
\noalign{\hrule} \noalign{\smallskip}                                           
\aleph & \\aleph & \ell & \\ell & \Re & \\Re & \Im & \\Im \cr                   
\partial & \\partial & \infty & \\infty & \! & \\! & \angle & \\angle \cr       
\nabla & \\nabla & \backslash &\\backslash & \forall & \\forall                 
              & \exists & \\exists \cr                                          
\neg & \\neg & \flat & \\flat & \sharp & \\sharp & \natural & \\natural \cr     
          }                                                                     
                                                                                
\toindex{aleph}                                                                 
\toindex{ell}                                                                   
\toindex{Re}                                                                    
\toindex{Im}                                                                    
\toindex{partial}                                                               
\toindex{infty}                                                                 
\toindex{!}                                                                     
\toindex{angle}                                                                 
\toindex{nabla}                                                                 
\toindex{backslash}                                                             
\toindex{forall}                                                                
\toindex{exists}                                                                
\toindex{neg}                                                                   
\toindex{flat}                                                                  
\toindex{sharp}                                                                 
\toindex{natural}                                                               
                                                                                
\exercise Typeset: $(\forall x\in \Re)(\exists y\in\Re)$ $y>x$.                 
                                                                                
\subsection{Fractions}                                                          
                                                                                
There are two methods of typesetting a fraction: it can be                      
typeset either in the form $1/2$ or in the form ${1\over2}$\null.               
The first case is just entered with no special control sequences,               
that is, {\tt \$1/2\$}\null.  The second case uses the control                  
word {\tt \\over} and the following pattern: {\tt \lb                           
<numerator> \\over <denominator>\rb}\null. Hence {\tt \$\$\lb a+b               
\\over c+d\rb.\$\$} gives $${a+b\over c+d}.$$                                   
\TeXref{139--140}                                                               
\toindex{over}                                                                  
                                                                                
\exercise Typeset the following: ${a+b\over c}\quad {a\over b+c}                
\quad {1\over a+b+c} \not= {1\over a}+{1\over b}+{1\over c}$.                   
                                                                                
\exercise Typeset: What are the points where                                    
${\partial \over \partial x} f(x,y) = {\partial \over \partial y}               
f(x,y) = 0$?                                                                    
                                                                                
\subsection{Subscripts and superscripts}                                        
                                                                                
Subscripts and superscripts are particularly easy to enter using                
\TeX\null. The characters {\tt \_{}} and {\tt \~{}} are used to                 
indicate that the next character is a subscript or a superscript.               
Thus {\tt \$x\~{}2\$} gives $x~2$ and {\tt \$x\_{}2\$} gives                    
$x_2$\null.  To get several characters as a subscript or                        
superscript, they are grouped together within braces. Hence we                  
can use {\tt \$x\~{}\lb 21\rb\$} to get $x~{21}$ and {\tt \$x\_{}\lb            
21\rb\$} to get $x_{21}$\null. Notice that the                                  
superscripts and subscripts are automatically typeset in a                      
smaller type size. The situation is only slightly more                          
complicated for a second layer of subscripts or superscripts.                   
You can {\sl not\/} use {\tt \$x\_{}2\_{}3\$} since this could                  
have two possible interpretations, namely, {\tt \$x\_{}\lb 2\_{}3\rb\$}         
or {\tt \$\lb x\_{}2\rb\_{}3\$}; this gives two                                 
different results: $x_{2_3}$ and ${x_2}_3$, the first of which is               
the usual mathematical subscript notation.  Thus you must put in                
the complete braces to describe multiple layers of subscripts and               
superscripts.  They may be done to any level.                                   
\TeXref{128--130}                                                               
                                                                                
To use both subscripts and superscripts on one symbol, you use                  
both the {\tt \_{}} and {\tt \~{}} in either order.  Thus either                
{\tt \$x\_{}2\~{}1\$} or {\tt\$x\~{}1\_{}2\$} will give $x_2~1$.                
                                                                                
\exercise                                                                       
Typeset each of the following: $e~x \quad e~{-x} \quad                          
e~{i\pi}+1=0 \quad x_0 \quad x_0~2 \quad {x_0}~2 \quad 2~{x~x}$.                
                                                                                
\exercise Typeset:                                                              
$\nabla~2 f(x,y) = {\partial~2 f \over\partial x~2}                             
+ {\partial~2 f \over\partial y~2}$.                                            
                                                                                
A similar method is used for summations and integrals.  The input               
of {\tt \$\\sum\_{}\lb k=1\rb\~{}n k\~{}2\$} will give $\sum_{k=1}~n            
k~2$, and {\tt \$\\int\_{}0\~{}x f(t) dt\$} will                                
give $\int_0~x f(t) dt$.                                                        
\TeXref{144--145}                                                               
\toindex{sum}                                                                   
\toindex{int}                                                                   
                                                                                
Another use of this type of input is for expressions                            
involving limits.  You can use                                                  
{\tt \$\\lim\_\lb n\\to \\infty\rb (\lb n+1 \\over n\rb)\~{}n = e\$}            
to get $\lim_{n\to \infty} ({n+1\over n})~n = e$.                               
\toindex{lim}                                                                   
                                                                                
\exercise Typeset the following expression: $\lim_{x\to 0}                      
(1+x)~{1\over x}=e$.                                                            
                                                                                
\exercise Typeset: The cardinality of $(-\infty, \infty)$ is $\aleph_1$.        
                                                                                
\exercise Typeset: $\lim_{x\to {0~+}} x~x = 1$.                                 
                                                                                
Here's a hint to make integrals look a little nicer: look at the                
difference between $\int_0~x f(t) dt$ and $\int_0~x f(t)\, dt$\null.            
In the second case there is a little extra space after                          
$f(t)$, and it looks nicer; {\tt \\,} was used to add the                       
additional space.                                                               
\toindex{,}                                                                     
                                                                                
\exercise Typeset the following integral: $\int_0~1 3x~2\,dx = 1$.              
                                                                                
\subsection{Roots, square and otherwise}                                        
                                                                                
To typeset a square root it is only necessary to use the construction           
{\tt \\sqrt\lb$\ldots$\rb}\null. Hence {\tt \$\\sqrt\lb x\~{}2+y\~{}2\rb\$}     
will give $\sqrt{x~2+y~2}$\null.  Notice that \TeX{} takes care of the          
placement of symbols and the height and length of the radical.                  
To make cube or other roots, the control words {\tt \\root} and {\tt            
\\of} are used.  You get $\root n \of {1+x~n}$ from the input                   
{\tt \$\\root n \\of \lb1+x\~{}n\rb\$}.                                         
\TeXref{130--131}                                                               
\toindex{root}                                                                  
\toindex{sqroot}                                                                
                                                                                
A possible alternative is to use the control word {\tt \\surd}; the             
input {\tt \$\\surd 2\$} will produce $\surd 2$.                                
\toindex{surd}                                                                  
                                                                                
\exercise Typeset the following: $\sqrt2 \quad \sqrt {x+y\over x-y}             
\quad \root 3 \of {10}$ \quad $e~{\sqrt x}$.                                    
                                                                                
\exercise Typeset: $\!x\! = \sqrt{x\cdot x}$.                                   
                                                                                
\exercise Typeset:                                                              
$\phi(t) = {1 \over \sqrt{2\pi}} \int_0~t e~{-x~2/2}\,dx$.                      
                                                                                
                                                                                
\subsection{Lines, above and below}                                             
                                                                                
Use the constructions {\tt \\overline\lb$\ldots$\rb} and                        
{\tt \\underline\lb$\ldots$\rb} to put lines above or below mathematical        
expressions. Hence {\tt \$\\overline\lb x+y\rb=\\overline x + \\overline        
y\$} gives $\overline{x+y}=\overline x + \overline y$\null.                     
But notice that the lines over the letters are at                               
different heights, and so some care is necessary. The use of {\tt               
\\overline\lb\\strut x\rb} will raise the height of the line over               
$x$\null.                                                                       
\TeXref{130--131}                                                               
\toindex{overline}                                                              
\toindex{underline}                                                             
                                                                                
To underline non-mathematical text, use {\tt \\underbar\lb$\dots$\rb}.          
\toindex{underbar}                                                              
                                                                                
\exercise Typeset the following: $\underline x \quad \overline y                
\quad \underline{\overline{x+y}}$.                                              
                                                                                
                                                                                
\subsection{Delimiters large and small}                                         
                                                                                
The most commonly used mathematical delimiters are brackets,                    
braces, and parentheses.  As we have seen, they may be produced                 
by using {\tt [ ] \\\lb\ \\\rb ( )} to get $[\>]\>\{\>\}\>(\>)\>$.              
Sometimes larger delimiters increase the clarity of                             
mathematical expressions, as in $$\bigl(a\times(b+c)\bigr) \bigl((a\times       
b)+c\bigr).$$ To make larger left delimiters the                                
control words {\tt \\bigl}, {\tt \\Bigl}, {\tt \\biggl}, and {\tt               
\\Biggl} are used in front of the delimiter; similarly, {\tt \\bigr},           
{\tt \\Bigr}, {\tt \\biggr}, and {\tt \\Biggr} are used                         
for the right delimiters. Hence {\tt \$\\Bigl[\$} and {\tt \$\\Bigr]\$}         
will produce $\Bigl[$ and $\Bigr]$.                                             
\TeXref{145--147}                                                               
                                                                                
\toindex{bigl}                                                                  
\toindex{Bigl}                                                                  
\toindex{biggl}                                                                 
\toindex{Biggl}                                                                 
\toindex{bigr}                                                                  
\toindex{Bigr}                                                                  
\toindex{biggr}                                                                 
\toindex{Biggr}                                                                 
                                                                                
Here is a table to compare the size of some of the delimiters.                  
                                                                                
%% \everycr can add 4 points between lines in the following table %%            
\everycr={\noalign{\vskip 4 pt}}                                                
\maketable [Delimiters of various sizes]                                        
\halign{                                                                        
\strut \hfil$#$ & \quad \tt# \hfil \quad\qquad                                  
      &\hfil$#$ & \quad \tt# \hfil \quad\qquad                                  
      &\hfil$#$ & \quad \tt# \hfil \quad\qquad                                  
      &\hfil$#$ & \quad \tt# \hfil \cr                                          
\noalign{\hrule} \noalign{\smallskip}                                           
\{ & \\\lb & \} & \\\rb & ( & ( & ) & )\cr                                      
\bigl\{ & \\bigl\lb & \bigr\} & \\bigr\rb & \bigl( & \\bigl( & \bigr) &         
\\bigr)\cr                                                                      
\Bigl\{ & \\Bigl\lb & \Bigr\} & \\Bigr\rb & \Bigl( & \\Bigl( & \Bigr) &         
\\Bigr)\cr                                                                      
\biggl\{ & \\biggl\lb & \biggr\} & \\biggr\rb & \biggl(                         
         & \\biggl( & \biggr) & \\biggr) \cr                                    
\Biggl\{ & \\Biggl\lb & \Biggr\} & \\Biggr\rb & \Biggl(                         
         & \\Biggl( & \Biggr) & \\Biggr)\cr                                     
       }                                                                        
                                                                                
\everycr={}                                                                     
                                                                                
If you want, you can let \TeX{} choose the size of delimiter by                 
using the control words {\tt \\left} and {\tt \\right} before                   
your delimiters.                                                                
\TeXref{148}                                                                    
Thus {\tt \\left[$\ldots$\\right]}  will cause the material to be               
enclosed by brackets that are appropriately big.  {\bf Note                     
well:} each use of a {\tt \\left} delimiter must have a matching                
{\tt \\right} delimiter. Hence {\tt \$\$\\left!\lb a+b \\over                   
c+d\rb\\right|.\$\$} gives $$\left|{a+b \over c+d}\right|.$$                    
                                                                                
\maketable [Mathematical delimiters]                                            
\halign{                                                                        
\strut \hfill$#$ & \quad \tt # \qquad\qquad &                                   
       \hfill$#$ & \quad \tt # \qquad\qquad &                                   
       \hfill$#$ & \quad \tt #  \cr                                             
\noalign{\hrule} \noalign{\smallskip}                                           
(       & (        & )          & )           & [        & [         \cr        
]       & ]        &\{          & \\\lb       & \}       & \\\rb     \cr        
\lfloor & \\lfloor &\rfloor     &\\rfloor     & \lceil   & \\lceil   \cr        
\rceil  & \\rceil  &\langle     & \\langle    & \rangle  & \\rangle  \cr        
/       & /        & \backslash & \\backslash &!         & !         \cr        
\!      & \\!      &\uparrow    & \\uparrow   & \Uparrow & \\Uparrow \cr        
\downarrow & \\downarrow & \Downarrow & \\Downarrow                             
                        & \updownarrow & \\updownarrow \cr                      
\Updownarrow & \\Updownarrow \cr                                                
       }                                                                        
                                                                                
\toindex{lfloor}                                                                
\toindex{rfloor}                                                                
\toindex{lceil}                                                                 
\toindex{rceil}                                                                 
\toindex{langle}                                                                
\toindex{rangle}                                                                
\toindex{!}                                                                     
\toindex{uparrow}                                                               
\toindex{Uparrow}                                                               
\toindex{downarrow}                                                             
\toindex{Downarrow}                                                             
\toindex{updownarrow}                                                           
\toindex{Updownarrow}                                                           
                                                                                
\exercise Typeset $\bigl \lceil \lfloor x \rfloor \bigr \rceil                  
              \leq \bigl \lfloor \lceil x \rceil \bigr \rfloor$.                
                                                                                
\subsection{Those special functions}                                            
                                                                                
There are several types of functions that appear frequently in                  
mathematical text.  In an equation like ``$\sin~2x + \cos~2x =                  
1$'' the trigonometric functions ``sin'' and ``cos'' are in roman               
rather than italic type. This is the usual mathematical                         
convention to indicate that it is a function being described and                
not the product of three variables.  The control words {\tt                     
\\sin} and {\tt \\cos} will use the right typeface automatically.               
Here is a table of these and some other special functions:                      
\TeXref{162}                                                                    
                                                                                
\maketable [Special mathematical functions]                                     
\halign{                                                                        
   \strut \tt {\\}#\hfil && \quad \tt {\\}#\hfil \cr                            
   \noalign{\hrule} \noalign{\smallskip}                                        
   sin    & cos    & tan  & cot  & sec  & csc & arcsin & arccos \cr             
   arctan & sinh   & cosh & tanh & coth & lim & sup    & inf    \cr             
   limsup & liminf & log  & ln   & lg   & exp & det    & deg    \cr             
   dim    & hom    & ker  & max  & min  & arg & gcd    & Pr     \cr             
       }                                                                        
                                                                                
\toindex{sin}                                                                   
\toindex{cos}                                                                   
\toindex{tan}                                                                   
\toindex{cot}                                                                   
\toindex{sec}                                                                   
\toindex{csc}                                                                   
\toindex{arcsin}                                                                
\toindex{arccos}                                                                
\toindex{arctan}                                                                
\toindex{sinh}                                                                  
\toindex{cosh}                                                                  
\toindex{tanh}                                                                  
\toindex{coth}                                                                  
\toindex{lim}                                                                   
\toindex{sup}                                                                   
\toindex{inf}                                                                   
\toindex{limsup}                                                                
\toindex{liminf}                                                                
\toindex{log}                                                                   
\toindex{ln}                                                                    
\toindex{lg}                                                                    
\toindex{exp}                                                                   
\toindex{det}                                                                   
\toindex{deg}                                                                   
\toindex{dim}                                                                   
\toindex{hom}                                                                   
\toindex{ker}                                                                   
\toindex{max}                                                                   
\toindex{min}                                                                   
\toindex{arg}                                                                   
\toindex{gcd}                                                                   
\toindex{Pr}                                                                    
                                                                                
\exercise Typeset: $\sin(2\theta) = 2\sin\theta\cos\theta \quad                 
\cos(2\theta) = 2\cos~2\theta - 1  $.                                           
                                                                                
\exercise Typeset: $$\int \csc~2x\, dx = -\cot x+ C                             
\qquad \lim_{\alpha\to 0} {\sin\alpha \over \alpha} = 1                         
\qquad \lim_{\alpha\to \infty} {\sin\alpha \over \alpha} = 0.$$                 
                                                                                
\exercise Typeset: $$\tan(2\theta) = {2\tan\theta \over                         
1-\tan~2\theta}.$$                                                              
                                                                                
                                                                                
\subsection{Hear ye, hear ye!}                                                  
                                                                                
There is a particular macro that is used in almost every                        
mathematical paper, and is different enough to require a special                
explanation. This is the {\tt \\proclaim} macro.  It is used when               
stating theorems, corollaries, propositions, and the like.  The                 
paragraph following {\tt \\proclaim} is broken into two parts:                  
the first part goes up to and including the first period that is                
followed by a space, and the second part is the rest of the                     
paragraph.\TeXref{202--203}  The idea is that the first part                    
should be something like ``Theorem 1.'' or ``Corollary B.'' The                 
second part is the statement of the theorem or corollary.  Here                 
is an example:                                                                  
\toindex{proclaim}                                                              
                                                                                
\beginliteral                                                                   
\proclaim Theorem 1 (H.~G.~Wells). In the country of the blind,                 
the one-eyed man is king.                                                       
@endliteral                                                                     
                                                                                
\noindent gives                                                                 
                                                                                
\proclaim Theorem 1 (H.~G.~Wells). In the country of the blind,                 
the one-eyed man is king.                                                       
                                                                                
The statement of the theorem may contain mathematical expressions, of           
course.                                                                         
                                                                                
\exercise Typeset:                                                              
\proclaim Theorem (Euclid). There exist an infinite number of                   
primes.                                                                         
                                                                                
\exercise Typeset:                                                              
\proclaim Proposition 1.                                                        
$\root n \of {\prod_{i=1}~n X_i} \leq {1 \over n} \sum_{i=1}~n X_i$             
with equality if and only if $X_1=\cdots=X_n$.                                  
                                                                                
\subsection{Matrices}                                                           
                                                                                
Matrices are typeset using combinations of the alignment                        
character {\tt \&} and the control word {\tt \\cr} to indicate                  
the end of the line.  Start with {\tt \$\$\\pmatrix\lb$\dots$\rb\$\$}.          
Into the space between the braces go the rows of the matrix, each               
one ended by {\tt \\cr}.  The entries are separated by the                      
{\tt \&}\null.  For example the input                                           
\TeXref{176--178}                                                               
                                                                                
\beginliteral                                                                   
$$\pmatrix{                                                                     
a & b & c & d \cr                                                               
b & a & c+d & c-d \cr                                                           
0 & 0 & a+b & a-b \cr                                                           
0 & 0 & ab  & cd \cr                                                            
}.$$                                                                            
@endliteral                                                                     
                                                                                
\noindent                                                                       
gives as printed output                                                         
$$\pmatrix{                                                                     
a & b & c & d \cr                                                               
b & a & c+d & c-d \cr                                                           
0 & 0 & a+b & a-b \cr                                                           
0 & 0 & ab  & cd \cr                                                            
}.$$                                                                            
\toindex{pmatrix}                                                               
                                                                                
The matrix entries in our examples have all been \centred{}                     
within their columns.  They can be made flush right or flush left               
by inserting {\tt \\hfill} before or after the entry. Notice the                
differences between the following example and the previous one.                 
                                                                                
\beginliteral                                                                   
$$\pmatrix{                                                                     
a & b & c \hfill  & \hfill d  \cr                                               
b & a & c+d      & c-d      \cr                                                 
0 & 0 & a+b      & a-b      \cr                                                 
0 & 0 & ab \hfill & \hfill cd \cr                                               
}.$$                                                                            
@endliteral                                                                     
                                                                                
\noindent                                                                       
gives as printed output                                                         
                                                                                
$$\pmatrix{                                                                     
a & b & c \hfill & \hfill d   \cr                                               
b & a & c+d     & c-d       \cr                                                 
0 & 0 & a+b     & a-b       \cr                                                 
0 & 0 & ab \hfill & \hfill cd \cr                                               
}.$$                                                                            
                                                                                
                                                                                
\vbox{                                                                          
\exercise Typeset                                                               
$$ I_4 = \pmatrix{ 1 &0 &0 &0 \cr                                               
                   0 &1 &0 &0 \cr                                               
                   0 &0 &1 &0 \cr                                               
                   0 &0 &0 &1 \cr}$$                                            
}                                                                               
                                                                                
It's possible to have matrices that use other delimiters. Using                 
{\tt \\matrix} instead of {\tt \\pmatrix} will leave off the                    
parentheses, so the delimiters must be explicitly included using                
{\tt \\left} and {\tt \\right}.  Here is how we can change the                  
matrix of our last example.                                                     
\toindex{matrix}                                                                
\toindex{left}                                                                  
\toindex{right}                                                                 
                                                                                
\beginliteral                                                                   
$$ \left !                                                                      
\matrix{                                                                        
a & b & c & d \cr                                                               
b & a & c+d & c-d \cr                                                           
0 & 0 & a+b & a-b \cr                                                           
0 & 0 & ab  & cd \cr                                                            
}                                                                               
\right ! $$                                                                     
@endliteral                                                                     
                                                                                
\noindent                                                                       
gives as printed output                                                         
$$ \left |                                                                      
\matrix{                                                                        
a & b & c & d \cr                                                               
b & a & c+d & c-d \cr                                                           
0 & 0 & a+b & a-b \cr                                                           
0 & 0 & ab  & cd \cr                                                            
}                                                                               
\right | $$                                                                     
                                                                                
It's even possible to use {\tt \\left.} and {\tt \\right.} to                   
indicate that the opening or closing delimiter is deleted (note                 
the use of the period).                                                         
                                                                                
\exercise Use a matrix construction to typeset                                  
                                                                                
$$ !x! = \left\{ \matrix{ x & x \ge 0 \cr                                       
                         -x & x \le 0 \cr} \right.$$                            
                                                                                
This exercise and more general constructions of this type                       
may also be typeset using the {\tt \\cases} macro.                              
\TeXref{175}                                                                    
                                                                                
Sometimes ellipses are used within matrices.  The control words                 
{\tt \\cdots}, {\tt \\vdots}, and {\tt \\ddots} can be used to                  
insert horizontal, vertical, and diagonal dots.                                 
                                                                                
Thus we can use                                                                 
\beginliteral                                                                   
$$ \left [                                                                      
\matrix{                                                                        
aa     & \cdots & az     \cr                                                    
\vdots & \ddots & \vdots \cr                                                    
za     & \cdots & zz     \cr                                                    
}                                                                               
\right ] $$                                                                     
@endliteral                                                                     
                                                                                
\noindent                                                                       
gives as printed output                                                         
$$ \left [                                                                      
\matrix{                                                                        
aa     & \cdots & az     \cr                                                    
\vdots & \ddots & \vdots \cr                                                    
za     & \cdots & zz     \cr                                                    
}                                                                               
\right ] $$                                                                     
                                                                                
                                                                                
Matrices may also be typeset in-line, but they are pretty ugly                  
unless they have a small number of rows.                                        
                                                                                
                                                                                
\subsection{Displayed equations}                                                
                                                                                
All of the mathematics covered so far has identical input whether               
it is to be typeset in-line or displayed.  At this point we'll                  
look at some situations that apply to displayed equations only.                 
                                                                                
The first is that of aligning multiline displays.  This is done with            
the alignment character {\tt \&} and the control words {\tt \\cr} and           
{\tt \\eqalign}. Starting with {\tt \$\\eqalign\lb$\dots$\rb\$},                
the equations to be aligned are entered with each one terminated by             
{\tt \\cr}.  In each equation there should be one alignment symbol              
{\tt \&} to indicate where the alignment should take place.  This is            
usually done at the equal signs, although it is not necessary to do             
so.  For example                                                                
\TeXref{190--192}                                                               
\toindex{eqalign}                                                               
                                                                                
\beginliteral                                                                   
$$\eqalign{                                                                     
a+b &= c+d \cr                                                                  
x &= w + y + z \cr                                                              
m + n + o + p &= q \cr                                                          
}$$                                                                             
@endliteral                                                                     
                                                                                
                                                                                
\noindent                                                                       
yields                                                                          
$$\eqalign{                                                                     
a+b &= c+d \cr                                                                  
x &= w + y + z \cr                                                              
m + n + o + p &= q \cr                                                          
}$$                                                                             
                                                                                
Displayed equations can be numbered at either the right or left                 
margin.  When the control word {\tt \\eqno} appears in a displayed              
equation, everything after the control word is put at the right                 
margin.  Hence {\tt \$\$ x+y=z. \\eqno (1)\$\$} yields                          
$$ x+y=z. \eqno (1)$$                                                           
To number an equation at the left margin, use {\tt \\leqno}                     
in place of {\tt \\eqno}.                                                       
\toindex{eqno}                                                                  
\toindex{leqno}                                                                 
                                                                                
It's possible to number aligned equations by using the control                  
word {\tt \\eqalignno}.  The alignment character {\tt \&} is                    
used to separate the equation from the equation number.                         
\beginliteral                                                                   
$$\eqalignno{                                                                   
a+b &= c+d & (1) \cr                                                            
x &= w + y + z \cr                                                              
m + n + o + p &= q & * \cr                                                      
}$$                                                                             
@endliteral                                                                     
                                                                                
\noindent                                                                       
yields                                                                          
$$\eqalignno{                                                                   
a+b &= c+d & (1) \cr                                                            
x &= w + y + z \cr                                                              
m + n + o + p &= q & * \cr                                                      
}$$                                                                             
                                                                                
Use {\tt \\leqalignno} to put the equation numbers on the left.                 
\TeXref{192--193}                                                               
\toindex{eqalignno}                                                             
\toindex{leqalignno}                                                            
                                                                                
Finally, suppose some text needs to appear in the middle of a                   
displayed equation.  This can be done by putting it in an hbox.                 
We will describe hboxes in more detail in the next section.                     
For now we want to use them to temporarily resume using ordinary                
roman type and to also allow the insertion of space between                     
words (remember that all spaces are ignored when typesetting                    
mathematics). Hence                                                             
{\tt \$\$X=Y \\hbox\lb{} if and only if \rb x=y.\$\$} will give                 
$$X=Y \hbox{ if and only if } x=y.$$                                            
Note carefully the spaces in the hbox.                                          
                                                                                
\exercise Do some of the challenge problems on pages 180--181 of The            
\TeX book.                                                                      
                                                                                
                                                                                
                                                                                
                                                                                
\section{All in a row}                                                          
                                                                                
It's not uncommon to want to put a table in the middle of some                  
text. Fortunately \TeX{} makes it easy to do this.  In fact there               
are two separate methods of aligning text.  The first is by using               
the tabbing environment.  This is similar to setting the tab                    
stops on a typewriter. Each line is handled individually,                       
according to set tab columns, but with greater flexibility than                 
that provided by a typewriter. The second is the horizontal                     
alignment environment which typesets the whole table at once                    
using a prescribed pattern.                                                     
                                                                                
\subsection{Picking up the tab}                                                 
                                                                                
To align material using the tabbing environment, you must first                 
set the tab positions using the {\tt \\settabs} control word.                   
Having done this, a line to use these tabs starts with the                      
control symbol {\tt \\+ } and ends with {\tt \\cr } (remember                   
that the actual spacing on lines in the input file in                           
unimportant). \toindex{settabs}                                                 
                                                                                
The easiest way to use the {\tt \\settabs} control word is to put               
the text into equal columns.                                                    
\TeXref{231}                                                                    
Using {\tt \\settabs 4 \\columns} will set the tabs that will                   
produce four equal columns.  The tabbing is then done by using                  
the alignment character {\tt \&} to move to the next tab stop.                  
So, for example, \toindex{columns}                                              
                                                                                
\beginliteral                                                                   
\settabs 4 \columns                                                             
\+ British Columbia & Alberta & Saskatchewan & Manitoba \cr                     
\+ Ontario & Quebec & New Brunswick & Nova Scotia \cr                           
\+ & Prince Edward Island & Newfoundland \cr                                    
@endliteral                                                                     
                                                                                
                                                                                
\noindent                                                                       
will produce the table                                                          
\vskip\baselineskip                                                             
                                                                                
\settabs 4 \columns                                                             
\+ British Columbia & Alberta & Saskatchewan & Manitoba \cr                     
\+ Ontario & Quebec & New Brunswick & Nova Scotia \cr                           
\+ & Prince Edward Island & Newfoundland \cr                                    
                                                                                
Notice that it is possible to skip over some tab positions, and                 
it is not necessary to use all of the tabs in a given line. To                  
make the same table using five columns, it is only necessary to                 
use {\tt \\settabs 5 \\columns} to reset the tab stops; then the                
same three lines from the last example will produce:                            
\vskip\baselineskip                                                             
                                                                                
\settabs 5 \columns                                                             
\+ British Columbia & Alberta & Saskatchewan & Manitoba \cr                     
\+ Ontario & Quebec & New Brunswick & Nova Scotia \cr                           
\+ & Prince Edward Island & Newfoundland \cr                                    
                                                                                
In this example, the columns are smaller, of course.  In fact                   
there are two overlapping entries in the last row.  This is                     
because \TeX{} will tab to the next tab position even if (unlike                
a typewriter) it means going backward on the page.                              
                                                                                
There is an interesting relationship between grouping and                       
tabbing. The {\tt \\settabs} values are only applicable to the                  
group in which it is defined, as would be expected.  Thus it is                 
possible to temporarily change the tab settings by grouping                     
within braces.  In addition, each table entry is in a group of                  
its own. Hence we may make a single entry boldface, for example,                
by using {\tt \\bf}  without braces. In addition, for any column                
but the last one it is possible to \centre{} the entry or to                    
align it either on the left or on the right, or to fill a column                
with a line or dots. Each entry has an implicit {\tt \\hfil} at                 
the end so that it will be at the left of the column by default.                
Adding {\tt \\hfil} at the beginning of the entry will then cause               
it to be \centred{}, just as with the {\tt \\line} control word.                
Adding {\tt \\hfill} to the beginning will cause the entries to                 
be pushed to the right ({\tt \\hfill} acts just like {\tt \\hfil}               
in that it absorbs excess space; when both {\tt \\hfil} and {\tt                
\\hfill} appear, the {\tt \\hfill} takes precedence).                           
\toindex{hfill}                                                                 
                                                                                
\beginliteral                                                                   
\settabs 4 \columns                                                             
\+ \hfil British Columbia & \hfill Alberta \qquad & \bf Saskatchewan            
@hskip 2.5in                                  & Manitoba \cr                    
\+ \hfil Ontario & \hfill Quebec \qquad & \bf New Brunswick                     
@hskip 2.5in                                 & Nova Scotia \cr                  
\+ \hfil ---  & \hfill * \qquad & \bf Newfoundland                              
@hskip 2.5in          & Prince Edward Island \cr                                
\+ \dotfill && \hrulefill & \cr                                                 
@endliteral                                                                     
                                                                                
\noindent                                                                       
will produce a table with the first column \centred{}, the second               
column flush left with a {\tt \\qquad} of padding, and the third                
column boldface. The control words {\tt \\dotfill} and {\tt \\hrulefill}        
give alternative column entries.                                                
\toindex{dotfill}                                                               
\toindex{hrulefill}                                                             
\vskip\baselineskip                                                             
                                                                                
\settabs 4 \columns                                                             
\+ \hfil British Columbia & \hfill Alberta \qquad & \bf Saskatchewan            
                                        & Manitoba \cr                          
\+ \hfil Ontario & \hfill Quebec \qquad & \bf New Brunswick                     
                                        & Nova Scotia \cr                       
\+ \hfil ---  & \hfill * \qquad & \bf Newfoundland &                            
                      Prince Edward Island \cr                                  
\+ \dotfill && \hrulefill & \cr                                                 
                                                                                
\exercise Take the table of Canadian provinces above and \centre{} each         
entry within its column.                                                        
                                                                                
The tab positions can be set with much more flexibility than just               
in equal columns.  The general pattern is to use a sample line of               
the form {\tt \\settabs \\+ $\ldots$ \& $\ldots$ \& $\ldots$ \\cr}.             
The spacing between the alignment characters {\tt \&}                           
determines the position of the tabs.  For example, {\tt \\settabs               
\\+ \\hskip 1 in \& \\hskip 2 in \& \\hskip 1.5 in \& \\cr} would               
set the first tab one inch from the left margin, the next another               
two inches further in, and the third 1.5 inches more.  It's also                
possible to use text to determine the distance between tabs. So,                
for example, another possible sample line is                                    
{\tt \\settabs \\+ \\quad Province \\quad \& \\quad Population \\quad           
\& \\quad Area \\quad \& \\cr}.                                                 
The tab column would then be just wide enough to accept the headings            
with a quad of space on each side.  Here's a more complete example:             
                                                                                
\beginuser \obeyspaces                                                          
\\settabs \\+ \\quad Year \\quad \& \\quad Price \\quad                         
                                            \& \\quad Dividend \& \\cr          
\\+ \\hfill Year \\quad \& \\quad Price  \\quad \& \\quad Dividend \\cr         
\\+ \\hfill 1971 \\quad \& \\quad 41--54 \\quad \& \\qquad \$2.60   \\cr        
\\+ \\hfill 2    \\quad \& \\quad 41--54 \\quad \& \\qquad \$2.70 \\cr          
\\+ \\hfill 3    \\quad \& \\quad 46--55 \\quad \& \\qquad \$2.87 \\cr          
\\+ \\hfill 4    \\quad \& \\quad 40--53 \\quad \& \\qquad \$3.24 \\cr          
\\+ \\hfill 5    \\quad \& \\quad 45--52 \\quad \& \\qquad \$3.40 \\cr          
\enduser                                                                        
                                                                                
\noindent                                                                       
gives \TeXref{247}                                                              
\vskip\baselineskip                                                             
                                                                                
\settabs \+ \quad Year \quad & \quad Price \quad & \quad                        
Dividend \quad & \cr                                                            
\+ \hfill Year \quad & \quad Price  \quad & \quad Dividend \cr                  
\+ \hfill 1971 \quad & \quad 41--54 \quad & \qquad \$2.60   \cr                 
\+ \hfill 2    \quad & \quad 41--54 \quad & \qquad \$2.70 \cr                   
\+ \hfill 3    \quad & \quad 46--55 \quad & \qquad \$2.87 \cr                   
\+ \hfill 4    \quad & \quad 40--53 \quad & \qquad \$3.24 \cr                   
\+ \hfill 5    \quad & \quad 45--52 \quad & \qquad \$3.40 \cr                   
                                                                                
\exercise Take the table given above and move it closer to the                  
\centre{} of the page.                                                          
                                                                                
\exercise One way to \centre{} a some text, possibly several                    
lines long, is to use: {\tt \$\$\\vbox\lb$\ldots$\rb\$\$}. Use                  
this to \centre{} the table given above.  Does the                              
{\tt \\settabs} line need to be included in the {\tt \\vbox}?                   
                                                                                
\exercise Improve your last result by putting a line under the                  
column heads.  The control word {\tt \\hrule} will insert a                     
horizontal line if introduced between two rows of a table.  Now                 
repeat with the control word {\tt \\strut} after the {\tt \\+ }                 
of the line containing the column heads. (A {\tt \\strut}                       
effectively makes the spacing between lines a little greater. The               
size can be altered from the default.)\TeXref{82} Note the extra                
space that results.                                                             
\toindex{strut}                                                                 
                                                                                
\exercise Make the following table with decimal alignment, that                 
is, with the decimal points above each other (think of the dollar               
figure as being right aligned and the cents figure as being left                
aligned against the decimal point):                                             
\medskip                                                                        
\settabs \+ \hskip 2 in & \hskip .75in & \hskip 1cm& \cr                        
\+ &Plums &\hfill\$1&.22 \cr                                                    
\+ &Coffee &\hfill1&.78 \cr                                                     
\+ &Granola &\hfill1&.98 \cr                                                    
\+ &Mushrooms & &.63 \cr                                                        
\+ &{Kiwi fruit} & &.39 \cr                                                     
\+ &{Orange juice} &\hfill1&.09 \cr                                             
\+ &Tuna &\hfill1&.29 \cr                                                       
\+ &Zucchini & &.64 \cr                                                         
\+ &Grapes &\hfill1&.69 \cr                                                     
\+ &{Smoked beef} & &.75 \cr                                                    
\+ &Broccoli &\hfill\underbar{\ \ 1}&\underbar{.09} \cr                         
\+ &Total &\hfill \$12&.55 \cr                                                  
                                                                                
                                                                                
\exercise Devise a method to make a rough table of contents by using            
{\tt settabs} and having entries looking something like:\hfil\break             
\leftline{\tt Getting Started \\dotfill \& \\hfill 1}                           
\leftline{\tt All Characters Great and Small \\dotfill \& \\hfill 9.}           
                                                                                
\subsection{Horizontal alignment with more sophisticated patterns}              
                                                                                
The {\tt \\settabs} environment is not difficult to use, and once               
the pattern is set, it can be used repeatedly in different                      
portions of the text that follows.  It does have some drawbacks,                
however. For one, the column size must be set before the entries                
are known. Also, even though in one case we wanted the third                    
column to be boldface, it had to be specified in each line.                     
These problems can be handled more easily by using the {\tt                     
\\halign} environment.                                                          
\TeXref{235--238}                                                               
\toindex{halign}                                                                
                                                                                
                                                                                
The general pattern in the {\tt \\halign} is as follows:                        
\beginuser                                                                      
\\halign\lb{} <template line> \\cr                                              
<first display line> \\cr                                                       
<second display line> \\cr                                                      
$\vdots$                                                                        
<last display line> \\cr                                                        
\rb                                                                             
\enduser                                                                        
                                                                                
Both the template line and the display lines are divided into                   
sections by the alignment symbol {\tt \&}\null. In the template                 
line each section uses control words in the same manner as does                 
{\tt \\line\lb\rb}\null.  The control word {\tt \\hfil}, for                    
example, can be used to display flush left, flush right, or \centred.           
Fonts can be changed using {\tt \\bf}, {\tt \\it}, etc.                         
Text may also be entered in the template line.  In addition the                 
special symbol {\tt \#} must appear once in each section.  Each                 
display line is then set by substituting each section of the                    
display line into its corresponding section of the template line                
at the occurrence of the {\tt \#}\null.                                         
                                                                                
Consider the following example:                                                 
                                                                                
\beginuser                                                                      
\\halign\lb\\hskip 2 in \$\#\$ \& \\hfil \\quad \# \\hfil \& \\qquad \$\#\$     
       \hskip 3in \& \\hfil \\quad \# \\hfil \\cr                               
\\alpha   \& alpha   \& \\beta  \& beta  \\cr                                   
\\gamma   \& gamma   \& \\delta \& delta \\cr                                   
\\epsilon \& epsilon \& \\zeta  \& zeta  \\cr                                   
\rb                                                                             
\enduser                                                                        
                                                                                
\noindent The template line indicates that the first section of                 
the typeset text will always be set two inches in from the left                 
and also be set as mathematics.  The second section will be \centred{}          
after adding a quad of space on the left.  The third                            
and fourth sections are handled similarly.  Here is the result:                 
\vskip\baselineskip                                                             
                                                                                
\halign{\hskip 2in $#$ & \hfil\quad # \hfil & \qquad $#$                        
                       & \hfil\quad # \hfil\cr                                  
\alpha   & alpha   & \beta  & beta \cr                                          
\gamma   & gamma   & \delta & delta \cr                                         
\epsilon & epsilon & \zeta  & zeta \cr                                          
}                                                                               
                                                                                
In this case the first display line is formed by substituting {\tt              
\\alpha} for the first {\tt \#} in the template line, {\tt alpha} for           
the second {\tt \#}, {\tt \\beta} for the third and {\tt beta} for              
the fourth.  The whole line is then saved for setting.  This                    
continues until all the lines are accumulated, and then they are set            
with each column being as wide as necessary to accept all of its                
entries (an implication of this accumulation process is that a table            
with too many entries could cause \TeX{} to run out of memory; it's             
better not to set tables that are more than a page or so long).                 
                                                                                
Hence the template line establishes the pattern for the table entries           
and the display lines insert the individual entries.                            
                                                                                
Sometimes horizontal and vertical lines are used to delimit                     
entries in a table.  To put in horizontal lines, we use {\tt \\hrule},          
just as we did in the {\tt \\settabs} environment.                              
However, we don't want the rule to be aligned according to the                  
template, so we use the control word {\tt \\noalign}.  Hence                    
horizontal lines are inserted by putting {\tt \\noalign\lb\\hrule\rb};          
vertical lines are inserted by putting {\tt \\vrule}                            
in either the template or the display line. But still all is not                
completely straightforward.  Suppose we take our last example and               
change the template to get vertical lines and also insert                       
horizontal lines.                                                               
\toindex{noalign}                                                               
                                                                                
\beginuser                                                                      
\\halign\lb\\hskip 2in\\vrule\\quad \$\#\$\\quad \& \\vrule \\hfil\\quad %      
\# \\hfil                                                                       
\hskip 2.5 in \& \\quad \\vrule \\quad \$\#\$ \\quad \\vrule                    
\hskip 2.5 in \& \\hfil \\quad \# \\quad \\hfil \\vrule \\cr                    
\\noalign\lb\\hrule\rb                                                          
\\alpha   \& alpha   \& \\beta  \& beta \\cr                                    
\\noalign\lb\\hrule\rb                                                          
\\gamma   \& gamma   \& \\delta \& delta \\cr                                   
\\noalign\lb\\hrule\rb                                                          
\\epsilon \& epsilon \& \\zeta  \& zeta \\cr                                    
\\noalign\lb\\hrule\rb                                                          
\rb                                                                             
\enduser                                                                        
                                                                                
\noindent                                                                       
doesn't give exactly what we want.                                              
                                                                                
\vskip\baselineskip                                                             
                                                                                
\halign{\hskip 2in\vrule\quad $#$\quad & \vrule \hfil\quad # \hfil              
& \quad \vrule \quad $#$ \quad \vrule & \hfil\quad # \quad \hfil                
\vrule \cr                                                                      
\noalign{\hrule}                                                                
\alpha   & alpha   & \beta  & beta \cr                                          
\noalign{\hrule}                                                                
\gamma   & gamma   & \delta & delta \cr                                         
\noalign{\hrule}                                                                
\epsilon & epsilon & \zeta  & zeta \cr                                          
\noalign{\hrule}                                                                
}                                                                               
                                                                                
There are several deficiencies: the most obvious is the extended                
horizontal lines, but also the text looks somewhat squashed into                
the boxes.  As in the {\tt \\settabs} environment, lines can be                 
made taller by including the control word {\tt \\strut} in the                  
template.\TeXref{82} A further problem can occur when the page is               
set since \TeX{} may spread lines apart slightly to improve the                 
appearance of the page.  This would leave a gap between the                     
vertical lines, so we use the control word {\tt \\offinterlineskip}             
within the {\tt \\halign} to avoid this.                                        
Finally we can get rid of the lines sticking out on the left by                 
deleting the {\tt \\hskip 2 in} from the template line.  To move                
the table to the same position we use {\tt \\moveright}.  Hence                 
we can improve our result by using                                              
\toindex{offinterlineskip}                                                      
\toindex{moveright}                                                             
                                                                                
\beginuser                                                                      
\\moveright 2 in                                                                
\\vbox\lb\\offinterlineskip                                                     
\\halign\lb\\strut\\vrule\\quad \$\#\$\\quad \& \\vrule \\hfil\\quad \# \\hfil  
\& \\quad \\vrule \\quad \$\#\$ \\quad \\vrule \& \\hfil\\quad %                
\# \\quad \\hfil \\vrule \\cr                                                   
\\noalign\lb\\hrule\rb                                                          
\\alpha   \& alpha   \& \\beta  \& beta \\cr                                    
\\noalign\lb\\hrule\rb                                                          
\\gamma   \& gamma   \& \\delta \& delta \\cr                                   
\\noalign\lb\\hrule\rb                                                          
\\epsilon \& epsilon \& \\zeta  \& zeta \\cr                                    
\\noalign\lb\\hrule\rb                                                          
\rb\rb                                                                          
\enduser                                                                        
                                                                                
\noindent                                                                       
to get                                                                          
\vskip\baselineskip                                                             
                                                                                
\moveright 2 in                                                                 
\vbox{\offinterlineskip                                                         
\halign{\strut\vrule\quad $#$\quad & \vrule \hfil\quad # \hfil                  
& \quad \vrule \quad $#$ \quad \vrule & \hfil\quad # \quad \hfil                
\vrule \cr                                                                      
\noalign{\hrule}                                                                
\alpha   & alpha   & \beta  & beta \cr                                          
\noalign{\hrule}                                                                
\gamma   & gamma   & \delta & delta \cr                                         
\noalign{\hrule}                                                                
\epsilon & epsilon & \zeta  & zeta \cr                                          
\noalign{\hrule}                                                                
}                                                                               
}                                                                               
                                                                                
In general, if we want to construct a table with boxed entries                  
that is \centred{} on the page, we can do so by putting the {\tt                
\\vbox} within a {\tt \\centerline\lb\rb}\null. But here is a                   
trick that will produce a nicer result.  If the {\tt \\vbox} is                 
put in between double dollar signs, it will be typeset as                       
displayed mathematics.  Of course, there is no actual                           
mathematics being displayed, but \TeX{} will put in a little                    
extra space above and below the table as is appropriate for a                   
display.  Hence a \centred{} table with this nice spacing may be                
formed using the following four steps:                                          
(1) put a {\tt \\vbox} between double dollar signs,                             
(2) put an {\tt \\offinterlineskip} and an {\tt \\halign} within                
the {\tt \\vbox},                                                               
(3) in the {\tt \\halign} put a template line with a {\tt \\strut}              
in the beginning,  and a {\tt \\vrule} surrounding each entry,                  
(4) each row of the table should be preceded and followed by                    
{\tt \\noalign\lb\\hrule\rb}.                                                   
                                                                                
Here is the pattern to be followed:                                             
                                                                                
\beginuser                                                                      
\$\$\\vbox\lb                                                                   
\\offinterlineskip                                                              
\\halign\lb                                                                     
\\strut \\vrule \# \& \\vrule \# \& \dots \& \\vrule \# \\vrule \\cr            
\\noalign\lb\\hrule\rb                                                          
<first column entry> \& <second column entry> \& \dots %                        
\& <last column entry> \\cr                                                     
\\noalign\lb\\hrule\rb                                                          
\dots                                                                           
\\noalign\lb\\hrule\rb                                                          
<first column entry> \& <second column entry> \& \dots %                        
\& <last column entry> \\cr                                                     
\\noalign\lb\\hrule\rb                                                          
\rb                                                                             
\rb\$\$                                                                         
\enduser                                                                        
                                                                                
                                                                                
                                                                                
\section{Rolling your own}                                                      
                                                                                
In this section we'll create new control words.  The making of                  
these new definitions, also called macros, is one of the most                   
powerful techniques available in \TeX\null.  For the first                      
application of this facility, we'll see how a new definition can                
save a lot of typing by substituting short strings for long ones.               
                                                                                
\subsection{The long and short of it}                                           
                                                                                
The control word {\tt \\def} is used to define new control words.               
The simplest form for doing this is {\tt \\def\\newname\lb$\ldots$\rb}.         
Then whenever {\tt \\newname} appears in your                                   
input file, it will be replaced by whatever is between the braces               
in the definition.  Of course {\tt \\newname} must satisfy the                  
convention for naming control sequences, that is, it must be a                  
control word (all letters) or a control symbol (exactly one                     
nonletter). So, for example, suppose you write a document that                  
contains the phrase ``University of Manitoba'' many times.  Then                
{\tt \\def\\um\lb University of Manitoba\rb} defines a new                      
control sequence {\tt \\um} which can then be used at any time.                 
The sentence {\tt I take courses at the \\um.}\ then makes                      
sense. If the control word exists, your new definition will                     
replace it (this includes the control words defined by \TeX{}, so               
a little care must be taken in the choice of name). Any                         
definition is, however,  local to the group in which it is                      
defined.  For example, \toindex{def}                                            
                                                                                
\beginuser                                                                      
\\def\\um\lb University of Manitoba\rb                                          
I took my first course at the \\um.                                             
\lb                                                                             
\\def\\um\lb Universit\\'e de Montr\\'eal\rb                                    
Then I took my next course at the \\um.                                         
\rb                                                                             
Finally I took my last course at the \\um.                                      
\enduser                                                                        
                                                                                
\noindent gives \medskip                                                        
                                                                                
\def\um{University of Manitoba}                                                 
I took my first course at the \um.                                              
{                                                                               
\def\um{Universit\'e de Montr\'eal}                                             
Then I took my next course at the \um.                                          
}                                                                               
Finally I took my last course at the \um.                                       
                                                                                
Remember that all spaces after a control word are absorbed; this                
includes the ones you define.                                                   
                                                                                
Once a new control sequence has been defined, it may be used in                 
new definitions. This is one way of making simple form letters.                 
First let's define a simple letter.                                             
                                                                                
\beginuser                                                                      
\\def\\letter\lb                                                                
\\par \\noindent                                                                
Dear \\name,                                                                    
                                                                                
This is a little note to let you know that your name is \\name.                 
                                                                                
\\hskip 2 in Sincerely yours,                                                   
\\vskip 2\\baselineskip                                                         
\\hskip 2 in The NameNoter                                                      
\\smallskip \\hrule                                                             
\rb                                                                             
\enduser                                                                        
                                                                                
\def\letter{                                                                    
\par \noindent                                                                  
Dear \name,                                                                     
                                                                                
This is a little note to let you know that your name is \name.                  
                                                                                
\hskip 2 in Sincerely yours,                                                    
\vskip 2\baselineskip                                                           
\hskip 2 in The NameNoter                                                       
\smallskip \hrule                                                               
}                                                                               
                                                                                
Now this letter uses the control word {\tt \\name}, which is                    
undefined at this point.  When {\tt \\letter} is used, the                      
current value of {\tt \\name} will appear in the body of the                    
letter. Hence                                                                   
                                                                                
\beginuser                                                                      
\\def\\name\lb Michael Bishop\rb                                                
\\letter                                                                        
\\def\\name\lb Michelle L\\'ev\\\~{}eque\rb                                     
\\letter                                                                        
\enduser                                                                        
                                                                                
\noindent will produce two copies of the letter, each with the                  
correct name, followed by a horizontal rule:                                    
                                                                                
\def\name{Michael Bishop}                                                       
\letter                                                                         
\def\name{Michelle L\'ev\~eque}                                                 
\letter                                                                         
                                                                                
We could have put anything between the braces in                                
{\tt \\def\\name\lb$\ldots$\rb}; it could be several paragraphs                 
long and use other control sequences (although in this context it               
would be a little strange). Of course it is possible to use {\tt                
\\vfill \\eject} as part of the definition of {\tt \\letter} to                 
eject the page when the letter is completed.                                    
                                                                                
\exercise Make a form letter that uses the control words {\tt \\name},          
{\tt \\address}, {\tt \\city}, {\tt \\\province}, and {\tt                      
\\\postcode}.                                                                   
                                                                                
\exercise An unnumbered list of items is often made                             
using {\tt \\item\lb\$\\bullet\$\rb}.  Define a macro {\tt \\bitem}             
that does this, and use it for several paragraphs. Now change the               
all the bullets to a dash (note that a simple change in the macro               
propagates all the necessary changes in all of the other                        
paragraphs).                                                                    
                                                                                
\exercise Suppose that you are going to have to format several                  
paragraphs in a paper using {\tt \\hangindent = 30 pt}, {\tt \\hangafter        
= 4}, and {\tt \\filbreak} (don't worry about what                              
these control sequences actually do; the only important thing                   
for now is that once they are set, they remain in effect for only               
one paragraph).  Define a single control sequence {\tt \\setpar}                
which can then be put in front of each paragraph that needs to be               
so formatted.                                                                   
                                                                                
                                                                                
\subsection{Filling in with parameters}                                         
                                                                                
It's possible to use macros in much greater generality by                       
allowing parameters to be passed.  The idea is somewhat similar                 
to the template line in the {\tt \\halign} environment.  First,                 
let's look at the case where there is one parameter.  In this                   
case a control sequence is defined by {\tt \\def\\newword\#1\lb                 
$\ldots$\rb}\null.  The symbol {\tt \#1} may appear between the                 
braces (several times) in the definition of {\tt \\newword}.  The               
material between the braces acts like a template. When {\tt                     
\\newword\lb$\ldots$\rb} appears in the text, it will use the                   
definition of {\tt \\newword} with the material between the                     
braces inserted into the template at every occurrence of {\tt                   
\#1} in the original definition. {\bf The spacing in the original               
definition is crucial here; there must be no spaces before the                  
opening brace}.                                                                 
                                                                                
As an example, we could use the form letter of the last section in              
the following way:                                                              
                                                                                
                                                                                
\beginuser                                                                      
\\def\\letter\#1\lb                                                             
\\par \\noindent                                                                
Dear \#1,                                                                       
                                                                                
This is a little note to let you know that your name is \#1.                    
                                                                                
\\hskip 2 in Sincerely yours,                                                   
\\vskip 2\\baselineskip                                                         
\\hskip 2 in The NameNoter                                                      
\\smallskip \\hrule                                                             
\rb                                                                             
\enduser                                                                        
                                                                                
\def\letter#1{                                                                  
\par \noindent                                                                  
Dear #1,                                                                        
                                                                                
This is a little note to let you know that your name is #1.                     
                                                                                
\hskip 2 in Sincerely yours,                                                    
\vskip 2\baselineskip                                                           
\hskip 2 in The NameNoter                                                       
\smallskip \hrule                                                               
}                                                                               
                                                                                
Now we can use                                                                  
                                                                                
\beginuser                                                                      
\\letter\lb Michael Bishop\rb                                                   
\\letter\lb Michelle L\\'ev\\\~{}eque\rb                                        
\enduser                                                                        
                                                                                
\noindent to get \medskip                                                       
\letter{Michael Bishop}                                                         
\letter{Michelle L\'ev\~eque}                                                   
                                                                                
\def\displaytext#1{$$\vbox{\hsize=12cm #1}$$}                                   
\bigskip                                                                        
\displaytext{                                                                   
Now let's define                                                                
{\tt\\def\\displaytext\#1\lb \$\$\\vbox\lb\\hsize = 12 cm \#1\rb\$\$\rb}        
as a new macro to display text.                                                 
Then {\tt \\displaytext\lb$\ldots$\rb} will cause the material                  
between the braces to be put in a paragraph with width 12 \centimetre           
s and then \centred{} with some space added above and                           
below as is appropriate for a display.  This paragraph was set                  
using this {\tt \\displaytext} macro.                                           
}                                                                               
                                                                                
The parameter of a macro can be no more than one paragraph long.                
If a new paragraph is encountered as part of a parameter, an                    
error will be generated.  This is a safety feature, for                         
otherwise the accidental omission of a closing brace would cause                
\TeX{} to eat up the rest of the file as the parameter.                         
                                                                                
\exercise Define a macro {\tt \\yourgrade} so that {\tt                         
\\yourgrade\lb89\rb} will cause the following sentence to be                    
typeset: The grade you received is 89\%\null. It should be able                 
to work with any other percentage, of course.                                   
                                                                                
It's not really any harder to use more that one parameter.  The form            
used to define a new control word with two parameters is                        
{\tt \\def\\newword\#1\#2\lb$\ldots$\rb}. The definition between the            
braces may have {\tt \#1} and {\tt \#2} occurring in it, perhaps                
several times.  When {\tt \\newword\lb$\ldots$\rb\lb$\ldots$\rb}                
appears in the text, the material between the first set of braces               
replaces {\tt \#1} in the definition and the material between the               
second set of braces replaces {\tt \#2} in the definition.  Here is             
an example followed by its result:                                              
                                                                                
\beginuser                                                                      
\\def\\talks\#1\#2\lb \#1 talks to \#2.\rb                                      
\\talks\lb John\rb\lb Jane\rb                                                   
\\talks\lb Jane\rb\lb John\rb                                                   
\\talks\lb John\rb\lb me\rb                                                     
\\talks\lb She\rb\lb Jane\rb                                                    
\enduser                                                                        
                                                                                
\def\talks#1#2{#1 talks to #2.}                                                 
\talks{John}{Jane}                                                              
\talks{Jane}{John}                                                              
\talks{John}{me}                                                                
\talks{She}{Jane}                                                               
                                                                                
\exercise In a manner similar to the previous exercise, define a                
macro {\tt \\yourgrade} so that {\tt \\yourgrade\lb89\rb\lb85\rb}               
causes the following sentence to be typeset: You received a grade               
of 89\% on your first exam and a grade of 85\% on your second                   
exam.                                                                           
                                                                                
\exercise Write a macro {\tt \\frac} so that {\tt \\frac\lb                     
a\rb\lb b\rb} will typeset the fraction ${a\over b}$.                           
                                                                                
It's important not to put any spaces before the first brace in                  
the definition. If you do, \TeX{} will interpret the definition                 
differently from the way described here. For more than two                      
parameters, the method of definition is similar. To define a                    
control word with three parameters, start with                                  
{\tt \\def\\newword\#1\#2\#3\lb$\ldots$\rb}. Then {\tt \#1},                    
{\tt \#2} and {\tt \#3} may occur between the braces. When                      
{\tt \\newword\lb$\ldots$\rb\lb$\ldots$\rb\lb$\ldots$\rb}                       
appears in the text, the material between each set of braces                    
replaces its corresponding symbol in the definition of the                      
control word. The parameters may go up to {\tt \#9}.                            
                                                                                
\subsection{By any other name}                                                  
                                                                                
Sometimes it's convenient to be able to give a control word an                  
alternative name.  For example, if you prefer a different spelling,             
you might want to call the control word {\tt \\centerline} by the               
name of {\tt \\centreline}.  This can be done by using the {\tt                 
\\let} control word.  The use of {\tt \\let \\centreline =                      
\\centerline} now makes a new (as well as the old) control word                 
available.  This can also be used with mathematical names as with               
{\tt \\let \\tensor = \\otimes}.  It is then possible to use                    
\TeXref{206--207}                                                               
                                                                                
\toindex{let}                                                                   
\toindex{centreline}                                                            
\toindex{tensor}                                                                
                                                                                
\let\tensor=\otimes                                                             
\beginuser                                                                      
\$\$ (A \\tensor B) (C \\tensor D) = AC \\tensor BD. \$\$                       
\enduser                                                                        
                                                                                
\noindent to get                                                                
                                                                                
$$ (A \tensor B) (C \tensor D) = AC \tensor BD. $$                              
                                                                                
                                                                                
                                                                                
\exercise Define control sequences {\tt \\ll}, {\tt \\cl}, and                  
{\tt \\rl} that are equivalent to {\tt \\leftline}, {\tt \\centerline},         
and {\tt \\rightline}.                                                          
                                                                                
The {\tt \\let} control word allows users to name their own                     
control sequences.  This allows a personalized set of control                   
sequences that may be used in place of the ones provided by \TeX{}              
when desired.                                                                   
                                                                                
\section{To err is human}                                                       
                                                                                
In some ways \TeX{} is not completely divine. \TeX{} will respond               
to invalid input by giving an error message, both to the screen                 
if you are using it interactively and also to the log file.                     
Because \TeX{} is very complicated, the actual point where the                  
error is detected may be deep within the program, so a full                     
report of the error may be rather long and involved.    Not only                
that, \TeX{} will try to recover from errors, and reports what                  
was done in that process.  For this reason the reading of error                 
messages may be a little difficult for the uninitiated.  The key                
is to know what is important from your perspective and what can                 
be safely ignored. So let's look at some typical errors and the                 
messages that they generate.                                                    
                                                                                
\subsection{The forgotten bye}                                                  
                                                                                
The first mistake that we'll look at is one that everyone makes                 
at some time, namely, the omission of {\tt \\bye} at the end of                 
the file. If you're using \TeX{} interactively, an asterisk \hfil               
\break                                                                          
\leftline{\tt *}                                                                
will be printed on the screen and nothing will happen since,                    
having not been told to finish up,  \TeX{} is waiting for input                 
(from your keyboard).  Whatever you type in will be appended to                 
whatever has been input from your files.  The usual response is                 
to type {\tt \\bye<CR>}\footnote{${}~1$}{{\tt <CR>} is the key                  
used to end a line of input.  It might be called the carriage                   
return, enter, or simply the return key on your terminal.                       
Sometimes it is indicated by a large left arrow.} since that will               
finish things up.                                                               
                                                                                
\subsection{The misspelled or unknown control sequence}                         
                                                                                
Using a misspelled or other control sequence unknown to \TeX{} is               
a common error.  If \TeX{} is being run as a batch job, an error                
message is printed and the job goes on ignoring the control                     
sequence.  When using \TeX{} interactively, it is possible to                   
repair errors (of course this does not change the original input                
file, so that must be done when the \TeX{} job is completed).                   
Suppose we have a \TeX{} input file consisting of the following                 
two lines:                                                                      
\beginuser                                                                      
\\line\lb The left side \\hfli the right side\rb                                
\\bye                                                                           
\enduser                                                                        
                                                                                
The control word should be {\tt \\hfil}, of course. Here is the                 
message that would be sent to your terminal:                                    
                                                                                
\beginuser                                                                      
\obeyspaces                                                                     
! Undefined  control sequence.                                                  
l.1 \\line\lb The left side \\hfli                                              
\                               the right side\rb                               
?                                                                               
\enduser                                                                        
                                                                                
The first line starts with {\tt !} and gives the error message.                 
Next comes the line number on which the error occurred and the                  
part of the line that was read successfully.  The next line gives               
the continuation of the line after the error. At this point the                 
question mark means that \TeX{} is waiting for a response.  There               
are several legal ones:                                                         
\nobreak                                                                        
\maketable [Responses to \TeX{} error messages]                                 
\halign{                                                                        
\strut \hfil # \hfil & \quad \hfil \tt # \hfil & # \hfil\cr                     
   \bf Desired response & \bf Input to \TeX{} & \bf \hfil Result\cr             
   \noalign{\hrule} \noalign{\smallskip}                                        
   Help   & h<CR>& Reason for stopping listed on terminal.\cr                   
   Insert & i<CR>& Next line inserted into \TeX{} input file.\cr                
   Exit   & x<CR>& Exit from \TeX\null. Completed pages to DVI file.\cr         
   Scroll & s<CR>& List message and continue after minor errors.\cr             
   Run    & r<CR>& List message and continue after any errors.\cr               
   Quiet  & q<CR>& All terminal listings suppressed.\cr                         
   Carry on &<CR> & \TeX{} continues as best it can.\cr                         
      }                                                                         
                                                                                
In our last example a reasonable response might be to enter {\tt                
h<CR>} to get a help message, then {\tt i<CR>} to insert more                   
text, (at which point \TeX{} responds with {\tt insert> } and                   
finally {\tt \\hfil} as the correct control word.  Here is the                  
result:                                                                         
                                                                                
                                                                                
\beginuser                                                                      
? h                                                                             
The control sequence at the end of the top line                                 
of your error message was never \\def'ed. If you have                           
misspelled it (e.g., `\\hobx'), type `I' and the correct                        
spelling (e.g., `I\\hbox'). Otherwise just continue,                            
and I'll forget about whatever was undefined.                                   
                                                                                
? i                                                                             
insert>\\hfil                                                                   
[1]                                                                             
\enduser                                                                        
                                                                                
The final {\tt [1]} means that the first (and only) page has                    
been completed and sent to the DVI file.  The original input file               
still needs to be fixed, of course.                                             
                                                                                
\subsection{The misnamed font}                                                  
                                                                                
A misspelled font name is an error similar to the misspelled                    
control sequence.  The error message is different and a little                  
confusing at first. Suppose for example the following appears in                
your input file:                                                                
                                                                                
\leftline{\tt \\font\\sf = cmss01}                                              
                                                                                
It should be {\tt cmss10}, that is, the numbers have been                       
transposed.  Here are the error and help messages:                              
                                                                                
\beginuser \obeyspaces                                                          
! Font \\sf=cmss01 not loadable: Metric (TFM) file not found.                   
<to be read again>                                                              
\                   \\par                                                       
\\bye ->\\par                                                                   
\            \\vfill \\supereject \\end                                         
l.2 \\bye                                                                       
                                                                                
? h                                                                             
I wasn't able to read the size data for this font,                              
so I will ignore the font specification.                                        
[Wizards can fix TFM files using TFtoPL/PLtoTF.]                                
You might try inserting a different font spec;                                  
e.g., type `I\\font<same font id>=<substitute font name>'.                      
\enduser                                                                        
                                                                                
The TFM (\TeX{} font metric) file is an auxiliary file that is                  
used by \TeX\null. So this strange message is just telling you                  
that the font you defined doesn't exist on your computer system.                
                                                                                
\subsection{Mismatched mathematics}                                             
                                                                                
Another common error is to use {\tt \$} or {\tt \$\$} to start a                
mathematical expression and then forget to use the second {\tt \$}              
or {\tt \$\$} when finished.  The text that follows is then                     
treated as mathematics, and to make matters worse, if more                      
mathematics is started by a new {\tt \$} or {\tt \$\$}, it will                 
then be treated as ordinary text.  Needless to say, error                       
messages galore may be generated. \TeX{} will attempt to correct                
the problem by inserting a new {\tt \$} or {\tt \$\$}; in any                   
case, the problem is corrected by the end of the paragraph since                
a new paragraph will automatically start as ordinary text.                      
                                                                                
Consider the following correct input and its output:                            
\beginuser                                                                      
Since \$f(x) > 0\$, \$a<b\$,  and \$f(x)\$ is continuous, we know that          
\$\\int\_{}a\~{}b f(x)\\,dx >0\$.                                               
\enduser                                                                        
                                                                                
Since $f(x) > 0$, $a<b$,  and $f(x)$ is continuous, we know that                
$\int_a~b f(x)\,dx >0$.                                                         
                                                                                
If we now leave out the second dollar sign in {\tt \$f(x)\$} we                 
then get the following error and help messages:                                 
                                                                                
\beginuser \obeyspaces                                                          
! Missing \$ inserted.                                                          
<inserted text>                                                                 
\               \$                                                              
<to be read again>                                                              
\                  \\intop                                                      
\\int ->\\intop                                                                 
\             \\nolimits                                                        
l.2 \$\\int                                                                     
\          \_{}a\~{}b f(x)\\,dx >0\$.                                           
? h                                                                             
I've inserted a begin-math/end-math symbol since I think                        
you left one out. Proceed, with fingers crossed.                                
                                                                                
?                                                                               
\enduser                                                                        
                                                                                
The line starting with {\tt !} tells us what has been done. The                 
line starting with {\tt l.2} shows us where we were in the input                
file when the error occurred.  As in our other examples, the part               
of the line successfully read, that is, through {\tt \\int},                    
appears on one line, and the continuation appears on the next                   
line. The remaining material may seem somewhat obscure.  These                  
intermediate messages show what was happening further in the guts               
of the \TeX{} program when the error occurred.  The newer user                  
may ignore them.                                                                
                                                                                
Here is what you get as output after \TeX{} tries to recover from               
the error.                                                                      
                                                                                
Since $f(x) > 0$, $a<b$,  and $f(x) is continuous, we know that                 
\int_a~b f(x)\,dx >0$.                                                          
                                                                                
There is a stretch of text that is italic with no spacing.  This                
is typical for normal text being processed as mathematics; if you               
see this in your output, you have almost certainly left out a                   
{\tt \$} or {\tt \$\$}.                                                         
                                                                                
\subsection{Mismatched braces}                                                  
                                                                                
It's easy to forget or mismatch the closing braces when making                  
groups.  The result may be a relatively benign error, or it may                 
be catastrophic. Suppose, for example, you have {\tt \lb\\bf A                  
bold title } in your text with the closing right brace omitted.                 
The result will be the same as if no opening brace were there;                  
that is, the rest of the paper will be boldface if no other font                
changes are made.  You will get the following message at the end                
of the file:                                                                    
                                                                                
{\tt (\\end occurred inside a group at level 1)}                                
                                                                                
If you had made the same mistake twice, then there would be two                 
more opening braces than closing braces, and you would get the                  
message:                                                                        
                                                                                
{\tt (\\end occurred inside a group at level 2)}                                
                                                                                
\TeX{} doesn't know that the closing brace is missing until it                  
reaches the end of the input file.  Hence the message doesn't                   
tell you where you went wrong.  If the location of the missing                  
brace isn't obvious, it's always possible to insert {\tt \\bye}                 
halfway through your document.  Running \TeX{} again will cause                 
only the first half to be processed, and if the error message                   
persists, you will know that the error is in the first half of                  
the document.  By moving the {\tt \\bye} to different places, the               
error can be localized.  Also, looking at the output often                      
reveals what has gone wrong.                                                    
                                                                                
Missing opening braces are much easier to spot.  Here is a two                  
line input file and the resulting error and help messages:                      
                                                                                
\beginuser                                                                      
\\bf Here is the start\rb, but there is the finish.                             
\\bye                                                                           
\enduser                                                                        
                                                                                
\beginuser \obeyspaces                                                          
! Too many \rb's.                                                               
l.1 \\bf Here is the start\rb                                                   
\                          , but there is the finish.                           
                                                                                
? h                                                                             
You've closed more groups than you opened.                                      
Such booboos are generally harmless, so keep going.                             
\enduser                                                                        
                                                                                
It's quite possible, of course, that line that is supposed to                   
have the missing left brace will not be on the line where \TeX{}                
catches the error.                                                              
                                                                                
A mismatched brace in the definition of a new control sequence                  
can cause a major error.  Since such a definition may include                   
several paragraphs, it may not be caught by the end of a                        
paragraph, but, rather will just keep piling more and more text                 
into the unfinished definition.  It's even possible for \TeX{} to               
run out of memory as it keeps eating up more text! This is called               
a ``runaway definition''.                                                       
\TeXref{206}                                                                    
Here is a two line input file with a runaway definition:                        
                                                                                
\beginuser                                                                      
\\def\\newword\lb the def                                                       
\\newword                                                                       
\\bye                                                                           
\enduser                                                                        
                                                                                
Here are the resulting error and help messages:                                 
                                                                                
\beginuser \obeyspaces                                                          
Runaway definition?                                                             
->the def                                                                       
! Forbidden control sequence found while scanning definition of \\newword.      
<inserted text>                                                                 
\                \rb                                                            
<to be read again>                                                              
\                   \\bye                                                       
l.3 \\bye                                                                       
                                                                                
? h                                                                             
I suspect you have forgotten a `\rb', causing me                                
to read past where you wanted me to stop.                                       
I'll try to recover; but if the error is serious,                               
you'd better type `E' or `X' now and fix your file.                             
                                                                                
?                                                                               
No pages of output.                                                             
\enduser                                                                        
                                                                                
This is obviously a serious error.  If it occurs at the beginning               
of a file (as in the previous example), there will be no output                 
at all!                                                                         
                                                                                
If a closing brace is left out while using a macro with                         
parameters, the runaway definition will be terminated at the end                
of the paragraph. So if {\tt \\def\\newword\#1\lb$\ldots$\rb} has               
been defined and you use {\tt \\newword\lb$\dots$ } without the                 
closing brace, then at most one paragraph will be ruined.                       
\TeXref{205}                                                                    
                                                                                
In short, when an error occurs, make a note of the line number to               
see how much of the input file has been read, and also the line                 
starting with an exclamation point to get a short description of                
the error. If the error is still not clear, ask \TeX{} for more                 
information by typing {\tt h<CR>}. For small errors, \TeX{} can                 
carry on quite a way if you just keep hitting the {\tt <CR>}.                   
                                                                                
                                                                                
                                                                                
                                                                                
\section{Digging a little deeper}                                               
                                                                                
In this section we look at a few topics that allow \TeX{} to be                 
used with greater flexibility or efficiency.  As the documents                  
being produced get longer, different techniques can help make                   
their creation easier.                                                          
                                                                                
\subsection{Big files, little files}                                            
                                                                                
\TeX{} can read and write files as it runs.  This makes it                      
possible to use files that are smaller and more convenient to                   
handle by creating a master file that reads the smaller files in                
the proper order. This document, for example, consists of ten                   
sections plus an introduction.  In addition, there are macros                   
that are used for all sections. The macros can be put in a file                 
called, say, {\tt macros.tex}, the introduction can be put in {\tt              
intro.tex}, and each section put in its own file.  The control                  
word {\tt \\input} is then used to read in a file.  In general,                 
{\tt \\input filename} will cause the file called {\tt                          
filename.tex} to be read in and processed immediately, just as if               
the text of {\tt filename.tex} had been part of the file that                   
read it in.  This file may input other files. In fact it's often                
convenient to make a single file that reads in smaller pieces,                  
perhaps as follows:                                                             
\toindex{input}                                                                 
                                                                                
\beginuser                                                                      
\\input macros                                                                  
\\input intro                                                                   
\\input sec1                                                                    
\\input sec2                                                                    
\\input sec3                                                                    
\\input sec4                                                                    
\\input sec5                                                                    
\\input sec6                                                                    
\\input sec7                                                                    
\\input sec8                                                                    
\\input sec9                                                                    
\\input sec10                                                                   
\enduser                                                                        
                                                                                
While the text is still being heavily edited, it's possible to                  
process only some of the files by putting a {\tt \%} at the                     
beginning of each line that contains a file to be skipped (this                 
is sometimes called ``commenting out'' the unwanted files).                     
                                                                                
The {\tt \\input} control word also allows the use of                           
predesigned macros.  The macros for a memorandum, for example,                  
might be put in a file called {\tt memo.tex}.  These macros might               
set up the right {\tt \\hsize}, {\tt \\vsize} and other                         
parameters, and might stamp the time and date.  Once this has                   
been set up, all memoranda may be started with {\tt \\input memo}               
to make them come out with a common format.                                     
                                                                                
\exercise Make a \TeX{} input file that reads in a second file.                 
Try reading in the second file twice using the {\tt \\input}                    
control word twice.                                                             
                                                                                
\subsection{Larger macro packages}                                              
                                                                                
Designing macros that can be used with many types of documents                  
is obviously useful.  Most universities, for example, have                      
specific and often complicated format requirements for theses.  A               
collection of macros, that is, a macro package that meets all                   
these specifications could be somewhat time consuming to design                 
and could be quite long.  It is possible to use the {\tt \\input}               
command to use such a macro package, just as it is possible to                  
use it with your own macros.  But \TeX{} has a better facility                  
for larger packages.                                                            
                                                                                
A macro package can be put in a special form that can be quickly                
read by \TeX\null.  These are called {\sl format files}, and                    
their exact form is of technical interest only.  The important                  
thing is that they allow \TeX{} to be run with many new control                 
sequences predefined. Certain commands called {\sl primitives\/}                
are part of the definition of \TeX\null.                                        
                                                                                
What we have described in this manual is sometimes called {\sl                  
plain \TeX\/}, and consists of the \TeX{} primitives plus a set                 
of macros in a format file (that is usually included in \TeX{}                  
automatically) called {\tt plain.fmt}. For the curious, any                     
control word can be viewed using {\tt \\show}.  The command {\tt                
\\show\\centerline} will display                                                
\beginliteral                                                                   
> \centerline=macro:                                                            
#1->\line {\hss #1\hss }.                                                       
@endliteral                                                                     
\noindent                                                                       
on the screen and in the log file. You can use {\tt \\show} with                
your own macros, too.  If you end up using several macro                        
packages, you can use the {\tt \\show} command to see if a                      
particular macro is defined.                                                    
                                                                                
Many computer \centre{}s have the \LaTeX{} macro package.  This                 
package allows the user to create an index, a table of contents,                
and a bibliography automatically.  It also has the ability to                   
insert some elementary graphic figures such as circles, ovals,                  
lines, and arrows. \LaTeX{} also uses style files to set up                     
specific page parameters. Many different style files are                        
available; some journals will accept papers on a magnetic medium                
for direct processing if they are prepared using \LaTeX{} and a                 
designated style file. It is not difficult to shift from \TeX{}                 
to \LaTeX\null. A user's guide by the author of the macro                       
package, Leslie Lamport, is available:                                          
{\bf \LaTeX{}: A document preparation system}%                                  
\footnote{${}~1$}{Addison-Wesley Reading, Massachusetts, 1986,                  
ISBN 0-201-15790-X.}.                                                           
                                                                                
The American Mathematical Society uses the \AMSTeX{} macro                      
package for its journals.  It is readily available from that                    
Society, and papers may be submitted to their journals on a                     
magnetic medium using \AMSTeX\null.  A manual by Michael                        
Spivak, {\bf The Joy of \TeX{}}, is available from the American                 
Mathematical Society.                                                           
                                                                                
Other macro packages exist, and undoubtedly more will be                        
developed. They are usually of modest cost and can be very                      
effective in some circumstances.  The \TeX{} Users Group                        
announces the existence of new macro packages in its                            
publications.                                                                   
                                                                                
\subsection{Horizontal and vertical lines}                                      
                                                                                
Making horizontal and vertical lines is easy using \TeX\null.                   
When typing in text, {\tt \\hrule } will cause the current                      
paragraph to end, will draw a horizontal line whose width is the                
current hsize, and then will continue on with a new paragraph.                  
It's possible to specify the width of the hrule as, for example,                
with {\tt \\hrule width 5 cm}; also you can use {\tt \\vskip} or                
{\tt \\bigskip} to put some space above or below the hrule.  Here               
is an example:                                                                  
                                                                                
\beginuser                                                                      
\\parindent = 0 pt \\parskip = 12 pt                                            
Here is the text before the hrule.                                              
\\bigskip                                                                       
\\hrule width 3 in                                                              
And here is some text after the hrule.                                          
\enduser                                                                        
                                                                                
\noindent that produces                                                         
                                                                                
{                                                                               
\parindent = 0 pt                                                               
Here is the text before the hrule.                                              
\bigskip                                                                        
\hrule width 3 in                                                               
And here is some text after the hrule.                                          
}                                                                               
                                                                                
In fact this hrule not only has width of three inches, but also                 
by default has a height (the amount by which the hrule extends                  
above the baseline on which the type is being set) of 0.4 points                
and a depth (the amount by which the hrule extends below the                    
baseline on which the type is being set) of 0 points.  Each of                  
these parameters can be individually set.  Thus if we change the                
last example to say                                                             
                                                                                
{\tt \\hrule width 3 in height 2 pt depth 3 pt } we get                         
                                                                                
{                                                                               
\parindent = 0 pt                                                               
Here is the text before the hrule.                                              
\bigskip                                                                        
\hrule width 3 in height 2 pt depth 3 pt                                        
And here is some text after the hrule.                                          
}                                                                               
                                                                                
The three parameters {\tt width}, {\tt height}, and {\tt depth} may             
be given in any order.                                                          
\toindex{hrule}                                                                 
\TeXref{221--222}                                                               
                                                                                
A vrule may be defined analogously to an hrule by specifying the                
{\tt width}, {\tt height}, and {\tt depth} if desired.                          
\TeXref{221--222}                                                               
But, unlike the hrule, the vrule will not automatically start a                 
new paragraph when it appears.  By default the vrule will be 0.4                
points wide, and will be as high as the line on which it is being               
set. Hence                                                                      
                                                                                
\toindex{vrule}                                                                 
                                                                                
\beginuser                                                                      
Here is some text before the vrule                                              
\\vrule\\                                                                       
and this follows the vrule.                                                     
\enduser                                                                        
                                                                                
\noindent will give                                                             
                                                                                
Here is some text before the vrule                                              
\vrule\                                                                         
and this follows the vrule.                                                     
                                                                                
\exercise Make three horizontal lines that are 15 points apart,                 
3 inches in length, and one inch in from the left margin.                       
                                                                                
Although we usually think of hrules and vrules as horizontal and                
vertical lines, they need not necessarily be used that way.  For                
example:                                                                        
                                                                                
\beginuser                                                                      
\\noindent                                                                      
Name: \\vrule height 0 pt depth 0.4 pt width 3 in                               
\enduser                                                                        
                                                                                
\noindent will give                                                             
                                                                                
\noindent                                                                       
Name: \vrule height 0 pt depth 0.4 pt width 3 in                                
                                                                                
\exercise Make the following grid (each box is 1 \centimetre{} square):         
\medskip                                                                        
\settabs \+ \hskip 1cm&\hskip 1 cm&\hskip 1 cm& \cr                             
\moveright 2 in                                                                 
\vbox{                                                                          
\hrule width 3 cm                                                               
\+  \vrule height 1 cm & \vrule height 1 cm & \vrule height 1 cm                
  & \vrule height 1 cm \cr                                                      
\hrule width 3 cm                                                               
\+  \vrule height 1 cm & \vrule height 1 cm & \vrule height 1 cm                
  & \vrule height 1 cm \cr                                                      
\hrule width 3 cm                                                               
\+  \vrule height 1 cm & \vrule height 1 cm & \vrule height 1 cm                
  & \vrule height 1 cm \cr                                                      
\hrule width 3 cm                                                               
}                                                                               
                                                                                
                                                                                
\subsection{Boxes within boxes}                                                 
                                                                                
We have already seen (in our discussion of line shapes) that                    
vboxes and hboxes are objects that may be overfull or underfull.                
In this section we will look at these boxes in a bit more detail.               
They may be stacked or lined up to allow a variety of positions                 
for text on the page.                                                           
                                                                                
An hbox is formed by using {\tt \\hbox\lb $\ldots$\rb}\null.                    
Once the material between the braces has been put into an hbox,                 
it is set and can not be further split (this means that material                
that must go on one line can be put into an hbox, and it will                   
then remain as one unit).  It's possible to specify the size of                 
an hbox.\TeXref{64--66} Thus {\tt \\hbox to 5 cm\lb contents of                 
the box\rb } will produce an hbox exactly five \centimetre{}s                   
wide containing the typeset text ``contents of the box''.                       
It's easy to get an underfull or overfull box in this way. An                   
underfull box can be avoided by using {\tt \\hfil} to absorb the                
extra space. When no dimension is given, an hbox is formed that                 
is just wide enough to hold the enclosed text.                                  
\toindex{hbox}                                                                  
                                                                                
Similarly, vboxes are formed using {\tt \\vbox\lb $\ldots$\rb}.                 
What makes these boxes interesting is that when a vbox contains                 
hboxes, these hboxes are stacked one above the other and set as a               
unit. Similarly, an hbox can contain vboxes, which will be set in               
a row. Suppose we take three hboxes and put them in a vbox:                     
\toindex{vbox}                                                                  
                                                                                
\beginuser \obeyspaces                                                          
\\vbox\lb                                                                       
\      \\hbox\lb{}Contents of box 1\rb                                          
\      \\hbox\lb{}Contents of box 2\rb                                          
\      \\hbox\lb{}Contents of box 3\rb                                          
\      \rb                                                                      
\enduser                                                                        
                                                                                
\noindent gives                                                                 
\vskip \baselineskip                                                            
\vbox{                                                                          
      \hbox{Contents of box 1}                                                  
      \hbox{Contents of box 2}                                                  
      \hbox{Contents of box 3}                                                  
      }                                                                         
                                                                                
Now suppose we take another vbox:                                               
                                                                                
\beginuser \obeyspaces                                                          
\\vbox\lb                                                                       
\      \\hbox\lb{}Contents of box 4\rb                                          
\      \\hbox\lb{}Contents of box 5\rb                                          
\      \rb                                                                      
\enduser                                                                        
                                                                                
These two vboxes can be put into an hbox; this will cause them to               
be placed side by side. In other words                                          
                                                                                
\beginuser \obeyspaces                                                          
\\hbox\lb                                                                       
\     \\vbox\lb                                                                 
\           \\hbox\lb{}Contents of box 1\rb                                     
\           \\hbox\lb{}Contents of box 2\rb                                     
\           \\hbox\lb{}Contents of box 3\rb                                     
\           \rb                                                                 
\     \\vbox\lb                                                                 
\           \\hbox\lb{}Contents of box 4\rb                                     
\           \\hbox\lb{}Contents of box 5\rb                                     
\           \rb                                                                 
\     \rb                                                                       
\enduser                                                                        
                                                                                
\noindent gives                                                                 
\vskip \baselineskip                                                            
\hbox{                                                                          
     \vbox{                                                                     
           \hbox{Contents of box 1}                                             
           \hbox{Contents of box 2}                                             
           \hbox{Contents of box 3}                                             
           }                                                                    
     \vbox{                                                                     
           \hbox{Contents of box 4}                                             
           \hbox{Contents of box 5}                                             
          }                                                                     
     }                                                                          
                                                                                
                                                                                
Notice that the two vboxes are aligned so that the bottoms are                  
level; also there is a little space between the vboxes.                         
Actually, the reason the space appears is rather subtle.                        
Unless a line ends in a control word, there is always a space                   
between the last entry in one line and the first one in the next                
line. For this reason the space comes from the end of the line                  
containing the closing brace of the first vbox.  This space can                 
be avoided by ``commenting out'' the end of the line, that is, by               
putting a {\tt \%} immediately after the closing brace of the                   
first vbox. If you try to put some vboxes together and accidently               
get extra space by forgetting to comment out the end of the line,               
you're in good company.  Some very able and experienced \TeX{}                  
users have done the same thing!                                                 
                                                                                
Extra space, say one \centimetre,  can be added by putting an                   
{\tt \\hskip 1 cm } between the vboxes.  They can be aligned so                 
that the tops are level by using {\tt \\vtop } instead of                       
{\tt \\vbox}.  Making these two changes results in:                             
                                                                                
\toindex{vtop}                                                                  
\vskip \baselineskip                                                            
\hbox{                                                                          
     \vtop{                                                                     
           \hbox{Contents of box 1}                                             
           \hbox{Contents of box 2}                                             
           \hbox{Contents of box 3}                                             
           }                                                                    
     \hskip 1 cm                                                                
     \vtop{                                                                     
           \hbox{Contents of box 4}                                             
           \hbox{Contents of box 5}                                             
          }                                                                     
     }                                                                          
                                                                                
                                                                                
We can combine vboxes, hboxes, vrules, and hrules to get boxed                  
text.  How might we construct such a box? One way is to take the                
material to be boxed and put it in an hbox preceded and followed                
by a vrule.  Then put this in a vbox with hrules above and below                
it. This gives us:                                                              
                                                                                
\beginuser \obeyspaces                                                          
\\vbox\lb                                                                       
\      \\hrule                                                                  
\      \\hbox\lb\\vrule{} The text to be boxed \\vrule\rb                       
\      \\hrule                                                                  
\     \rb                                                                       
\enduser                                                                        
                                                                                
\noindent which results in                                                      
\vskip \baselineskip                                                            
\vbox{                                                                          
      \hrule                                                                    
      \hbox{\vrule{} The text to be boxed \vrule}                               
      \hrule                                                                    
     }                                                                          
                                                                                
This produces boxed material, but there is no margin around it                  
and so it looks very cramped (of course \TeX{} is just giving us                
what we asked for).  We can improve the spacing by putting a {\tt               
\\strut} at the beginning of the hbox to make it a little taller                
and deeper.  This gives us:                                                     
\vskip \baselineskip                                                            
\vbox{                                                                          
      \hrule                                                                    
      \hbox{\strut \vrule{} The text to be boxed \vrule}                        
      \hrule                                                                    
     }                                                                          
                                                                                
\def\boxtext#1{%                                                                
\vbox{%                                                                         
      \hrule                                                                    
      \hbox{\strut \vrule{} #1 \vrule}%                                         
      \hrule                                                                    
     }%                                                                         
}                                                                               
                                                                                
\exercise Use the method of boxing material to put text \centred{}              
in a box which extends from the left to the right margin.                       
                                                                                
\exercise Make the following magic square:                                      
\vskip\baselineskip                                                             
                                                                                
\moveright 2 in \vbox{\offinterlineskip                                         
\hbox{\boxtext 6\boxtext 1\boxtext 8}                                           
\hbox{\boxtext 7\boxtext 5\boxtext 3}                                           
\hbox{\boxtext 2\boxtext 9\boxtext 4}                                           
}                                                                               
                                                                                
\def\boxtext#1 {%                                                               
\vbox{%                                                                         
      \hrule                                                                    
      \hbox{\strut \vrule #1\vrule}%                                            
      \hrule                                                                    
     }                                                                          
}                                                                               
                                                                                
\exercise Write a macro { \tt \\boxtext\#1\lb$\ldots$\rb } which                
will take the text between the braces and put a box around it.  Test            
your macro by making up a sentence with every other word boxed.                 
I'm \boxtext not quite \boxtext sure why \boxtext someone                       
would \boxtext do this \boxtext since the \boxtext result is \boxtext           
pretty strange. Note how the baseline and the bottom of                         
the surrounding boxes align.                                                    
                                                                                
\def\boxtext#1 {%                                                               
\lower 3.5pt \hbox{%                                                            
    \vbox{%                                                                     
         \hrule                                                                 
         \hbox{\strut \vrule #1\vrule}%                                         
         \hrule                                                                 
        }                                                                       
    }                                                                           
}                                                                               
                                                                                
It's easy to move boxes up, down, left, or right on the page. A                 
{\tt \\vbox} can be moved to the right one inch by using {\tt                   
\\moveright 1 in \\vbox\lb\dots\rb}.  To move it to the left,                   
use {\tt \\moveleft}.  Similarly, an {\tt hbox} can be moved up                 
or down using {\tt \\raise} or {\tt \\lower}.                                   
\toindex{moveright}                                                             
\toindex{moveleft}                                                              
\toindex{raise}                                                                 
\toindex{lower}                                                                 
                                                                                
\exercise Rewrite the {\tt \\boxtext} macro from the previous                   
exercise so that all of the text is aligned (by default the                     
depth of a strut is 3.5 points).  This would give a sentence like               
the following: I'm \boxtext not quite \boxtext sure why \boxtext                
someone would \boxtext do this \boxtext since the \boxtext result               
is \boxtext pretty strange.                                                     
                                                                                
It's possible to fill a box with either an hrule or with dots.                  
The idea is to use {\tt \\hrulefill } or {\tt \\dotfill } in the                
hbox.                                                                           
                                                                                
\beginuser                                                                      
\\hbox to 5 in\lb Getting Started\\hrulefill 1\rb                               
\\hbox to 5 in\lb All Characters Great and Small\\hrulefill 9\rb                
\\hbox to 5 in\lb The Shape of Things to come\\hrulefill 17\rb                  
\\hbox to 5 in\lb No Math Anxiety Here!\\hrulefill 30\rb                        
\enduser                                                                        
                                                                                
\noindent gives                                                                 
\vskip \baselineskip                                                            
                                                                                
\hbox to 5 in{Getting Started\hrulefill 1}                                      
\hbox to 5 in{All Characters Great and Small\hrulefill 9}                       
\hbox to 5 in{The Shape of Things to come\hrulefill 17}                         
\hbox to 5 in{No Math Anxiety Here!\hrulefill 30}                               
                                                                                
If {\tt \\hrulefill} is replaced by {\tt \\dotfill} we get                      
\vskip \baselineskip                                                            
                                                                                
\hbox to 5 in{Getting Started\dotfill 1}                                        
\hbox to 5 in{All Characters Great and Small\dotfill 9}                         
\hbox to 5 in{The Shape of Things to come\dotfill 17}                           
\hbox to 5 in{No Math Anxiety Here!\dotfill 30}                                 
                                                                                
\exercise Make a boxed headline appear at the top of the page                   
that is like the one used in this manual.                                       
                                                                                
                                                                                
                                                                                
\section{Control word list}                                                     
                                                                                
Here is a list of the control words given in this manual.                       
If you want more detail about these words than is given here,                   
check the index of {\bf The \TeX book}.                                         
                                                                                
\vskip 2\baselineskip                                                           
\centerline{Control symbols}                                                    
\vskip\baselineskip                                                             
                                                                                
\settabs \+ \hskip 1.5 in & \hskip 1.65in & \hskip 1.3in & \cr                  
{\tt                                                                            
\+  \\\sp{} 5 & \\!\ 33  & \\" 11 & \\' 11    \cr                               
\+  \\, 33    & \\.\ 11  & \\/ 15 & \\; 33    \cr                               
\+  \\= 11    & \\> 33   & \\\# 10&  \\\$ 7  \cr                                
\+  \\\% 7    & \\\& 10   & \\\char '173{} 10  & \\\char '175{} 10 \cr          
\+  \\\underbar{ } 10 & \\` 11 & \\{\accent "7E } 10 & \\{\accent "7E } 10 \cr  
\+  \\{\accent 94 } 10 & \\! 39 \cr                                             
}                                                                               
                                                                                
\vskip 2\baselineskip                                                           
\centerline{Control words}                                                      
\vskip\baselineskip                                                             
                                                                                
{\tt                                                                            
\+ \\AA 12 & \\aa 12 & \\acute 35 & \\AE 12 \cr                                 
\+ \\ae 12 & \\aleph 36 & \\alpha 34 & \\angle 36 \cr                           
\+ \\approx 36 & \\arccos 40 & \\arcsin 40 & \\arctan 40 \cr                    
\+ \\arg 40 & \\ast 35 & \\b 11 & \\backslash 36 \cr                            
\+ \\bar 35 & \\baselineskip 21 & \\beta 34 & \\bf 15 \cr                       
\+ \\biggl 39 & \\Biggl 39 & \\biggr 39 & \\Biggr 39 \cr                        
\+ \\bigl 39 & \\Bigl 39 & \\bigr 39 & \\Bigr 39 \cr                            
\+ \\bigskip 25 & \\break 25 & \\breve 35 & \\bullet 35 \cr                     
\+ \\bye 5 & \\c 11 & \\cap 35 & \\cdot 35 \cr                                  
\+ \\centerline 25 & \\centreline 59 & \\check 35 & \\chi 34 \cr                
\+ \\circ 35 & \\columns 47 & \\cos 40 & \\cosh 40 \cr                          
\+ \\cot 40 & \\coth 40 & \\csc 40& \\cup 35 \cr                                
\+ \\d 11 & \\ddag 26 & \\ddot 35 & \\def 54 \cr                                
\+ \\deg 40 & \\delta 34 & \\Delta 34 & \\det 40 \cr                            
\+ \\diamond 35 & \\dim 40 & \\div 35 & \\dot 35 \cr                            
\+ \\dotfill 48 & \\dots 14 & \\downarrow 40 & \\Downarrow 40 \cr               
\+ \\eject 19 & \\ell 36 & \\endinsert 25 & \\epsilon 34 \cr                    
\+ \\eqalign 44 & \\eqalignno 45 & \\eqno 45 & \\equiv 36 \cr                   
\+ \\eta 34 & \\exists 36 & \\exp 40 & \\flat 36 \cr                            
\+ \\folio 27 & \\font 15 & \\footline 27 & \\footnote 26 \cr                   
\+ \\forall 36 & \\gamma 34 & \\Gamma 34 & \\gcd 40 \cr                         
\+ \\geq 36 & \\grave 35 & \\H 11 & \\halign 50 \cr                             
\+ \\hang 22 & \\hangafter 22 & \\hangindent 22 & \\hat 35 \cr                  
\+ \\hbaddness 28 & \\hbox 70 & \\headline 27 & \\hfil 26 \cr                   
\+ \\hfill 25 & \\hfuzz 28 & \\hoffset 19 & \\hom 40 \cr                        
\+ \\hrule 69 & \\hrulefill 48 & \\hsize 19 & \\hskip 26 \cr                    
\+ \\hyphenation 28 & \\i 11 & \\Im 36 & \\in 36 \cr                            
\+ \\inf 40 & \\infty 36 & \\input 66 & \\int 37 \cr                            
\+ \\iota 34 & \\it 16 & \\item 23 & \\itemitem 23 \cr                          
\+ \\j 11 & \\kappa 34 & \\ker 40 & \\L 12 \cr                                  
\+ \\l 12 &  \\lambda 34 & \\Lambda 34 & \\langle 40 \cr                        
\+ \\lceil 40 &  \\left 43 & \\leftline 25 & \\leftskip 22 \cr                  
\+ \\leq 36 &  \\leqalignno 45 & \\leqno 45 & \\let 59 \cr                      
\+ \\lfloor 40 &  \\lg 40 & \\lim 37 & \\liminf 40 \cr                          
\+ \\limsup 40 &  \\line 25 & \\ln 40 & \\log 40 \cr                            
\+ \\lower 73 &  \\magnification 20 & \\magstep 15 & \\matrix 43 \cr            
\+ \\max 40 &  \\medskip 25 & \\min 40 & \\moveleft 73 \cr                      
\+ \\moveright 73 &  \\mu 34 & \\nabla 36 & \\narrower 22 \cr                   
\+ \\natural 36 &  \\neg 36 & \\ni 36 & \\noalign 51 \cr                        
\+ \\noindent 21 &  \\nopagenumbers 6 & \\not 35 & \\nu 34 \cr                  
\+ \\O 12 &  \\o 12 & \\odot 35 & \\OE 12 \cr                                   
\+ \\oe 12 &  \\offinterlineskip 52 & \\omega 34 & \\Omega 34 \cr               
\+ \\ominus 35 &  \\oplus 35 & \\otimes 35 & \\over 36 \cr                      
\+ \\overfullrule 28 &  \\overline 38 & \\P 26 & \\pageno 27 \cr                
\+ \\par 8 &  \\parallel 36 & \\parindent 21 & \\parshape 23 \cr                
\+ \\parskip 21 &  \\partial 36 & \\perp 36 & \\phi 34 \cr                      
\+ \\Phi 34 &  \\pi 34 & \\Pi 34 & \\pmatrix 42 \cr                             
\+ \\Pr 40 &  \\proclaim 41 & \\psi 34 & \\Psi 34 \cr                           
\+ \\qquad 33 &  \\quad 33 & \\raggedright 26 & \\raise 73 \cr                  
\+ \\rangle 40 &  \\rceil 40 & \\Re 36 & \\rfloor 40 \cr                        
\+ \\rho 34 &  \\right 43 & \\rightline 26 & \\rightskip 22 \cr                 
\+ \\rm 15 &  \\root 38 & \\S 26 & \\scaled 15 \cr                              
\+ \\sec 40 &  \\settabs 47 & \\sharp 36 & \\sigma 34 \cr                       
\+ \\Sigma 34 &  \\sim 36 & \\simeq 36 & \\sin 40 \cr                           
\+ \\sinh 40 &  \\sl 15 & \\smallskip 25 & \\sqroot 38 \cr                      
\+ \\ss 12 &  \\star 35 & \\strut 49 & \\subset 36 \cr                          
\+ \\subseteq 36 &  \\sum 37 & \\sup 40 & \\supset 36 \cr                       
\+ \\supseteq 36 &  \\surd 37 & \\t 11 & \\tan 40 \cr                           
\+ \\tanh 40 &  \\tau 35 & \\tensor 59 & \\TeX{} 5 \cr                          
\+ \\tensor 59 &  \\the 27 & \\theta 34 & \\Theta 34 \cr                        
\+ \\tilde 35 &  \\times 35 & \\tolerance 28  & \\topinsert 25 \cr              
\+ \\tt 15 &  \\u 11 & \\underbar 39 & \\underline 38 \cr                       
\+ \\uparrow 40 &  \\Uparrow 40 & \\updownarrow 40 & \\Updownarrow 40 \cr       
\+ \\upsilon 34 &  \\Upsilon 34 & \\v 11 & \\varepsilon 34 \cr                  
\+ \\varphi 34 &  \\varrho 34 & \\varsigma 34 & \\vartheta 34 \cr               
\+ \\vbadness 29 &  \\vbox 70 & \\vec 35 & \\vee 35 \cr                         
\+ \\vfill 19 &  \\vglue 24 & \\voffset 19 & \\vrule 69 \cr                     
\+ \\vsize 19 & \\vtop 71 &  \\wedge 35 & \\widehat 35 \cr                      
\+ \\widetilde 35 & \\xi 34 &  \\Xi 34 & \\zeta 34 \cr                          
}                                                                               
                                                                                
\ifwritinganswers                                                               
   \let\next=\relax                                                             
\else                                                                           
   \let\next=\endinput                                                          
   \vfill                                                                       
   \datestamp                                                                   
\fi                                                                             
                                                                                
\next                                                                           
                                                                                
\def\beginliteral{                                                              
\vskip\baselineskip                                                             
\begingroup                                                                     
\obeylines                                                                      
\tt                                                                             
\catcode`\@=0\catcode`\~=12                                                     
\catcode`\$=12\catcode`\&=12\catcode`\~=12\catcode`\#=12                        
\catcode`\_=12\catcode`\=12                                                     
\def\par{\leavevmode\endgraf}                                                   
\catcode`\{=12\catcode`\}=12\catcode`\%=12                                      
\catcode`\\=12}                                                                 
\def\endliteral{\nobreak \vskip 6pt \endgroup}                                  
                                                                                
                                                                                
\section{I get by with a little help}                                           
Many of the exercises can be answered in several ways. If you                   
like your way better than the way given below, by all means use                 
it!                                                                             
\vskip 2\baselineskip                                                           
                                                                                
\parskip=0pt \parindent=0pt                                                     
\raggedright                                                                    
\hbadness=10000                                                                 
                                                                                
\hrule                                                                          
\beginliteral                                                                   
I like \TeX!                                                                    
Once you get the hang of it, \TeX\  is really easy to use.                      
You just have to master the \TeX nical aspects.                                 
@endliteral                                                                     
I like \TeX! Once you get the hang of it, \TeX\  is really easy                 
to use. You just have to master the \TeX nical aspects.                         
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
Does \AE schylus understand \OE dipus?                                          
@endliteral                                                                     
Does \AE schylus understand \OE dipus?                                          
\vskip \baselineskip \hrule                                                     
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
The smallest internal unit of \TeX\  is about 53.63\AA.                         
@endliteral                                                                     
The smallest internal unit of \TeX\  is about 53.63\AA.                         
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
 They took some honey and plenty of money wrapped up in a {\it \$}5 note.       
@endliteral                                                                     
 They took some honey and plenty of money wrapped up in a {\it \$}5 note.       
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
 \'El\`eves, refusez vos le\c cons! Jetez vos cha\~\i nes!                      
@endliteral                                                                     
 \'El\`eves, refusez vos le\c cons! Jetez vos cha\~\i nes!                      
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
 Za\v sto tako polako pijete \v caj?                                            
@endliteral                                                                     
 Za\v sto tako polako pijete \v caj?                                            
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
 Mein Tee ist hei\ss.                                                           
@endliteral                                                                     
 Mein Tee ist hei\ss.                                                           
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
 Peut-\~etre qu'il pr\'ef\`ere le caf\'e glac\'e.                               
@endliteral                                                                     
 Peut-\~etre qu'il pr\'ef\`ere le caf\'e glac\'e.                               
\vskip \baselineskip \hrule                                                     
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\vskip\baselineskip                                                             
\begingroup                                                                     
\obeylines                                                                      
\tt                                                                             
\catcode`\@=0                                                                   
%%%%%\def\par{\leavevmode\endgraf}                                              
\catcode`\\=12                                                                  
{?}`Por qu\'e no bebes vino blanco? {!}`Porque est\'a avinagrado!               
@endliteral                                                                     
?`Por qu\'e no bebes vino blanco? !`Porque est\'a avinagrado!                   
\vskip \baselineskip \hrule                                                     
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
 M\'\i\'\j n idee\"en worden niet be\"\i nvloed.                                
@endliteral                                                                     
 M\'\i\'\j n idee\"en worden niet be\"\i nvloed.                                
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
 Can you take a ferry from \"Oland to \AA land?                                 
@endliteral                                                                     
 Can you take a ferry from \"Oland to \AA land?                                 
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
 T\"urk\c ce konu\c san ye\u genler nasillar?                                   
@endliteral                                                                     
 T\"urk\c ce konu\c san ye\u genler nasillar?                                   
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
 I entered the room and---horrors---I saw both my father-in-law and my          
mother-in-law.                                                                  
@endliteral                                                                     
 I entered the room and---horrors---I saw both my father-in-law and my          
mother-in-law.                                                                  
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
The winter of 1484--1485 was one of discontent.                                 
@endliteral                                                                     
The winter of 1484--1485 was one of discontent.                                 
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
His ``thoughtfulness'' was impressive.                                          
@endliteral                                                                     
His ``thoughtfulness'' was impressive.                                          
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
 Frank wondered, ``Is this a girl that can't say `No!'?''                       
@endliteral                                                                     
 Frank wondered, ``Is this a girl that can't say `No!'?''                       
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
 He thought, ``\dots and this goes on forever, perhaps to the last recorded     
syllable.''                                                                     
@endliteral                                                                     
 He thought, ``\dots and this goes on forever, perhaps to the last recorded     
syllable.''                                                                     
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
 Have you seen Ms.~Jones?                                                       
@endliteral                                                                     
 Have you seen Ms.~Jones?                                                       
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
Prof.~Smith and Dr.~Gold flew from                                              
Halifax N.~S. to Montr\'eal, P.~Q. via Moncton, N.~B.                           
@endliteral                                                                     
Prof.~Smith and Dr.~Gold flew from                                              
Halifax N.~S. to Montr\'eal, P.~Q. via Moncton, N.~B.                           
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
\line{left end \hfil left tackle \hfil left guard \hfil \centre{} \hfil         
right guard \hfil right tackle \hfil right end}                                 
@endliteral                                                                     
\line{left end \hfil left tackle \hfil left guard \hfil \centre{} \hfil         
right guard \hfil right tackle \hfil right end}                                 
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
\line{left \hfil \hfil right-\centre{} \hfil right}                             
@endliteral                                                                     
\line{left \hfil \hfil right-\centre{} \hfil right}                             
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
\line{\hskip 1 in ONE \hfil TWO \hfil THREE}                                    
@endliteral                                                                     
\line{\hskip 1 in ONE \hfil TWO \hfil THREE}                                    
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
i{f}f if{}f if{f}                                                               
@endliteral                                                                     
i{f}f if{}f if{f}                                                               
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
$C(n,r) = n!/(r!\,(n-r)!)$                                                      
@endliteral                                                                     
$C(n,r) = n!/(r!\,(n-r)!)$                                                      
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
$a+b=c-d=xy=w/z$                                                                
$$a+b=c-d=xy=w/z$$                                                              
@endliteral                                                                     
$a+b=c-d=xy=w/z$                                                                
$$a+b=c-d=xy=w/z$$                                                              
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
$(fg)' = f'g + fg'$                                                             
$$(fg)' = f'g + fg'$$                                                           
@endliteral                                                                     
$(fg)' = f'g + fg'$                                                             
$$(fg)' = f'g + fg'$$                                                           
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
$\alpha\beta=\gamma+\delta$                                                     
$$\alpha\beta=\gamma+\delta$$                                                   
@endliteral                                                                     
$\alpha\beta=\gamma+\delta$                                                     
$$\alpha\beta=\gamma+\delta$$                                                   
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
$\Gamma(n) = (n-1)!$                                                            
$$\Gamma(n) = (n-1)!$$                                                          
@endliteral                                                                     
$\Gamma(n) = (n-1)!$                                                            
$$\Gamma(n) = (n-1)!$$                                                          
\vskip \baselineskip \hrule                                                     
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
$x\wedge (y\vee z) = (x\wedge y) \vee (x\wedge z)$                              
@endliteral                                                                     
$x\wedge (y\vee z) = (x\wedge y) \vee (x\wedge z)$                              
\vskip \baselineskip \hrule                                                     
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
$2+4+6+\cdots +2n = n(n+1)$                                                     
@endliteral                                                                     
$2+4+6+\cdots +2n = n(n+1)$                                                     
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
$\vec x\cdot \vec y  = 0$ if and only if $\vec x \perp \vec y$.                 
@endliteral                                                                     
$\vec x\cdot \vec y  = 0$ if and only if $\vec x \perp \vec y$.                 
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
$\vec x\cdot \vec y \not= 0$ if and only if $\vec x \not\perp \vec y$.          
@endliteral                                                                     
$\vec x\cdot \vec y \not= 0$ if and only if $\vec x \not\perp \vec y$.          
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
$(\forall x\in \Re)(\exists y\in\Re)$ $y>x$.                                    
@endliteral                                                                     
$(\forall x\in \Re)(\exists y\in\Re)$ $y>x$.                                    
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
${a+b\over c}\quad {a\over b+c}\quad {1\over a+b+c} \not= {1\over a}+           
{1\over b}+{1\over c}$.                                                         
@endliteral                                                                     
${a+b\over c}\quad {a\over b+c}\quad {1\over a+b+c} \not= {1\over a}+           
{1\over b}+{1\over c}$.                                                         
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
What are the points where ${\partial \over \partial x} f(x,y) = {\partial \over 
\partial y} f(x,y) = 0$?                                                        
@endliteral                                                                     
What are the points where ${\partial \over \partial x} f(x,y) = {\partial \over 
\partial y} f(x,y) = 0$?                                                        
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
$e~x \quad e~{-x} \quad e~{i\pi}+1=0 \quad x_0 \quad x_0~2                      
\quad {x_0}~2 \quad 2~{x~x}$.                                                   
@endliteral                                                                     
$e~x \quad e~{-x} \quad e~{i\pi}+1=0 \quad x_0 \quad x_0~2                      
\quad {x_0}~2 \quad 2~{x~x}$.                                                   
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
$\nabla~2 f(x,y) = {\partial~2 f \over\partial x~2}+ {\partial~2 f \over        
\partial y~2}$.                                                                 
@endliteral                                                                     
$\nabla~2 f(x,y) = {\partial~2 f \over\partial x~2}+ {\partial~2 f \over        
\partial y~2}$.                                                                 
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
$\lim_{x\to 0} (1+x)~{1\over x}=e$.                                             
@endliteral                                                                     
$\lim_{x\to 0} (1+x)~{1\over x}=e$.                                             
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
The cardinality of $(-\infty, \infty)$ is $\aleph_1$.                           
@endliteral                                                                     
The cardinality of $(-\infty, \infty)$ is $\aleph_1$.                           
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
$\lim_{x\to {0~+}} x~x = 1$.                                                    
@endliteral                                                                     
$\lim_{x\to {0~+}} x~x = 1$.                                                    
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
$\int_0~1 3x~2\,dx = 1$.                                                        
@endliteral                                                                     
$\int_0~1 3x~2\,dx = 1$.                                                        
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
$\sqrt2 \quad \sqrt {x+y\over x-y} \quad \root 3 \of {10}$ \quad $e~{\sqrt x}$. 
@endliteral                                                                     
$\sqrt2 \quad \sqrt {x+y\over x-y} \quad \root 3 \of {10}$ \quad $e~{\sqrt x}$. 
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
$\!x\! = \sqrt{x\cdot x}$.                                                      
@endliteral                                                                     
$\!x\! = \sqrt{x\cdot x}$.                                                      
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
$\phi(t) = {1 \over \sqrt{2\pi}} \int_0~t e~{-x~2/2}\,dx$.                      
@endliteral                                                                     
$\phi(t) = {1 \over \sqrt{2\pi}} \int_0~t e~{-x~2/2}\,dx$.                      
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
$\underline x \quad \overline y \quad \underline{\overline{x+y}}$.              
@endliteral                                                                     
$\underline x \quad \overline y \quad \underline{\overline{x+y}}$.              
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
$\bigl \lceil \lfloor x \rfloor \bigr \rceil \leq \bigl \lfloor \lceil x \rceil 
\bigr \rfloor$.                                                                 
@endliteral                                                                     
$\bigl \lceil \lfloor x \rfloor \bigr \rceil \leq \bigl \lfloor \lceil x \rceil 
\bigr \rfloor$.                                                                 
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
$\sin(2\theta) = 2\sin\theta\cos\theta                                          
\quad \cos(2\theta) = 2\cos~2\theta - 1  $.                                     
@endliteral                                                                     
$\sin(2\theta) = 2\sin\theta\cos\theta                                          
\quad \cos(2\theta) = 2\cos~2\theta - 1  $.                                     
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
$$\int \csc~2x\, dx = -\cot x+ C                                                
\qquad \lim_{\alpha\to 0} {\sin\alpha \over \alpha} = 1                         
\qquad \lim_{\alpha\to \infty} {\sin\alpha \over \alpha} = 0.$$                 
@endliteral                                                                     
$$\int \csc~2x\, dx = -\cot x+ C                                                
\qquad \lim_{\alpha\to 0} {\sin\alpha \over \alpha} = 1                         
\qquad \lim_{\alpha\to \infty} {\sin\alpha \over \alpha} = 0.$$                 
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
$$\tan(2\theta) = {2\tan\theta \over 1-\tan~2\theta}.$$                         
@endliteral                                                                     
$$\tan(2\theta) = {2\tan\theta \over 1-\tan~2\theta}.$$                         
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
\proclaim Theorem (Euclid). There exist an infinite number of primes.           
@endliteral                                                                     
\proclaim Theorem (Euclid). There exist an infinite number of primes.           
                                                                                
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
\proclaim Proposition 1.                                                        
$\root n \of {\prod_{i=1}~n X_i} \leq                                           
{1 \over n} \sum_{i=1}~n X_i$ with equality if and only if $X_1=\cdots=X_n$.    
@endliteral                                                                     
\proclaim Proposition 1.                                                        
$\root n \of {\prod_{i=1}~n X_i} \leq                                           
{1 \over n} \sum_{i=1}~n X_i$ with equality if and only if $X_1=\cdots=X_n$.    
                                                                                
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
$$ I_4 = \pmatrix{                                                              
1 &0 &0 &0 \cr                                                                  
0 &1 &0 &0 \cr                                                                  
0 &0 &1 &0 \cr                                                                  
0 &0 &0 &1 \cr}$$                                                               
@endliteral                                                                     
$$ I_4 = \pmatrix{                                                              
1 &0 &0 &0 \cr                                                                  
0 &1 &0 &0 \cr                                                                  
0 &0 &1 &0 \cr                                                                  
0 &0 &0 &1 \cr}$$                                                               
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
$$ !x! = \left\{ \matrix{                                                       
x & x \ge 0 \cr                                                                 
-x & x \le 0 \cr} \right.$$                                                     
@endliteral                                                                     
$$ !x! = \left\{ \matrix{                                                       
x & x \ge 0 \cr                                                                 
-x & x \le 0 \cr} \right.$$                                                     
\vskip \baselineskip \hrule                                                     
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
\settabs \+ \hskip 2 in & \hskip .75in & \hskip 1cm& \cr                        
\+ &Plums &\hfill\$1&.22 \cr                                                    
\+ &Coffee &\hfill1&.78 \cr                                                     
\+ &Granola &\hfill1&.98 \cr                                                    
\+ &Mushrooms & &.63 \cr                                                        
\+ &{Kiwi fruit} & &.39 \cr                                                     
\+ &{Orange juice} &\hfill1&.09 \cr                                             
\+ &Tuna &\hfill1&.29 \cr                                                       
\+ &Zucchini & &.64 \cr                                                         
\+ &Grapes &\hfill1&.69 \cr                                                     
\+ &{Smoked beef} & &.75 \cr                                                    
\+ &Broccoli &\hfill\underbar{\ \ 1}&\underbar{.09} \cr                         
\+ &Total &\hfill \$12&.55 \cr                                                  
@endliteral                                                                     
\settabs \+ \hskip 2 in & \hskip .75in & \hskip 1cm& \cr                        
\+ &Plums &\hfill\$1&.22 \cr                                                    
\+ &Coffee &\hfill1&.78 \cr                                                     
\+ &Granola &\hfill1&.98 \cr                                                    
\+ &Mushrooms & &.63 \cr                                                        
\+ &{Kiwi fruit} & &.39 \cr                                                     
\+ &{Orange juice} &\hfill1&.09 \cr                                             
\+ &Tuna &\hfill1&.29 \cr                                                       
\+ &Zucchini & &.64 \cr                                                         
\+ &Grapes &\hfill1&.69 \cr                                                     
\+ &{Smoked beef} & &.75 \cr                                                    
\+ &Broccoli &\hfill\underbar{\ \ 1}&\underbar{.09} \cr                         
\+ &Total &\hfill \$12&.55 \cr                                                  
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
\leftline{Getting Started \dotfill 1}                                           
\leftline{All Characters Great and Small \dotfill 9}                            
@endliteral                                                                     
\leftline{Getting Started \dotfill 1}                                           
\leftline{All Characters Great and Small \dotfill 9}                            
\vskip \baselineskip \hrule                                                     
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
\settabs \+ \hskip 1cm&\hskip 1 cm&\hskip 1 cm& \cr                             
\moveright 2 in                                                                 
\vbox{                                                                          
\hrule width 3 cm                                                               
\+  \vrule height 1 cm & \vrule height 1 cm & \vrule height 1 cm                
  & \vrule height 1 cm \cr                                                      
\hrule width 3 cm                                                               
\+  \vrule height 1 cm & \vrule height 1 cm & \vrule height 1 cm                
  & \vrule height 1 cm \cr                                                      
\hrule width 3 cm                                                               
\+  \vrule height 1 cm & \vrule height 1 cm & \vrule height 1 cm                
  & \vrule height 1 cm \cr                                                      
\hrule width 3 cm                                                               
}                                                                               
@endliteral                                                                     
\settabs \+ \hskip 1cm&\hskip 1 cm&\hskip 1 cm& \cr                             
\moveright 2 in                                                                 
\vbox{                                                                          
\hrule width 3 cm                                                               
\+  \vrule height 1 cm & \vrule height 1 cm & \vrule height 1 cm                
  & \vrule height 1 cm \cr                                                      
\hrule width 3 cm                                                               
\+  \vrule height 1 cm & \vrule height 1 cm & \vrule height 1 cm                
  & \vrule height 1 cm \cr                                                      
\hrule width 3 cm                                                               
\+  \vrule height 1 cm & \vrule height 1 cm & \vrule height 1 cm                
  & \vrule height 1 cm \cr                                                      
\hrule width 3 cm                                                               
}                                                                               
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\goodbreak \vskip 2pt \hrule                                                    
\beginliteral                                                                   
\def\boxtext#1{%                                                                
\vbox{%                                                                         
      \hrule                                                                    
      \hbox{\strut \vrule{} #1 \vrule}%                                         
      \hrule                                                                    
     }%                                                                         
}                                                                               
\moveright 2 in \vbox{\offinterlineskip                                         
\hbox{\boxtext 6\boxtext 1\boxtext 8}                                           
\hbox{\boxtext 7\boxtext 5\boxtext 3}                                           
\hbox{\boxtext 2\boxtext 9\boxtext 4}                                           
}                                                                               
@endliteral                                                                     
\def\boxtext#1{%                                                                
\vbox{%                                                                         
      \hrule                                                                    
      \hbox{\strut \vrule{} #1 \vrule}%                                         
      \hrule                                                                    
     }%                                                                         
}                                                                               
\moveright 2 in \vbox{\offinterlineskip                                         
\hbox{\boxtext 6\boxtext 1\boxtext 8}                                           
\hbox{\boxtext 7\boxtext 5\boxtext 3}                                           
\hbox{\boxtext 2\boxtext 9\boxtext 4}                                           
}                                                                               
\vskip \baselineskip \hrule                                                     
                                                                                
                                                                                
\vfill                                                                          
\datestamp                                                                      
                                                                                
\bye                                                                            
                                                                                
                                                                                
