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+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%                                                                    %
+% Project Gutenberg's Introduction to Infinitesimal Analysis, by     %
+% Oswald Veblen.                                                     %
+%                                                                    %
+%                                                                    %
+% This eBook is for the use of anyone anywhere at no cost and with   %
+% almost no restrictions whatsoever. You may copy it, give it away   %
+% or re-use it under the terms of the Project Gutenberg License      %
+% included with this eBook or online at www.gutenberg.net            %
+%                                                                    %
+%                                                                    %
+% Producer's Comments                                                %
+%                                                                    %
+% Since the illustrations have been provided in pdf format, it is    %
+% easiest to compile using pdflatex. However, running latex then     %
+% dvips will also work if the graphics are converted to eps,         %
+% provided that the package eepic is substituted for eepicemu, and   %
+% pdftex driver option to graphicx and  hyperref is changed to       %
+% dvips.                                                             %
+%                                                                    %
+%                                                                    %
+% Things to Check:                                                   %
+%                                                                    %
+% Spellcheck: OK                                                     %
+% LaCheck: OK, false positives:                                      %
+%     Book used \ldots never \cdots;                                 %
+%     ! for factorial,                                               %
+%     decimal points                                                 %
+%     deliberate space after {                                       %
+% Lprep/gutcheck: OK                                                 %
+% PDF pages, excl. Gutenberg boilerplate: 214                        %
+% PDF pages, incl. Gutenberg boilerplate: 225                        %
+% ToC page numbers: OK                                               %
+% Index: OK                                                          %
+% Images: 8 PDF (in /images), 14 LaTeX (embedded).                   %
+% Fonts:                                                             %
+% Longtable (at back): aligned                                       %
+%                                                                    %
+%                                                                    %
+% Compile history:                                                   %
+%                                                                    %
+% 20th June 2006: LW compiled with pdflatex (tetex under MacOSX)     %
+%                                                                    %
+% pdflatex infinitesimal                                             %
+% makeindex infinitesimal                                            %
+% pdflatex infinitesimal                                             %
+% pdflatex infinitesimal                                             %
+%                                                                    %
+% 2nd July  2006: JT compiled with pdflatex (MiKTeX / WinXP)         %
+%                                                                    %
+% pdflatex 18741-t                                                   %
+% makeindex 18741-t                                                  %
+% pdflatex 18741-t                                                   %
+% pdflatex 18741-t                                                   %
+%                                                                    %
+% Front- and back-matter give pdfTeX warnings - this is a known      %
+% issue with documents that restart numbering, and is safe to        %
+% ignore. Also warns of math in section titles.                      %
+% 3 Overfulls, 2 Underfulls.                                         %
+%                                                                    %
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PACKAGES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\documentclass[a4paper,12pt]{book}[2004/02/16]
+
+\usepackage{amsmath, amsthm}% Required
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+\usepackage{longtable}% Required for Wiley catalogue at back.
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+%\newcommand{\rinterval}[2]{(#1,#2]}
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+
+%%%%%%%% Sections formatting
+\renewcommand{\thesection}{\S~\arabic{section}}
+\renewcommand{\sectionmark}[1]{}
+\renewcommand{\chaptermark}[1]{\markboth{INFINITESIMAL ANALYSIS.}{#1}}
+
+%%%%%%%% Theorems formatting (book also used parindent but it's ugly!)
+\newtheoremstyle{itheorem}{}{}{\itshape}{}{\bfseries}{.}{ }{#1\if!#3!\else\ \fi\thmnote{#3}}
+\newtheoremstyle{icorollary}{}{}{}{}{\itshape}{.---}{0pt}{#1}
+\newtheoremstyle{numcorollary}{}{}{}{}{\itshape}{.}{ }{#1\if!#3!\else\ \fi\thmnote{#3}}
+\newtheoremstyle{idefinition}{}{}{}{}{\bfseries}{.---}{0pt}{}
+\newtheoremstyle{ilemma}{}{}{\itshape}{}{\bfseries}{.---}{0pt}{#1\if!#3!\else\ \fi\thmnote{#3}}
+\newtheoremstyle{iother}{}{}{\itshape}{}{\bfseries}{.---}{0pt}{\thmnote{#3}}
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+\theoremstyle{itheorem}
+\newtheorem{theorem}{Theorem}
+\theoremstyle{iother}
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+\theoremstyle{icorollary}
+\newtheorem{corollary}{Corollary}
+\theoremstyle{numcorollary}
+\newtheorem{ncorollary}{Corollary}
+\theoremstyle{idefinition}
+\newtheorem*{definition}{Definition}
+\newtheorem*{definitions}{Definitions}
+\newtheorem*{defnorder}{Definition of Order}
+
+%%%%%%%% Proof environment
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+\renewcommand{\qedsymbol}{}% For the default square box at the end
+% of proofs, remove this line.  For any other text or symbol at
+% end of proofs, insert it in the second curly bracket.
+
+%%%%%%%% Miscellaneous
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+% for fractions, but it makes text ugly!
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+\newcommand{\plainindexname}{\gdef\indexname{INDEX.}}
+
+
+\makeindex
+%%%%%%%%%%%%%%%%%%%%%%%% START OF DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{document}
+\thispagestyle{empty}
+\small
+\begin{verbatim}
+Project Gutenberg's Introduction to Infinitesimal Analysis
+by Oswald Veblen and N. J. Lennes
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.net
+
+
+Title: Introduction to Infinitesimal Analysis
+       Functions of one real variable
+
+Author: Oswald Veblen and N. J. Lennes
+
+Release Date: July 2, 2006 [EBook #18741]
+
+Language: English
+
+Character set encoding: TeX
+
+*** START OF THIS PROJECT GUTENBERG EBOOK INFINITESIMAL ANALYSIS  ***
+
+
+
+
+Produced by K.F. Greiner, Joshua Hutchinson, Laura Wisewell, 
+Owen Whitby and the Online Distributed Proofreading Team at 
+http://www.pgdp.net (This file was produced from images 
+generously made available by Cornell University Digital 
+Collections.)
+
+
+
+\end{verbatim}
+\normalsize
+\frontmatter
+\begin{titlepage}
+{\setlength{\fboxsep}{10pt}
+\framebox{%
+\begin{minipage}{0.9\textwidth}
+\setlength{\fboxsep}{1pt}\sf
+\label{transnotes}\pdfbookmark[0]{Transcriber's Notes.}{transnotes}
+\textbf{Transcriber's Notes.}\medskip\par
+A large number of printer errors have been corrected.  These are 
+shaded \correction{like this}{}, and details can be found in the 
+source code in the syntax \texttt{\textbackslash correction\{corrected\}\{original\}}.  
+In addition, the formatting of a few lemmas, corollaries etc.\ has 
+been made consistent with the others.
+\medskip\par
+The unusual inequality sign $\qqge$ used a few times in the book 
+in addition to $\geqq$ has been preserved, although it may reflect 
+the printing rather than the author's intention.  The notation 
+$\interval{a}{b}$ for intervals is not in common use today, and 
+the reader able to run \LaTeX{} will find it easy to redefine this 
+macro to give a modern equivalent.  Similarly, the original did not 
+mark the ends of proofs in any way and so nor does this version, 
+but the reader who wishes can easily redefine \texttt{\textbackslash qedsymbol} 
+in the source.
+\end{minipage}
+}}
+\end{titlepage}
+%-----File: 001.png---
+%[Blank page]
+%-----File: 002.png---
+%[Blank Page]
+%-----File: 003.png---
+% [Library stamp]
+%
+% Cornell University Library
+%
+%
+% BOUGHT WITH THE INCOME
+% FROM THE
+% SAGE ENDOWMENT FUND
+% THE GIFT OF
+% Henry W. Sage
+% 1891
+%
+%-----File: 004.png---
+%[Blank Page]
+%-----File: 005.png---Folio i-------
+\title{\label{titlepage}\pdfbookmark[0]{Title Page.}{titlepage}%
+INTRODUCTION\\
+{\small TO}\\
+{\Huge INFINITESIMAL ANALYSIS\\[1ex]}
+{\Large FUNCTIONS OF ONE REAL VARIABLE}
+}
+
+\author{{\small BY}\\
+OSWALD VEBLEN\\
+\textit{Preceptor in Mathematics, Princeton University}\\
+{\small \textsc{And}}\\
+N.~J. LENNES\\
+\textit{Instructor in Mathematics in the Wendell Phillips High School, Chicago}
+}
+\date{%
+\textit{FIRST EDITION}\\
+{\small FIRST THOUSAND}\\
+\vspace{0.2\textheight}
+NEW YORK\\
+JOHN WILEY \& SONS\\
+\textsc{London: CHAPMAN \& HALL, Limited}\\
+1907
+}
+\maketitle
+%-----File: 006.png---Folio ii-------
+\begin{center}
+\vspace*{0.4\textheight}
+Copyright, 1907\\
+\textsc{by\\
+OSWALD VEBLEN and N.~J. LENNES\\}
+\vfill
+ROBERT DRUMMOND, PRINTER, NEW YORK
+\end{center}
+\newpage
+%-----File: 007.png---Folio iii------
+
+
+
+
+\chapter*{PREFACE}
+
+
+A course dealing with the fundamental theorems of infinitesimal
+calculus in a rigorous manner is now recognized as an
+essential part of the training of a mathematician. It appears
+in the curriculum of nearly every university, and is taken by
+students as ``Advanced Calculus'' in their last collegiate year,
+or as part of ``Theory of Functions'' in the first year of graduate
+work. This little volume is designed as a convenient reference
+book for such courses; the examples which may be considered
+necessary being supplied from other sources. The book may
+also be used as a basis for a rather short theoretical course on
+real functions, such as is now given from time to time in some
+of our universities.
+
+The general aim has been to obtain rigor of logic with a
+minimum of elaborate machinery. It is hoped that the systematic
+use of the Heine-Borel theorem has helped materially
+toward this end, since by means of this theorem it is possible
+to avoid almost entirely the sequential division or ``pinching''
+process so common in discussions of this kind. The definition
+of a limit by means of the notion ``value approached'' has
+simplified the proofs of theorems, such as those giving necessary
+and sufficient conditions for the existence of limits, and in
+general has largely decreased the number of $\varepsilon$'s and
+$\delta$'s. The theory of limits is developed for multiple-valued
+functions, which gives certain advantages in the treatment of the
+definite integral.
+
+In each chapter the more abstract subjects and those which
+can be omitted on a first reading are placed in the concluding
+%-----File: 008.png---Folio iv-------
+sections. The last chapter of the book is more advanced in
+character than the other chapters and is intended as an introduction
+to the study of a special subject. The index at the
+end of the book contains references to the pages where technical
+terms are first defined.
+
+When this work was undertaken there was no convenient
+source in English containing a rigorous and systematic treatment
+of the body of theorems usually included in even an elementary
+course on real functions, and it was necessary to refer
+to the French and German treatises. Since then one treatise,
+at least, has appeared in English on the Theory of Functions
+of Real Variables. Nevertheless it is hoped that the present
+volume, on account of its conciseness, will supply a real want.
+
+The authors are much indebted to Professor E.~H. Moore
+of the University of Chicago for many helpful criticisms and
+suggestions; to Mr.~E.~B. Morrow of Princeton University for
+reading the manuscript and helping prepare the cuts; and to
+Professor G.~A. Bliss of Princeton, who has suggested several
+desirable changes while reading the proof-sheets.
+%-----File: 009.png---Folio v--------
+\tableofcontents
+
+%-----File: 010.png---Folio vi-------
+%-----File: 011.png---Folio vii------
+%-----File: 012.png---
+%[Blank Page]
+%-----File: 013.png---Folio 1--------
+\mainmatter
+\chapter{THE SYSTEM OF REAL NUMBERS.}\hypertarget{chapI}{}%[I]
+\section{Rational and Irrational Numbers.}\hypertarget{chIsec1}{}%[1]
+\index{Number}
+The real number system may be classified as follows:
+\begin{enumerate}
+
+\item[(1)]\hypertarget{item1p1}{} All integral numbers, both positive and negative, including
+zero.
+
+\item[(2)]\hypertarget{item2p1}{} All numbers $\frac mn$, where $m$ and $n$ are integers
+($n\neq 0$).
+
+\item[(3)]\hypertarget{item3p1}{} Numbers not included in either of the above classes,
+such as $\sqrt{2}$ and $\pi$.\footnote{%
+    It is clear that there is no number $\frac mn$ such that
+    $\frac{m^2}{n^2}=2$, for if $\frac{m^2}{n^2}=2$, then
+    $m^2=2n^2$, where $m^2$ and $2n^2$ are integral numbers, and
+    $2n^2$ is the square of the integral number $m$. Since in the
+    square of an integral number every prime factor occurs an even
+    number of times, the factor 2 must occur an even number of
+    times both in $n^2$ and $2n^2$, which is impossible because of
+    the theorem that an integral number has only one set of prime
+    factors.}
+\end{enumerate}
+
+Numbers of classes \hyperlink{item1p1}{(1)} and \hyperlink{item2p1}{(2)} are called rational or commensurable
+numbers, while the numbers of class~\hyperlink{item3p1}{(3)} are called \index{Rational!numbers}\index{Number!irrational}\index{Irrational!number}irrational or
+incommensurable numbers.
+
+As an illustration of an irrational number consider the
+square root of $2$. One ordinarily says that $\sqrt{2}$ is $1.4+$, or
+%-----File: 014.png---Folio 2--------
+$1.41+$, or $1.414+$, etc. The exact meaning of these statements is
+expressed by the following inequalities:\footnote{%
+  $a<b$ signifies
+  that $a$ is less than $b$. $a>b$ signifies that $a$ is greater than
+  $b$.}
+\begin{align*}
+&(1.4)^2 < 2 < (1.5)^2, \\
+&(1.41)^2 < 2 < (1.42)^2, \\
+&(1.414)^2 < 2 < (1.415)^2,\\
+&\qquad\mbox{etc.}
+\end{align*}
+Moreover, by the foot-note above no terminating decimal is equal to
+the square root of 2. Hence Horner's Method, or the usual algorithm
+for extracting the square root, leads to an infinite sequence of
+rational numbers which may be denoted by $a_1, a_2, a_3, \ldots,
+a_n,\ldots$ (where $a_1 = 1.4$, $a_2 = 1.41$, etc.), and which has the
+property that for every positive integral value of $n$
+\begin{align*}
+  a_n &\le a_{n+1},
+  &a_n^2 &< 2 < \left(a_n + \frac{1}{10^n}\right)^2.
+\end{align*}
+
+Suppose, now, that there is a \emph{least} number $a$ greater than
+every $a_n$. We easily see that if the ordinary laws of arithmetic as
+to equality and inequality and addition, subtraction, and
+multiplication hold for $a$ and $a^2$, then $a^2$ is the rational
+number 2. For if $a^2<2$, let $2-a^2 = \varepsilon$, whence $2=a^2 +
+\varepsilon$. If $n$ were so taken that $\frac{1}{10^n} <
+\frac\varepsilon5$, we should have from the last inequality\footnote{%
+    This involves the assumption that for every number,
+    $\varepsilon$, however small there is a positive
+    \correction{integer}{integrer} $n$ such that $\frac{1}{10^n} <
+    \frac\varepsilon5$. This is of course obvious when
+    $\varepsilon$ is a rational number. If $\varepsilon$ is an
+    irrational number, however, the statement will have a definite
+    meaning only after the irrational number has been fully
+    defined.}
+\[
+  2< \left(a_n + \frac{1}{10^n}\right)^2 = a_n^2 +
+  2a_n\cdot\frac{1}{10^n}+
+  \left(\frac{1}{10^n}\right)^2 < a_n^2 + 4\frac\varepsilon5 +
+  \frac\varepsilon5 < a^2 + \varepsilon,
+\]
+so that we should have both $2=a^2 + \varepsilon$ and
+$2<a^2+\varepsilon$. On the
+%-----File: 015.png---Folio 3--------
+other hand, if $a^2 > 2$, let $a^2-2 = \varepsilon'$ or $2 +
+\varepsilon' = a^2$. Taking $n$ such that $\frac{1}{10^n} <
+\frac\varepsilon5$, we should have
+\[
+  \left(a_n +\frac{1}{10^n} \right)^2 < (a_n^2) +\varepsilon'
+  < 2+\varepsilon' < a;
+\]
+and since $a_n + \frac{1}{10^n}$ is greater than $a_k$ for all values
+of $k$, this would contradict the hypothesis that $a$ is the
+\emph{least} number greater than every number of the sequence
+$a_1,a_2,a_3,\ldots$ We also see without difficulty that $a$ is the
+only number such that $a^2 =2$.
+
+\section{Axiom of Continuity.}\hypertarget{chIsec2}{}%[2]
+
+The essential step in passing from ordinary rational numbers
+to the number corresponding to the symbol $\sqrt{2}$ is thus
+made to depend upon an assumption of the existence of a
+number $a$ bearing the unique relation just described to the
+sequence $a_1,a_2$,\correction{$a_3$}{$a_n$},$\ldots$
+In order to state this hypothesis in
+general form we introduce the following definitions:
+
+\begin{definition}\index{Numbers!sets of}\index{Sets of numbers}
+The notation $[x]$ denotes a \textit{set},\footnote{%
+    Synonyms of set are class\index{Class}, aggregate, collection, assemblage,
+    etc.}
+any element of which is denoted by $x$ alone, with or
+without an index or subscript.
+
+A \index{Upper bound!of a set of numbers}\index{Lower bound!of a set of numbers}\index{Bounds!upper and lower}set of numbers $[x]$ is said to have an \emph{upper bound},
+$M$, if there exists a number $M$ such that there is no number
+of the set greater than $M$. This may be denoted by $M \geqq [x]$.
+
+A set of numbers $[x]$ is said to have a \emph{lower bound}, $m$, if
+there exists a number $m$ such that no number of the set is
+less than $m$. This we denote by $m \leqq [x]$.
+\end{definition}
+
+Following are examples of sets of numbers:
+\begin{enumerate}
+\item[(1)] $1,2,3$.
+\item[(2)] $2,4,6,\ldots,2k,\ldots$
+\item[(3)] $1/2,1/{2^2},1/{2^3},\ldots,1/{2^n},\ldots$
+\item[(4)] All rational numbers less than $1$.
+\item[(5)]\hypertarget{item5p3}{} All rational numbers whose squares are less than $2$.
+\end{enumerate}
+%-----File: 016.png---Folio 4--------
+
+Of the first set $1$, or any smaller number, is a lower bound and $3$,
+or any larger number, is an upper bound. The second set has no upper
+bound, but $2$, or any smaller number, is a lower bound. The number
+$3$ is the least upper bound of the first set, that is, the smallest
+number which is an upper bound.  The \index{Least upper bound}least upper and the \index{Greatest lower bound}greatest
+lower bounds of a set of numbers $[x]$ are called by some writers the
+upper and lower limits respectively. We shall denote them by
+$\overline{B}[x]$ and $\underline{B}[x]$ respectively.  By what
+precedes, the set~\hyperlink{item5p3}{(5)} would have no least upper bound unless
+$\sqrt{2}$ were counted as a number.
+
+We now state our hypothesis of continuity in the following
+form:
+\begin{other}[Axiom K]\hypertarget{axiomK}{}\index{Axioms!of continuity}\index{Continuity!axioms of}
+If a set $[r]$ of rational numbers having an upper
+bound has no rational least upper bound, then there exists one and
+only one number $\overline{B}[r]$ such that
+\begin{enumerate}
+\item[(a)] $\overline{B}[r] > r'$, where $r'$ is any number of $[r]$
+or any rational number less than some number of $[r]$.
+\item[(b)] $\overline{B}[r] < r''$, where $r''$ is any rational upper
+bound of $[r]$.%
+\footnote{%
+  This axiom implies that the new (irrational) numbers have relations
+  of order with all the rational numbers, but does not explicitly
+  state relations
+  of order among the irrational numbers themselves. Cf.\ Theorem~\hyperlink{thm2}{2}.}
+\end{enumerate}
+\end{other}
+
+\begin{definition}\index{Continuous!real number system}\index{Real number system}\index{Number!system}\index{Continuum, linear}\index{Linear continuum}
+The number $\overline{B}[r]$ of \hyperlink{axiomK}{axiom~K} is called the
+least upper bound of $[r]$, and as it cannot be a rational number it
+is called an \index{Number!irrational}\index{Irrational!number}\textit{irrational} number. The set of all rational and
+irrational numbers so defined is called the \textit{continuous real
+number system}. It is also called \textit{the linear continuum.} The
+set of all real numbers between any two real numbers is likewise
+called a linear continuum.
+\end{definition}
+\begin{theorem}[1]\hypertarget{thm1}{}
+If two sets of rational numbers $[r]$ and
+$[s]$, having upper bounds, are such that no $r$ is greater than every
+$s$ and no $s$ greater than every $r$, then $\overline{B}[r]$ and
+$\overline{B}[s]$ are the same; that is, in symbols,
+\[
+  \overline{B}[r] = \overline{B}[s].
+\]
+\end{theorem}
+\begin{proof}
+If $\overline{B}[r]$ is rational, it is evident,
+and if $\overline{B}[r]$ is irrational, it is a consequence of \hyperlink{axiomK}{Axiom~K}
+that
+\[
+  \overline{B}[r] > s',
+\]
+%-----File: 017.png---Folio 5--------
+where $s'$ is any rational number not an upper bound of $[s]$.
+Moreover, if $s''$ is rational and greater than every $s$, it is
+greater than every $r$. Hence
+\[
+  \overline{B}[r] < s'',
+\]
+where $s''$ is any rational upper bound of $[s]$. Then, by the
+definition of $\overline{B}[s]$,
+\[
+  \overline{B}[r] = \overline{B}[s],\qedhere
+\]
+\end{proof}
+\begin{definition}
+If a number $x$ (in particular an irrational
+number) is the least upper bound of a set of rational numbers $[r]$,
+then the set $[r]$ is said to \textit{determine} the number $x$.
+\end{definition}
+\begin{ncorollary}[1]\label{cor1p5}\hypertarget{cor1p5}{}
+The irrational numbers $i$ and $i'$ determined by the two sets $[r]$
+and $[r']$ are equal if and only if there is no number in either set
+greater than every number in the other set.
+\end{ncorollary}
+\begin{ncorollary}[2]\label{cor2p5}\hypertarget{cor2p5}{}
+Every irrational number is determined by some set of rational numbers.
+\end{ncorollary}
+\begin{definition}
+If $i$ and $i'$ are two irrational numbers determined
+respectively by sets of rational numbers $[r]$ and $[r']$
+and if some number of $[r]$ is greater than every number of $[r']$,
+then
+\[
+  i>i' \text{ and } i'<i.
+\]
+\end{definition}
+From these definitions and the order relations among the
+rational numbers we prove the following theorem:
+\begin{theorem}[2]\hypertarget{thm2}{}
+If $a$ and $b$ are any two distinct real numbers, then $a<b$ or $b<a$;
+if $a<b$, then not $b<a$; if $a<b$ and $b<c$, then $a<c$.
+\end{theorem}
+\begin{proof}
+Let $a$, $b$, $c$ all be irrational and let $[x]$, $[y]$, $[z]$ be
+sets of rational numbers determining $a$, $b$, $c$. In the two sets
+$[x]$ and $[y]$ there is either a number in one set greater than every
+number of the other or there is not. If there is no number in either
+set greater than every number in the other, then, by Theorem~\hyperlink{thm1}{1},
+$a=b$. If there is a number in $[x]$ greater than every number in
+$[y]$, then no number in $[y]$ is greater than every number in
+$[x]$. Hence the first part of the theorem is
+%-----File: 018.png---Folio 6--------
+proved, that is, either $a=b$ or $a<b$ or $b<a$, and if one of these,
+then neither of the other two. If a number $y_1$ of $[y]$ is greater
+than every number of $[x]$, and a number $z_1$ of $[z]$ is greater
+than every number of $[y]$, then $z_1$ is greater than every number
+of $[x]$. Therefore if $a<b$ and $b<c$, then $a<c$.
+\end{proof}
+
+We leave to the reader the proof in case one or two of the
+numbers $a$, $b$, and $c$ are rational.
+\begin{lemma}
+If $[r]$ is a set of rational numbers determining an
+irrational number, then there is no number $r_1$ of the set $[r]$
+which is greater than every other number of the set.
+\end{lemma}
+This is an immediate consequence of \hyperlink{axiomK}{axiom~K}.
+\begin{theorem}[3]\hypertarget{thm3}{}
+If $a$ and $b$ are any two distinct numbers, then
+there exists a rational number $c$ such that $a<c$ and $c<b$, or $b<c$
+and $c<a$.
+\end{theorem}
+\begin{proof}
+Suppose $a<b$. When $a$ and $b$ are both rational
+$\frac{b-a}{2}$ is a number of the required type. If $a$ is rational
+and $b$ irrational, then the theorem follows from the lemma and
+Corollary~\hyperlink{cor2p5}{2}, page~\pageref{cor2p5}. If $a$ and $b$ are both
+irrational, it follows from Corollary~\hyperlink{cor1p5}{1}, page~\pageref{cor1p5}. If $a$
+is irrational and $b$ rational, then there are rational numbers less
+than $b$ and greater than every number of the set $[x]$ which
+determines $a$, since otherwise $b$ would be the smallest rational
+number which is an upper bound of $[x]$, whereas by definition there
+is no least upper bound of $[x]$ in the set of rational numbers.
+\end{proof}
+\begin{corollary}
+A rational number $r$ is the least upper bound of
+the set of all numbers which are less than $r$, as well as of the
+set of all rational numbers less than $r$.
+\end{corollary}
+\begin{theorem}[4]\hypertarget{thm4}{}
+Every set of numbers $[x]$ which has an upper bound, has a least upper
+bound.
+\end{theorem}
+\begin{proof}
+Let $[r]$ be the set of all rational numbers such that
+no number of the set $[r]$ is greater than every number of the set
+$[x]$. Then $\overline{B}[r]$ is an upper bound of $[x]$, since if
+there were a number $x_1$ of $[x]$ greater than $\overline{B}[r]$,
+then, by Theorem~\hyperlink{thm3}{3}, there would be a rational number less than $x_1$
+and greater than $\overline{B}[r]$, which would be contrary to the
+definition of $[r]$ and $\overline{B}[r]$.
+%-----File: 019.png---Folio 7--------
+Further, $\overline{B}[r]$ is the \textit{least} upper bound of $[x]$,
+since if a number $N$ less than $\overline{B}[r]$ were an upper bound
+of $[x]$, then by Theorem~\hyperlink{thm3}{3} there would be rational numbers greater
+than $N$ and less than $\overline{B}[r]$, which again is contrary to
+the definition of $[r]$.
+\end{proof}
+\begin{theorem}[5]\hypertarget{thm5}{}
+Every set $[x]$ of numbers which has a lower
+bound has a greatest lower bound.
+\end{theorem}
+\begin{proof}
+The proof may be made by considering the least
+upper bound of the set $[y]$ of all numbers, such that every number of
+$[y]$ is less than every number of $[x]$. The details are left to the
+reader.
+\end{proof}
+\begin{theorem}[6]\hypertarget{thm6}{}
+If all numbers are divided into two sets $[x]$ and
+$[y]$ such that $x<y$ for every $x$ and $y$ of $[x]$ and $[y]$, then
+there is a greatest $x$ or a least $y$, but not both.
+\end{theorem}
+\begin{proof}
+The proof is left to the reader.
+\end{proof}
+
+The proofs of the above theorems are very simple, but experience has
+shown that not only the beginner in this kind of reasoning but even
+the expert mathematician is likely to make mistakes. The beginner is
+advised to write out for himself every detail which is omitted from
+the text.
+
+Theorem~\hyperlink{thm4}{4} is a form of the continuity axiom due to Weierstrass, and \hyperlink{thm6}{6}
+is the so-called \index{Dedekind cut}\emph{Dedekind Cut Axiom}. Each of Theorems \hyperlink{thm4}{4}, \hyperlink{thm5}{5}, and
+\hyperlink{thm6}{6} expresses the \emph{continuity} of the real number system.
+
+\section{Addition and Multiplication of Irrationals.}\hypertarget{chIsec3}{}%[3]
+
+It now remains to show how to perform the operations of addition,
+subtraction, multiplication, and division on these numbers. A
+definition of addition of irrational numbers is suggested by the
+following theorem: ``If $a$ and $b$ are rational numbers and $[x]$ is
+the set of all rational numbers less than $a$, and $[y]$ the set of
+all rational numbers less than $b$, then $[x+y]$ is the set of all
+rational numbers less than $a+b$.'' The proof of this theorem is left
+to the reader.
+
+\begin{definition}\index{Sum of irrational numbers}
+If $a$ and $b$ are not both rational and $[x]$ is the
+set of all rationals less than $a$ and $[y]$ the set of all rationals
+less
+%-----File: 020.png---Folio 8--------
+than $b$, then $a+b$ is the least upper bound of $[x+y]$, and is
+called \index{Irrational!numbers!sum of}\emph{the sum} of $a$ and $b$.
+\end{definition}
+
+It is clear that if $b$ is rational, $[x+b]$ is the same set as
+$[x+y]$; for a given $x+b$ is equal to $x'+(b-(x'-x))=x'+y'$, where
+$x'$ is any rational number such that $x<x'<a$; and conversely, any
+$x+y$ is equal to $(x-b+y)+b=x''+b$. It is also clear that $a+b=b+a$,
+since $[x+y]$ is the same set as $[y+x]$. Likewise $(a+b)+c=a+(b+c)$,
+since $[(x+y)+z]$ is the same as $[x+(y+z)]$.  Furthermore, in case
+$b<a$, $c=\overline{B}[x'-y']$, where $a<x'<b$ and $a<y'<b$, is such
+that $b+c=a$, and in case $b<a$, $c=\underline{B}[x'-y']$ is such that
+$b+c=a$; $c$ is denoted by $a-b$ and called the \index{Irrational!numbers!difference of}\index{Difference of irrational numbers}\emph{difference}
+between $a$ and $b$. The \emph{negative} of $a$, or $-a$, is simply
+$0-a$.  We leave the reader to verify that if $a>0$, then $a+b>b$, and
+that if $a<0$, then $a+b<b$ for irrational numbers as well as for
+rationals.
+
+The theorems just proved justify the usual method of adding
+infinite decimals. For example: $\pi$ is the least upper bound
+of decimals like $3.1415$, $3.14159$, etc. Therefore $\pi+2$ is the
+least upper bound of such numbers as $5.1415$, $5.14159$, etc.
+Also $e$ is the least upper bound of $2.7182818$, etc. Therefore
+$\pi+e$ is the least upper bound of $5$, $5.8$, $5.85$, $5.859$, etc.
+
+The definition of multiplication is suggested by the following
+theorem, the proof of which is also left to the reader.
+
+Let $a$ and $b$ be rational numbers not zero and let $[x]$ be the
+set of all rational numbers between $0$ and $a$, and $[y]$ be the set
+of all rationals between $0$ and $b$. Then if
+\[
+\begin{array}{cccccc}
+a>0,\ b>0, &\text{\ \ it\ follows\ that\ }&ab=\overline{B}[xy];\\
+a<0,\ b<0, &``\hspace{2em}``\hspace{2em}``&ab=\overline{B}[xy];\\
+a<0,\ b>0, &``\hspace{2em}``\hspace{2em}``&ab=\underline{B}[xy];\\
+a>0,\ b<0, &``\hspace{2em}``\hspace{2em}``&ab=\underline{B}[xy].
+\end{array}
+\]
+
+\begin{definition}\index{Irrational!numbers!product of}\index{Product of irrational numbers}
+If $a$ and $b$ are not both rational and $[x]$ is the
+set of all rational numbers between $0$ and $a$, and $[y]$ the set of
+all rationals between $0$ and $b$, then if $a>0$, $b>0$, $ab$ means
+$\overline{B}[xy]$; if $a<0$, $b<0$, $ab$ means $\overline{B}[xy]$; if
+$a<0$, $b>0$, $ab$ means $\underline{B}[xy]$; if $a>0$, $b<0$, $ab$
+means $\underline{B}[xy]$. If $a$ or $b$ is zero, then $ab=0$.
+\end{definition}
+%-----File: 021.png---Folio 9--------
+
+It is proved, just as in the case of addition, that $ab=ba$, that
+$a(bc) = (ab)c$, that if $a$ is rational $[ay]$ is the same set as
+$[xy]$, that if $a>0$, $b>0$, $ab>0$. Likewise the \index{Quotient of irrational numbers}\index{Irrational!numbers!quotient of}quotient
+$\frac{a}{b}$ is defined as a number $c$ such that $ac=b$, and it is
+proved that in case $a>0$, $b>0$, then
+$c=\overline{B}\bigl[\frac{x}{y'}\bigr]$, where $[y']$ is the set of
+all rationals greater than $b$. Similarly for the other
+cases. Moreover, the same sort of reasoning as before justifies the
+usual method of multiplying non-terminated decimals.
+
+To complete the rules of operation we have to prove what
+is known as the distributive law, namely, that
+\[
+  a(b+c)=ab+ac.
+\]
+To prove this we consider several cases according as $a$, $b$, and $c$
+are positive or negative. We shall give in detail only the case where
+all the numbers are positive, leaving the other cases to be proved by
+the reader. In the first place we easily see that for positive numbers
+$e$ and $f$, if $[t]$ is the set of all the rationals between 0 and
+$e$, and $[T]$ the set of all rationals less than $e$, while $[u]$ and
+$[U]$ are the corresponding sets for $f$, then
+\[
+  e+f = \overline{B}[T+U]=\overline{B}[t+u].
+\]
+Hence if $[x]$ is the set of all rationals between 0 and $a$, $[y]$
+between 0 and $b$, $[z]$ between 0 and $c$,
+\[
+  b + c = \overline{B}[y+z] \quad \text{and hence} \quad
+  a(b+c) = \overline{B}[x(y + z)].
+\]
+On the other hand $ab=\overline{B}[xy]$, $ac=\overline{B}[xz]$, and
+therefore $ab+ac= \overline{B}[(xy+xz)]$. But since the distributive
+law is true for rationals, $x(y+z)=xy+xz$. Hence
+$\overline{B}[x(y+z)]=\overline{B}[(xy+xz)]$ and hence
+\[
+  a(b+c)=ab+ac.
+\]
+
+We have now proved that the system of rational and irrational numbers
+is not only continuous, but also is such that we may perform with
+these numbers all the operations of arithmetic.  We have indicated the
+method, and the reader may
+%-----File: 022.png---Folio 10-------
+detail that every rational number may be represented by a terminated
+decimal,
+\[
+  a_k 10^k+a_{k-1}10^{k-1}+\ldots+a_0+\frac{a_{-1}}{10}
+  +\ldots+\frac{a_{-n}}{10^n}
+  = a_k a_{k-1}\ldots a_0 a_{-1}a_{-2}\ldots a_{-n},
+\]
+or by a circulating decimal,
+\[
+  a_k a_{k-1}\ldots a_0 a_{-1}a_{-2}\ldots
+  a_{-i}\ldots a_{-j}a_{-i}\ldots a_{-j}\ldots,
+\]
+where $i$ and $j$ are any positive integers such that $i<j$; whereas
+every irrational number may be represented by a non-repeating infinite
+decimal,
+\[
+  a_k a_{k-1}\ldots a_0 a_{-1}a_{-2}\ldots a_{-n}\ldots
+\]
+The operations of raising to a power or extracting a root on
+irrational numbers will be considered in a later chapter (see
+page~\pageref{s4p53}). An example of elementary reasoning with the
+symbol $\overline{B}[x]$ is to be found on pages \pageref{t7p17} and
+\pageref{endpf18}. For the present we need only that $x^n$, where $n$
+is an integer, means the number obtained by multiplying $x$ by itself
+$n$ times.
+
+It should be observed that the essential parts of the definitions and
+arguments of this section are based on the assumption of continuity
+which was made at the outset. A clear understanding of the irrational
+number and its relations to the rational number was first reached
+during the latter half of the last century, and then only after
+protracted study and much discussion. We have sketched only in brief
+outline the usual treatment, since it is believed that the importance
+and difficulty of a full discussion of such subjects will appear more
+clearly after reading the following chapters.
+
+Among the good discussions of the irrational number in the English
+language are: \textsc{H.~P. Manning}, \textit{Irrational Numbers and
+their Representation by Sequences and Series}, Wiley \& Sons, New
+York; \textsc{H.~B. Fine}, \textit{College Algebra}, Part~I, Ginn \&
+Co., Boston;
+%-----File: 023.png---Folio 11-------
+\textsc{Dedekind}, \textit{Essays on the Theory of Number} (translated
+from the German), Open Court Pub.\ Co., Chicago; \textsc{J.~Pierpont},
+\textit{Theory of Functions of Real Variables}, Chapters I and II,
+Ginn \& Co., Boston.
+
+\section{General Remarks on the Number System.}\hypertarget{chIsec4}{}%[4]
+
+Various modes of treatment of the problem of the number system as a
+whole are possible. Perhaps the most elegant is the following: Assume
+the existence and defining properties of the positive integers by
+means of a set of postulates or axioms.  From these postulates it is
+not possible to argue that if $p$ and $q$ are prime there exists a
+number $a$ such that $a\cdot p=q$ or \correction{$a=\frac
+qp$}{$a=\frac pq$}, i.e., in the field of positive integers the
+operation of division is not always possible. The set of all pairs of
+integers $\{m,n\}$, if $\{mk, nk\}$ ($k$ being an integer) is regarded
+as the same as $\{m,n\}$, form an example of a set of objects which
+can be added, subtracted, and multiplied according to the laws holding
+for positive integers, provided addition, subtraction, and
+multiplication are defined by the equations,\footnote{%
+    The details needed to show that these integer pairs satisfy the
+    algebraic laws of operation are to be found in Chapter~I, pages
+    5--12, of \textsc{Pierpont's} \textit{Theory of Real
+    Functions}. \textsc{Pierpont's} exposition differs from that
+    indicated above, in that he says that the integer pairs actually
+    \emph{are} the fractions.}
+\begin{align*}
+  \{m, n\}\otimes\{p, q\} &= \{mp, nq\} \\
+  \{m, n\}\oplus \{p, q\} &= \{mq+np, nq\}.
+\end{align*}
+The operations with the subset of pairs $\{m,1\}$ are exactly the same
+as the operations with the integers.
+
+This example shows that no contradiction will be introduced by adding
+a further axiom to the effect that besides the integers there are
+numbers, called fractions, such that in the extended system division
+is possible. Such an axiom is added and the order relations among the
+fractions are defined as follows:
+\[
+  \frac pq <\frac mn\quad \text{if}\quad pn<qm.
+\]
+%-----File: 024.png---Folio 12-------
+
+By an analogous example\footnote{%
+  Cf.\ \textsc{Pierpont}, loc.\ cit., pages 12--19.}
+the possibility of negative numbers is shown and an axiom assuming
+their existence is justified.  This completes the rational number
+system and brings the discussion to the point where this book begins.
+
+Our \hyperlink{axiomK}{Axiom~K}, which completes the real number system, assuming that
+every bounded set has a least upper bound, should, as in the previous
+cases, be accompanied by an example to show that no contradiction with
+previous axioms is introduced by \hyperlink{axiomK}{Axiom~K}. Such an example is the set
+of all lower segments, a \index{Segment!lower}\index{Lower segment}lower segment, $S$, being defined as any
+bounded set of rational numbers such that if $x$ is a number of $S$,
+every rational number less than $x$ is in $S$. For instance, the set
+of all rational numbers less than a rational number $a$ is a lower
+segment. Of two lower segments one is always a subset of the other. We
+may denote that $S$ is a subset of $S'$ by the symbol
+\[
+  S \olessthan S'.
+\]
+
+According to the order relation, $\olessthan$, every bounded set of
+lower segments $[S]$ has a least upper bound, namely the lower
+segment, consisting of every number in any $S$ of $[S]$. If $S$ and
+$T$ are lower segments whose least upper bounds are $s$ and $t$, we
+may define
+\[
+  S \oplus T
+\]
+and
+\[
+  S \otimes T
+\]
+as those lower segments whose least upper bounds are $s+t$ and $s
+\times t$ respectively. It is now easy to see that the set of lower
+segments contains a subset that satisfies the same conditions as the
+rational numbers, and that the set as a whole satisfies \hyperlink{axiomK}{axiom~K}\@. The
+legitimacy of \hyperlink{axiomK}{axiom~K} from the logical point of view is thus
+established, since our example shows that it cannot contradict any
+previous theorem of arithmetic.
+
+Further axioms might now be added, if desired, to postulate the
+existence of imaginary numbers, e.g.\ of a number $x$ for
+%-----File: 025.png---Folio 13-------
+each triad of real numbers $a$, $b$, $c$, such that $ax^2 + bx +
+c=0$.  These axioms are to be justified by an example to show that
+they are not in contradiction with previous assumptions. The theory of
+the complex variable is, however, beyond the scope of this book.
+
+\section{Axioms for the Real Number System.}\hypertarget{chIsec5}{}%[5]
+\index{Real number system}
+A somewhat more summary way of dealing with the problem is to set down
+at the outset a set of postulates for the system of real numbers as a
+whole without distinguishing directly between the rational and the
+irrational number. Several sets of postulates of this kind have been
+published by \textsc{E.~V.  Huntington} in the 3d, 4th, and 5th
+volumes of the Transactions of the American Mathematical Society. The
+following set is due to \textsc{Huntington}.\footnote{%
+    Bulletin of the American Mathematical Society,
+    Vol.~XII, page~228.}
+
+The system of real numbers is a set of elements related to one another
+by the rules of addition ($+$), multiplication ($\times$), and
+magnitude or order ($<$) specified below.
+\begin{itemize}\index{Axioms!of the real number system}
+\item[A 1.] Every two elements $a$ and $b$ determine uniquely an
+element $a+b$ called their \emph{sum}.
+
+\item[A 2.] $(a+b) + c = a + (b+c)$.
+
+\item[A 3.] $(a+b)=(b+a)$.
+
+\item[A 4.] If $a+x=a+y$, then $x=y$.
+
+\item[A 5.] There is an element $z$, such that $z+z=z$. (This element
+$z$ proves to be unique, and is called 0.)
+
+\item[A 6.] For every element $a$ there is an element $a'$, such that
+$a+a'=0$.
+
+\item[M 1.] Every two elements $a$ and $b$ determine uniquely an
+element $ab$ called their \emph{product}; and if $a \neq 0$ and $b
+\neq 0$, then $ab \ne 0$.\footnote{%
+    The latter part of M~1 may be omitted from the list of axioms, since
+    it can be proved as a theorem from A~4 and A~M~1.}
+
+\item[M 2.] $(ab)c=a(bc)$.
+
+\item[M 3.] $ab = ba$.
+
+\item[M 4.] If $ax=ay$, and $a \neq 0$, then $x=y$.
+%-----File: 026.png---Folio 14-------
+\item[M 5.] There is an element $u$, different from 0, such that
+$uu=u$. This element proves to be uniquely determined, and is called
+1.
+
+\item[M 6.] For every element $a$, not 0, there is an element $a''$,
+such that $aa'' = 1$.
+
+\item[A M 1.] $a(b+c)=ab+ac$.
+
+\item[O 1.] If $a \neq b$, then either $a<b$ or $b<a$.
+
+\item[O 2.] If $a<b$, then $a \neq b$.
+
+\item[O 3.] If $a<b$ and $b<c$, then $a<c$.
+
+\item[O 4.] (Continuity.) If $[x]$ is any set of elements such that
+for a certain element $b$ and every $x$, $x<b$, then there exists an
+element $\overline{B}$ such that---
+\begin{enumerate}
+\item[(1)] For every $x$ of $[x]$, $x < \overline{B}$;
+
+\item[(2)] If $y < \overline{B}$, then there is an $x_1$ of $x$ such
+that $y<x_1$.
+\end{enumerate}
+\item[A O 1.] If $x<y$, then $a+x<a+y$.
+
+\item[M O 1.] If $a>0$ and $b>0$, then $ab>0$.
+\end{itemize}
+
+These postulates may be regarded as summarizing the properties of the
+real number system. Every theorem of real analysis is a logical
+consequence of them. For convenience of reference later on we
+summarize also the rules of operation with the symbol\index{Absolute value} $|x|$, which
+indicates the ``numerical'' or ``absolute'' value of $x$. That is, if
+$x$ is positive, $|x| = x$, and if $x$ is negative, $|x| =-x$.
+\begin{align*}
+  |x| + |y| &\geqq|x+y|.
+\tag{1}\\
+  \therefore\quad\sum_{k=1}^n|x_k| &\geqq
+  \Bigl|\sum_{k=1}^n x_k \Bigr|,
+\tag{2}
+\end{align*}
+where $\sum_{k=1}^n x_k = x_1 + x_2 + \ldots + x_n$.
+\begin{align*}
+  \bigl| |x|-|y| \bigr|\leqq|x-y|
+  &= |y-x|\leqq|x| +|y|.
+\tag{3}\\
+  |x\cdot y| &= |x| \cdot|y|.
+\tag{4}\\
+  \frac{|x|}{|y|} &= \left|\frac xy\right|.
+\tag{5}
+\end{align*}
+\[\text{If }
+  |x-y| < e_1,\
+  |y-z| < e_2, \text{ then }
+  |x-z| < e_1 + e_2.
+\tag{6}
+\]
+%-----File: 027.png---Folio 15-------
+
+If $[x]$ is any bounded set,
+\[
+  \overline{B}[x]-\underline{B}[x]
+  = \overline{B}[|x_1-x_2 |].
+\tag{7}
+\]
+
+\section[The Number $e$.]{The Number $\boldsymbol e$.}\hypertarget{chIsec6}{}%[6]
+
+In the theory of the exponential and logarithmic functions (see
+page~\pageref{s4p97}) the irrational number $e$ plays an important
+r\^ole.  This number may be defined as follows:
+\hypertarget{eq1p16}{\[
+  e = \overline{B}[E_n],
+\tag{1}
+\]}
+where
+\[
+  E_n = 1+\frac{1}{1!}+\frac{1}{2!}+\ldots +\frac{1}{n!},
+\]
+where $[n]$ is the set of all positive integers, and
+\[
+  n!= 1 \cdot 2 \cdot 3 \ldots n.
+\]
+
+It is obvious that \hyperlink{eq1p16}{(1)} defines a finite number and not infinity,
+since
+\[
+  E_n=1+\frac{1}{1!}+\frac{1}{2!}+ \ldots + \frac{1}{n!}
+  < 1+1+\frac12+\frac{1}{2^2}+ \ldots +\frac{1}{2^{n-1}}
+  = 3-\frac{1}{2^{n-1}}.
+\]
+The number $e$ may very easily be computed to any number of decimal
+places, as follows:
+%-----File: 028.png---Folio 16-------
+\begin{align*}
+         E_0 &= 1 \\
+\frac{1}{1!} &= 1 \\
+\frac{1}{2!} &= \;.5 \\
+\frac{1}{3!} &= \;.166666+\\
+\frac{1}{4!} &=\phantom{1}.041666+ \\
+\frac{1}{5!} &=\phantom{1}.008333+ \\
+\frac{1}{6!} &=\phantom{1}.001388+ \\
+\frac{1}{7!} &=\phantom{1}.000198+ \\
+\frac{1}{8!} &=\phantom{1}.000024+ \\
+\frac{1}{9!} &=\phantom{1}.000002+ \\
+             &\quad\ \rule{5em}{0.5pt}\\
+E_9 &= 2.7182\ldots
+\end{align*}
+
+\begin{lemma}
+If $k>e$, then $E_k > e-\frac{1}{k!}$.
+\end{lemma}
+\begin{proof}
+From the definitions of $e$ and $E_n$ it follows that
+\[
+  e-E_k = \overline{B} \left[
+  \frac{1}{(k+1)!} +
+  \frac{1}{(k+2)!} +
+  \ldots
+  \frac{1}{(k+l)!} \right],
+\]
+where $[l]$ is the set of all positive integers. Hence
+\[
+  e-E_k = \frac{1}{(k+1)!} \cdot
+  \overline{B} \biggl[1+
+  \frac{1}{k+2} +
+  \frac{1}{(k+2)(k+3)} +
+  \ldots
+  + \frac{1}{(k+2) \ldots (k+l)} \biggr],
+\]
+or
+\[
+  e-E_k < \frac{1}{(k+1)!} \cdot e.
+\]
+If $k>e$, this gives
+\[
+  E_k > e-\frac{1}{k!}.\qedhere
+\]
+\end{proof}
+%-----File: 029.png---Folio 17-------
+\begin{theorem}[7]\hypertarget{thm7}{}\label{t7p17}
+\[
+  e=\overline{B}\left[\left(1+\frac1n\right)^n\right],
+\]
+where $[n]$ is the set of all positive integers.
+\end{theorem}
+\begin{proof}
+By the binomial theorem for positive integers
+\[
+  \left(1+\frac1n \right)^n
+  = 1 + n\left(\frac1n \right)
+  + \frac{n(n-1)}{2!} \cdot \left(\frac1n \right)^2 + \ldots
+  + \left(\frac1n \right)^n.
+\]
+Hence
+\begin{align*}
+   E_n-\left(1+\frac1n \right)^n
+  &= \sum^n_{k=2}
+  \left(\frac{1}{k!}-\frac{n(n-1) \ldots (n-k+1)}{k!\, n^k} \right)
+\\\hypertarget{eqap17}{%
+  &= \sum^n_{k=2}
+  \frac{n^k-n(n-1) \ldots (n-k+1)}{k!\,n^k},
+\tag{\textit{a}}}
+\\
+  &< \sum^n_{k=2}
+  \frac{n^k-(n-k+1)^k}{k!\,n^k}.
+\end{align*}
+Hence by factoring
+\begin{align*}
+  E_n-\left(1+\frac1n \right)^n
+  &< \sum^n_{k=2}
+  \frac{(k-1)(n^{k-1} + n^{k-2}(n-k+1) + \ldots
+       + (n-k+1)^{k-1}) }{k!\,n^k}
+\\
+  &< \sum^n_{k=2}
+  \frac{(k-1)k n^{k-1}}{k!\,n^k}
+\\
+  &< \frac1n \sum^n_{k=2}
+  \frac{(k-1)k}{k!}
+\end{align*}
+i.e.,
+\hypertarget{eqbp17}{\[
+  E_n-\left(1+\frac1n \right)^n
+  < \frac1n\left(1+\sum^{n-2}_{l=1} \frac{1}{l!} \right)
+  < \frac en.
+\tag{\textit{b}}
+\]}
+From \hyperlink{eqap17}{(\textit{a})}
+\begin{align*}
+\hypertarget{eq1p17}{\tag{1}
+  E_n &> \left(1+\frac1n \right)^n\\}
+  \intertext{and from \hyperlink{eqbp17}{(\textit{b})}}
+\tag{2}
+  \left(1+\frac1n \right)^n > E_n-\frac en,
+\end{align*}
+%-----File: 030.png---Folio 18-------
+whence by the lemma
+\hypertarget{eq3p18}{\[
+  \left(1+\frac1n \right)^n > e-\frac{1}{n!}-\frac en.
+\tag{3}
+\]}
+
+From \hyperlink{eq1p17}{(1)} it follows that $e$ is an upper bound of
+\[
+  \left[\left(1+\frac1n \right)^n\right],
+\]
+and from \hyperlink{eq3p18}{(3)} it follows that no smaller number can be an upper
+bound. Hence
+\[
+  \overline{B}\left[\left(1+\frac1n \right)^n\right] = e.
+\]\label{endpf18}
+\end{proof}
+
+\section{Algebraic and Transcendental Numbers.}\hypertarget{chIsec7}{}%[7]
+\index{Algebraic!numbers}\index{Transcendental!numbers}\index{Number!algebraic}\index{Numbers!transcendental}
+The distinction between rational and irrational numbers, which is a
+feature of the discussion above, is related to that between
+\textit{algebraic} and \textit{transcendental} numbers. A number is
+algebraic if it may be the root of an algebraic equation,
+\[
+  a_0x^n + a_1x^{n-1} + \ldots + a_{n-1}x + a_n = 0,
+\]
+where $n$ and $a_0,a_1,\ldots,a_n$ are integers and $n>0$. A number is
+transcendental if not algebraic. Thus every rational number $\frac mn$
+is algebraic because it is the root of the equation
+\[
+  nx-m=0,
+\]
+while every transcendental number is irrational. Examples of
+transcendental numbers are, $e$, the base of the system of natural
+logarithms, and $\pi$, the ratio of the circumference of a circle to
+its diameter.
+
+The proof that these numbers are transcendental follows on
+page~\pageref{s8p19}, though it makes use of infinite series which
+will
+%-----File: 031.png---Folio 19-------
+not be defined before page~\pageref{dp71}, and the function $e^x$,
+which is defined on page~\pageref{dp57}.
+
+The existence of transcendental numbers was first proved by
+\textsc{J.~Liouville}, Comptes Rendus, 1844. There are in fact an
+infinitude of transcendental numbers between any two numbers.  Cf.\
+\textsc{H.~Weber}, \textit{Algebra}, Vol.~2, p.~822. No
+\textit{particular} number was proved transcendental till, in 1873,
+\textsc{C.~Hermite} (Crelle's Journal, Vol.~76, p.~303) proved $e$ to
+be transcendental.  In 1882 \textsc{E.~Lindemann} (Mathematische
+Annalen, Vol.~20, p.~213) showed that $\pi$ is also transcendental.
+
+The latter result has perhaps its most interesting application in
+geometry, since it shows the impossibility of solving the classical
+problem of constructing a square equal in area to a given circle by
+means of the ruler and compass. This is because any construction by
+ruler and compass corresponds, according to analytic geometry, to the
+solution of a special type of algebraic equation. On this subject, see
+\textsc{F.~Klein}, \textit{Famous Problems of Elementary Geometry}
+(Ginn \& Co., Boston), and \textsc{Weber} and \textsc{Wellstein},
+\textit{Encyclop\"adie der Elementarmathematik}, Vol.~1, pp.~418--432
+(B.~G.~Teubner, Leipzig).
+
+\section[The Transcendence of $e$.]{The Transcendence of $\boldsymbol e$.}\hypertarget{chIsec8}{}%[8]
+\label{s8p19}
+
+\begin{theorem}[8]\hypertarget{thm8}{}
+If $c,c_1,c_2,c_3,\ldots,c_n$ are integers (or
+zero but $c \neq 0$), then
+\hypertarget{eq1p19}{\[
+  \tag{1}
+  c+c_1e+c_2e^2+\ldots+c_ne^n \neq 0.
+\]}
+\end{theorem}
+
+\begin{proof}
+The scheme of proof is to find a number such that
+when it is multiplied into \hyperlink{eq1p19}{(1)} the product becomes equal to a whole
+number distinct from zero plus a number between $+1$ and $-1$, a sum
+which surely cannot be zero. To find this number $N$, we study the
+series\footnote{%
+  Cf.~pages \pageref{dp71} and \pageref{t58p99}.
+  }
+for $e^k$, where $k$ is an integer $\qqle n$:
+\[
+  e^k = 1 + \frac{k}{1!} + \frac{k^2}{2!} + \frac{k^3}{3!} + \ldots.
+\]
+%-----File: 032.png---Folio 20-------
+
+Multiplying this series successively by the arbitrary factors $i!\cdot
+b_i$, we obtain the following equations:
+\hypertarget{eq2p20}{\[
+ \left.
+ \begin{array}{l}
+  e^k\cdot 1!\cdot b_1 = b_1 \cdot 1! + b_1k
+  \left(1 + \frac k2 + \frac{k^2}{2\cdot 3} + \ldots \right);
+\\
+  e^k\cdot 2! \cdot b_2 = b_2 \cdot 2!
+  \left(1+\frac k1\right) +
+  b_2\cdot k^2\left(1 + \frac k3 + \frac{k^2}{3\cdot 4}
+  + \ldots \right);
+\\
+  e^k\cdot 3! \cdot b_3 = b_3 \cdot 3!
+  \left(1+\frac{k}{1!}+\frac{k^2}{2!}\right) +
+  b_3\cdot k^3\left(1 + \frac k4 + \frac{k^2}{4\cdot 5}
+  + \ldots \right);
+\\
+\hdotsfor[10]{1}
+\\
+  e^k\cdot s! \cdot b_s = b_s \cdot
+  s!\left(1 + \frac{k}{1!} + \frac{k^2}{2!} + \ldots
+  + \frac{k^{s-1}}{(s-1)!} \right)
+\\
+  \hfill + b_s\cdot k^s\left(1 + \frac{k}{s+1}
+  + \frac{k^2}{(s+1)(s+2)} + \ldots \right).
+ \end{array}
+ \right\}
+\tag{2}
+\]}
+
+For the sake of convenience in notation the numbers $b_1\ldots b_s$
+may be regarded as the coefficients of an arbitrary polynomial
+\[
+  \phi(x) + b_0 + b_1x + b_2x^2 + \ldots
+  + \mbox{\correction{$b_sx^s$}{$b_sx_s$}},
+\]
+the successive derivatives of which are
+\begin{gather*}
+\begin{array}{c}
+  \phi'(x) = b_1 + 2\cdot b_2x + \ldots + s\cdot b_s \cdot x^{s-1},
+\\
+\hdotsfor[10]{1}
+\end{array}
+\\
+\begin{array}{c}
+  \phi^{(m)}(x) = b_m\cdot m! + b_{m+1} \cdot \frac{(m+1)!}{1!}\cdot x
++ \ldots + b_s\cdot \frac{s!}{(s-m)!} \cdot x^{s-m};
+\\
+\hdotsfor[10]{1}
+\end{array}
+\end{gather*}
+
+The diagonal in \hyperlink{eq2p20}{(2)} from $\text{\correction{$b_1$}{$b$}}\cdot 1!$ to $b_s\cdot
+s!\frac{k^{s-1}}{(s-1)!}$ is obviously $\phi'(k)$, the next lower
+diagonal is $\phi''(k)$, etc. Therefore by adding equations~\hyperlink{eq2p20}{(2)} in
+this notation we obtain
+%-----File: 033.png---Folio 21-------
+\hypertarget{eq3p21}{\begin{align*}
+  e^k(1!\,b_1+2!\,b_2+\ldots+s!\,b_s)=\phi'(k) &+\phi''(k)+\ldots\\
+  &+\phi^{(s)}(k)+\sum_{m=1}^sb_m\cdot k^m\cdot R_{km}, \tag{3}
+\end{align*}}
+in which
+\[
+  R_{km}=1+\frac{k}{m+1}+\frac{k^2}{(m+1)(m+2)}+ \ldots.
+\]
+
+Remembering that $\phi(x)$ is perfectly arbitrary, we note that if it
+were so chosen that
+\[
+  \phi'(k)=0,\quad \phi''(k)=0,\ldots,\quad \phi^{(p-1)}(k)=0,
+\]
+for every $k$ ($k=1, 2, 3, \ldots, n$) then equations~\hyperlink{eq2p20}{(2)} and \hyperlink{eq3p21}{(3)}
+could be written in the form
+\hypertarget{eq4p21}{\begin{align*}
+  e^k(1!\text{\correction{$\cdot$}{}} b_1+2!\cdot b_2+\ldots+s!\cdot b_s)
+  & = \sum_{m=1}^sb_m\cdot k^m\cdot R_{km} \\
+  & + b_p\cdot p! \\
+  & + b_{p+1}\cdot (p+1)!\cdot \left(1+\frac{k}{1!}\right) \\
+  & + \ldots \\
+  & + b_s\cdot s!\left(1+\frac{k}{1!}+\frac{k^2}{2!}+\ldots
+  +\frac{k^{s-p}}{\text{\correction{$(s-p)!$}{$(s-p)$}}}\right).
+\tag{4}
+\end{align*}}
+A choice of $\phi(x)$ satisfying the required conditions is
+\hypertarget{eq5p21}{\[
+  \phi(x)=(a_0+a_1x+a_2x^2+ \ldots +a_nx^n)^p
+  \cdot \frac{x^{p-1}}{(p-1)!}=\frac{(f(x))^p\cdot x^{p-1}}{(p-1)!},
+\tag{5}
+\]}
+where $f(x) = (x-1)(x-2)(x-3) \ldots (x-n)$.
+%-----File: 034.png---Folio 22-------
+
+Every $k$ ($k = 1, 2, \ldots, n$) is a $p$-tuple root of \hyperlink{eq5p21}{(5)}. Here $p$
+is still perfectly arbitrary, but the degree $s$ of $\phi(x)$ is
+$np+p-1$.  If $\phi(x)$ is expanded and the result compared with
+\[
+  \phi(x) = b_0 + b_1x + \ldots + b_s x^s,
+\]
+it is plain that
+\[
+  b_0 = 0,\  b_1=0,\  \ldots,\  b_{p-2}=0,
+\]
+on account of the factor $x^{p-1}$, and
+\[
+  b_{p-1} = \frac{a_0^p}{(p-1)!},\
+  b_p = \frac{I_p}{(p-1)!},\  \ldots,\
+  b_s = \frac{I_s}{(p-1)!},
+\]
+where $I_p,I_{p+1},\ldots,I_s$, are all integers. The coefficient of
+$e^k$ in the left-hand member of \hyperlink{eq4p21}{(4)} is therefore
+\[
+  N_p = a_0^p + \frac{I_p}{(p-1)!}\cdot p!
+              + \frac{I_{p+1}}{(p-1)!}\cdot (p+1)!+ \ldots
+              + \frac{I_s }{(p-1)!}\cdot s!
+\]
+
+Whenever the arbitrary number $p$ is prime and greater than $a_0$,
+$N_p$ is the sum of $a_0^p$, which cannot contain $p$ as a factor,
+plus other integers each of which does contain the factor $p$.  $N_p$
+is therefore \emph{not zero and not divisible by $p$}.
+
+Further, since
+\[
+  \frac{(p+t)!}{(p-1)!\cdot r!}=p\frac{(p+1)(p+2)\ldots(p+t)}{r!}
+\]
+is an integer divisible by $p$ when $r\leqq t$, it follows that all
+the coefficients of the last block of terms in \hyperlink{eq4p21}{(4)} contain $p$ as a
+factor. Since $k$ is also an integer, \hyperlink{eq4p21}{(4)} evidently reduces to
+\[
+  N_p\cdot e^k=pW_{kp}+\sum_{m=1}^s b_m\cdot k^m\cdot R_{km},
+\]
+%-----File: 035.png---Folio 23-------
+where $W_{kp}$ is an integer or zero, and this may be abbreviated to
+the form
+\hypertarget{eq6p23}{\[
+\label{eq1onp23} \tag{6}
+  N_p\cdot e^k = pW_{kp} + r_{kp}.
+\]}
+Before completing our proof we need to show that by choosing the
+arbitrary prime number $p$ sufficiently large, $r_{kp}$ can be made as
+small as we please. If $\alpha$ is a number greater than $n$,
+\begin{align*}
+  |R_{km}|
+  &= \left|1+ \frac{k}{m+1} + \frac{k^2}{(m+1)(m+2)} + \ldots
+  \right|
+\\
+  &< \left|1+ \frac{\alpha}{m+1} + \frac{\alpha^2}{(m+1)(m+2)}
+  + \ldots \right|
+\\
+  &< \left|1+ \frac\alpha1 + \frac{\alpha^2}{2!} + \ldots
+  \right|
+\\
+  &< e^\alpha
+\end{align*}
+for all integral values of $m$ and of $k \qqle n$.
+\[
+  |r_{kp}|
+  = \left|\sum_{m=1}^s b_m \cdot k^m \cdot R_{km} \right|
+  \leqq \sum_{m=1}^s|b_m|\cdot k^m \cdot|R_{k,m}|.
+\]
+
+Since the number $b_m$ is the coefficient of $x^m$ in $\phi(x)$ and
+since each coefficient of $\phi(x)$ is numerically less than or equal
+to the corresponding coefficient of
+\[
+  \frac{x^{p-1}}{(p-1)!}
+  \left(|a_0|+ |a_1|x + |a_2|x^2
+       + \ldots + |a_n|x^n \right)^p,
+\]
+it follows that
+\begin{align*}
+  |r_{kp}|
+  &< e^\alpha \cdot \frac{\alpha^{p-1}}{(p-1)!}
+   \left(|a_0|+ |a_1|\alpha + \ldots
+        + |a_n|\alpha^n \right)^p
+\\
+  &< \frac{Q^p}{(p-1)!} \cdot e^\alpha,
+\end{align*}
+%-----File: 036.png---Folio 24-------
+where
+\[
+  Q = \alpha (|a_0|+ |a_1|\alpha + \ldots
+             + |a_n|\alpha^n)
+\]
+is a constant not dependent on $p$. The expression $
+\frac{Q^p}{(p-1)!}$ is the $p$th term of the series for $Qe^Q$, and
+therefore by choosing $p$ sufficiently large $r_{kp}$, may be made as
+small as we please.
+
+If now $p$ is chosen as a prime number, greater than $\alpha$ and
+$\alpha_0$ and so great that for every $k$,
+\[
+  r_{kp} < \frac{1}{n\cdot d},
+\]
+where $d$ is the greatest of the numbers
+\[
+  c,\; c_1,\; c_2,\; c_3,\; \ldots,\; c_n,
+\]
+the equations~\hyperlink{eq6p23}{(6)} evidently give
+\begin{align*}
+  N_p(c + c_1 e + c_2 e^2 + \ldots &+ c_n e^n)\\
+  &= N_p c + p(c_1 W_{1p} + c_2 W_{2p} + \ldots + c_n W_{np}) \\
+  &\hspace*{2cm}+c_1 r_{1p} + c_2 r_{2p} + \ldots + c_n r_{np},\\
+  &= N_p c + pW + R,\tag{8}
+\end{align*}
+where $W$ is an integer or zero and $R$ is numerically less than
+unity. Since $N_p c$ is not divisible by $p$ and is not zero, while
+$pW$ is divisible by $p$, this sum is numerically greater than or
+equal to zero. Hence
+\[
+  N_p (c + c_1 e + c_2 e^2 + \ldots + c_n e^n ) \neq 0.
+\]
+Hence
+\[
+  c + c_1 e + c_2 e^2 + \ldots + c_n e^n \neq 0,
+\]
+and $e$ is a transcendental number.
+\end{proof}
+%-----File: 037.png---Folio 25-------
+\section{The Transcendence of $\pi$.}\hypertarget{chIsec9}{}%[9]
+
+The definition of the number $\pi$ is derived from \textsc{Euler}'s
+formula
+\[
+  e^{x \sqrt{-1}} = \cos x + \sqrt{-1} \sin x;
+\]
+by replacing $x$ by $\pi$,
+\hypertarget{eq1p25}{\[
+\label{p25eq1}
+  e^{\pi \sqrt{-1}} =-1.\tag{1}
+\]}
+If $\pi$ is assumed to be an algebraic number, $\pi\sqrt{-1}$ is also
+an algebraic number and is the root of an irreducible algebraic
+equation $F(x)=0$ whose coefficients are integers. If the roots of
+this equation are denoted by $z_1, z_2, z_3,\ldots, z_n$, then, since
+$\pi \sqrt{-1}$ is one of the $z$'s, it follows as a consequence of
+\hyperlink{eq1p25}{(1)} that
+\hypertarget{eq2p25}{\[
+  (e^{z_1}+1) (e^{z_2}+1) (e^{z_3}+1) \ldots (e^{z_n}+1) =0.\tag{2}
+\]}
+By expanding \hyperlink{eq2p25}{(2)}
+\[
+  1 + \sum e^{z_i} + \sum e^{z_i+z_j} + \sum e^{z_i+z_j+z_k} + \ldots = 0.
+\]
+Among the exponents zero may occur a number of times e.g., $(c-1)$
+times. If then
+\[
+   z_i, \quad z_i + z_j, \quad z_i + z_j + z_k,\quad \ldots,
+\]
+be designated by $x_1, x_2, x_3, \ldots, x_n$, the equation becomes
+\hypertarget{eq3p25}{\[
+  c + e^{x_1} + e^{x_2} + \ldots + e^{x_n} =0,\tag{3}
+\]}
+where $c$ is a positive number at least unity and the numbers $x_i$
+are algebraic. These numbers, by an argument for which the reader is
+referred to \textsc{Weber} and \textsc{Wellstein}'s
+\emph{Encyclop\"{a}die der Elementarmathematik}, p.~427 et seq., may
+be shown to be the roots of an algebraic equation
+\[
+  f(x) = a_0 + a_1 x + a_2 x^2 + \ldots + a_n x^n = 0,\tag{$3'$}
+\]
+%-----File: 038.png---Folio 26-------
+the coefficients being integers and $a_0\neq0$ and $a_n\neq0$. The
+rest of the argument consists in showing that equation~\hyperlink{eq3p25}{(3)} is
+impossible when $x_1,x_2$, \ldots, $x_n$ are roots of ($3'$). The
+process is analogous to that in \hyperlink{chIsec8}{\S~8}.
+\hypertarget{eq4p26}{\[
+\left.
+\begin{array}{l}\displaystyle
+  e^{x_k}\cdot1!\,b_1 = b_1\cdot1! + b_1x_k \left(1 + \frac{x_k}{2}
+  + \frac{x_k^2}{2\cdot3} +\ldots \right),
+\\ \displaystyle
+  e^{x_k}\cdot2!\,b_2 = b_2\cdot2!\left(1 + \frac{x_k}{1!} \right)
+  + b_2x_k^2 \left(1 + \frac{x_k}{3} + \frac{x_k^2}{3\cdot4}
+  + \ldots\right),
+\\ \displaystyle
+  e^{x_k}\cdot3!\,b_3 = b_3\cdot3!\left(1 + \frac{x_k}{1!}
+  + \frac{x_k^2}{2!} \right) + b_3x_k^3\left(1 + \frac{x_k}{4}
+  + \frac{x_k^2}{4\cdot5} + \ldots \right),
+\\
+\hdotsfor[10]{1}
+\\ \displaystyle
+  e^{x_k}\cdot s!\,b_s = b_s\cdot s!\left(1 + \frac{x_k}{1!}
+  + \ldots + \frac{x_k^{s-1}}{(s-1)!} \right)
+\\ \displaystyle
+  \hfill + b_s x_k^s\left(1 + \frac{x_k}{s+1}
+  + \frac{x_k^2}{(s+1)(s+2)} + \ldots \right).
+\end{array}
+\right\}
+\tag{4}
+\]}
+
+The numbers $b_1,\ldots,b_{\text{\correction{$s$}{$n$}}}$ may be regarded as the coefficients of an
+arbitrary polynomial
+\[
+  \phi(x)=b_0+b_1x+b_2x^2+\ldots+b_s x^s,
+\]
+for which
+\[
+  \phi^{(m)}(x)=b_m\cdot m!+b_{m+1}\cdot\frac{(m+1)!}{1!}\cdot x
+  + \ldots + b_s\frac{s!}{(s-m)!}\cdot x^{s-m}.
+\]
+
+The diagonal in equations~\hyperlink{eq4p26}{(4)} from $b_1\cdot1!$ to $b_s\cdot
+s!\frac{{x_k}^{s-1}}{(s-1)\text{\correction{$!$}{}}}$ is obviously $\phi'(x_k)$, and the next
+lower diagonal $\phi''(x_k)$, etc.  Therefore, by adding equations~\hyperlink{eq4p26}{(4)},
+\hypertarget{eq5p26}{\begin{multline*}
+  e^{x_k}(1!\,b_1+2!\,b_2+\ldots+s!\,b_s) = \phi'(x_k)+\phi''(x_k)+\ldots
+\\
+  +\phi^{(s)}(x_k)+\sum_{m=1}^s b_m\cdot x_k^mR_{km},
+\tag{5}
+\end{multline*}}
+%-----File: 039.png---Folio 27-------
+in which
+\[
+  R_{km} = 1 + \frac{x_k}{m+1} + \frac{x_k^2}{(m+1)(m+2)}+ \ldots
+\]
+
+Remembering that $\phi(x)$ is perfectly arbitrary, let it be so chosen
+that
+\[
+  \phi'(x_k) = 0,\; \phi''(x_k)=0,\; \phi'''(x_k)=0,
+  \; \ldots, \phi^{(p-1)}(x_k)=0
+\]
+for every $x_k$.
+
+Equation~\hyperlink{eq5p26}{(5)} may then be written as follows:
+\hypertarget{eq6p27}{\begin{align*}
+  e^{x_k}(1!\,b_1 + 2!\,b_2 + \ldots + s!\,b_s)
+  &= \sum_{m=1}^s b_m \cdot (x_k)^m \cdot R_{\text{\correction{$km$}{$k,m$}}}
+\\
+  &+ b_p \cdot p! \\
+  &+ b_{p+1} \cdot (p+1)! \left(1+\frac{x_k}{1!} \right) \\
+  &+\ldots \\
+\tag{6}
+  &+ b_s \cdot s! \left(1 + \frac{x_k}{1!} + \frac{x^2_k}{2!} + \ldots
+                     + \frac{x_k^{s-p}}{(s-p)!} \right).
+\end{align*}}
+
+A choice of $\phi(x)$ satisfying the required conditions is
+\begin{align*}
+  \phi(x) &= \frac{a_n^{np-1} \cdot x^{p-1}}{(p-1)!}
+             (a_0 + a_1 x + a_2 x^2 + \ldots + a_n x^n)^p
+\\
+  &= \frac{a_n^{np-1} \cdot x^{p-1}}{(p-1)!} (f(x))^p,
+\end{align*}
+of which every $x_k$ is a $p$-tuple root. If $\phi(x)$ is expanded and
+the result compared with
+\[
+  \phi(x) = b_0 + b_1 x + \ldots + b_s x^s,
+\]
+it is plain that $b_0=0$, $b_1=0$, \ldots, $b_{p-2}=0$, on account of
+the factor $x^{p-1}$; and
+\[
+  b_{p-1} = \frac{a_0^p a_n^{np-1}}{(p-1)!}, \quad
+  b_p = \frac{I_p \cdot a_n^{np-1}}{(p-1)!} \quad \ldots, \quad
+  b_s = \frac{I_s \cdot a_n^{np-1}}{(p-1)!},
+\]
+%-----File: 040.png---Folio 28-------
+where $I_p,\ldots,I_s$, are all integers. The coefficient of $e^{x_k}$ in
+\hyperlink{eq6p27}{(6)} may now be written
+\[
+  N_p = a_n^{np-1} \left(a_0^p +
+  \frac{I_p}{(p-1)!}\cdot p! +
+  \frac{I_{p+1}}{(p-1)!}(p+1)! +
+  \ldots +
+  \frac{I_s}{(p-1)!} \cdot s! \right)\text{\correction{.}{}}
+\]
+
+If the arbitrary number $p$ is chosen as a prime number greater than
+$a_0$ and $a_n$, $N_p$ becomes the sum of $a_0^pa_n^{np-1}$, which
+cannot contain $p$ as a factor, and a number of other integers each of
+which is divisible by $p$. $N_p$ therefore is \textit{not zero and not
+divisible by $p$}.
+
+Further, since, $\dfrac{(p+t)!}{(p-1)!\cdot r!}$ is an integer
+divisible by $p$ when $r \leqq t$, it follows that all of the
+coefficients of the last block of terms in \hyperlink{eq6p27}{(6)} contain $p$ as a
+factor. If then \hyperlink{eq6p27}{(6)} is added by columns,
+\hypertarget{eq7p28}{\[
+  N_pe^{\text{\correction{$x_k$}{$xk$}}} = pa_n^{np-1} \left[
+  P_0 + P_1x_k + P_2x_k^2 + \ldots + P_{s-p}x_k^{s-p} \right]
+  + \sum_{m=1}^s b_m\cdot x_k^m \cdot R_{km}
+\tag{7}
+\]}
+where $P_0,P_1,\ldots,P_{s-p}$ are integers.
+
+It remains to show that $\sum_{m=1}^s b_m \cdot x_k^m \cdot R_{km}$
+can be made small at will by a suitable choice of the arbitrary
+$p$. As in the proof of the transcendence of $e$, it follows that
+\[
+  \left|r_{kp} \right|
+  = \left|\sum_{m=1}^s b_m \cdot x_k^m \cdot R_{km} \right|
+  < \frac{Q^p}{(p-1)!} \cdot e^\alpha,
+\]
+where
+\[
+  Q = |a_n^n|
+      \alpha(|a_0|+ |a_1|\alpha + \ldots
+            + |a_n|\alpha ),
+\]
+and $\alpha$ is the largest of the absolute values of $x_k$ ($k=1,
+\ldots, n$).  If now $p$ is chosen as a prime number, greater than
+unity, greater than $a_0 \ldots a_n$ and greater than $c$, and so
+great also that $|r_{kp}|< \dfrac{1}{n}$, it follows directly from
+equation~\hyperlink{eq7p28}{(7)} that
+%-----File: 041.png---Folio 29-------
+\begin{multline*}
+\hypertarget{eq8p29}{\tag{8}
+   N_p(c + e^{x_1} + e^{x_2} + \ldots + e^{x_n})}
+\\
+  = N_pc + p{a_n}^{np-1}
+  (P_0S_0 + P_1S_1 + \ldots + P_{s-p}S_{s-p}) +
+  \sum_{k=1}^n r_{kp},
+\end{multline*}
+where
+\[
+  |r_{kp}|
+  = \left|\sum_{m=1}^s b_m \cdot x_k^m \cdot R_{km} \right|
+  < \frac1n,
+\]
+$S_0=n$,  and  $S_i=x_1^i + x_2^i + x_3^i + \ldots +
+  x_n^i$, and therefore
+\begin{align*}
+  S_1 &=-\frac{a_{n-1}}{a_n},
+  &S_2 &= \frac{a_{n-1}^2}{a_n^2}-\frac{2a_{n-2}}{a_n},\ldots,
+\footnotemark
+\end{align*}
+\footnotetext{%
+      Cf.~\textsc{Burnside} and \textsc{Panton} \textit{Theory of
+      Equations}, Chapter~VIII, Vol.~I.}
+and therefore it follows that ${a_n}^{np-1}S_1$,
+${a_n}^{np-1}S_2$,\ldots, are all whole numbers or zero. The term
+\[
+   pa_n^{np-1} \cdot \sum_{i=0}^{s-p} P_iS_i
+\]
+is therefore an integer divisible by $p$, while, on the contrary,
+$N_p$ and $c$ are not divisible by $p$. The sum of these terms is
+therefore a whole number $\geqq +1$ or $\leqq-1$, and since
+$\displaystyle \sum_{k=1}^nr_{kp} < 1$, the entire right-hand member
+of \hyperlink{eq8p29}{(8)} is not zero, and hence \hyperlink{eq3p25}{(3)} is not zero. Therefore---
+\begin{theorem}[9]\hypertarget{thm9}{}
+The number $\pi$ is transcendental.
+\end{theorem}
+%-----File: 042.png---Folio 30-------
+
+\chapter{SETS OF POINTS AND OF SEGMENTS.}\hypertarget{chapII}{}%[II]
+
+\section{Correspondence of Numbers and Points.}\hypertarget{chIIsec1}{}%[1]
+
+The system of real numbers may be set into \index{One-to-one correspondence}one-to-one correspondence
+with the points of a straight line. That is, a scheme may be devised
+by which every number corresponds to one and only one point of the
+line and vice versa. The point $0$ is chosen arbitrarily, and the
+points $1, 2, 3, 4, \ldots$ are at regular intervals to the right of
+$0$ in the order $1, 2, 3, 4, \ldots$ from left to right, while the
+points $-1,-2,-3,\ldots$ follow at regular intervals in the order $0,
+-1,-2,-3,\ldots$ from right to left. The points which correspond to
+fractional numbers are at intermediate positions as
+follows:\footnote{%
+  It is convenient to think of numbers in this case as simply a
+  notation
+  for points. In view of the correspondence of points and numbers the
+  numbers
+  furnish a complete notation for all points.}
+
+To fix our ideas we obtain a point corresponding to a particular
+decimal of a finite number of digits, say $1.32$.
+\begin{figure}[!hbtp]\label{fig01}\hypertarget{fig01}{}
+\centering
+\setlength{\unitlength}{0.06\textwidth}
+\begin{picture}(10,1.4)(-5,-0.6)
+\scriptsize
+\put(-5,0){\line(1,0){10}}
+\put(-5,0){\line(0,1){0.25}}
+\put(-4,0){\line(0,1){0.25}}
+\put(-3,0){\line(0,1){0.25}}
+\put(-2,0){\line(0,1){0.25}}
+\put(-1,0){\line(0,1){0.25}}
+\put(0,0){\line(0,1){0.25}}
+\put(5,0){\line(0,1){0.25}}
+\put(-1.8,0){\line(0,1){0.65}}
+\put(-1.8,0.8){\makebox(0,0)[cc]{$1.32$}}
+\put(-5,0.5){\makebox(0,0)[cc]{$1$}}
+\put(-4,0.5){\makebox(0,0)[cc]{$.1$}}
+\put(-3,0.5){\makebox(0,0)[cc]{$.2$}}
+\put(-2,0.5){\makebox(0,0)[cc]{$.3$}}
+\put(-1,0.5){\makebox(0,0)[cc]{$.4$}}
+\put(0,0.5){\makebox(0,0)[cc]{$.5$}}
+\put(5,0.5){\makebox(0,0)[cc]{$1$}}
+\normalsize
+\put(0,-0.3){\makebox(0,0)[tc]{\sc Fig.~1}}
+\end{picture}
+\end{figure}
+Divide the segment $\overline{1\ 2}$ into ten equal parts. Then divide
+the segment \correction{$\overline{.3\ .4}$}{$\overline{3\ 4}$} of this division into ten equal
+parts. The point marked $2$ by the last division is the point
+corresponding to $1.32$.
+
+If the decimal is not terminating, we simply obtain an infinite
+sequence of points, such that any one is to the right of all that
+precede it, in case of a positive number, or to the
+%-----File: 043.png---Folio 31-------
+left in case of a negative number. The first few points of the
+sequence for the number $\pi$ are the points corresponding to the
+numbers $3$, $3.1$, $3.14$, $3.141$. This set of numbers is bounded,
+$4$, for instance, being an upper bound. Hence the points
+corresponding to these numbers all lie to the left of the point
+corresponding to the number $4$. To show that there exists a definite
+point corresponding to the least upper bound \correction{$\overline{B}$}{$B$} of the set of numbers
+$3$, $3.1$, $3.14$, $3.141$, etc., use is made of the following:
+
+\begin{other}[Postulate of Geometric Continuity]\index{Axioms!of continuity}\index{Continuity!axioms of}If a set $[x]$
+of points of a line has a right bound, that is, if there exists a
+point $B$ on the line such that no point of the set $[x]$ is to the
+right of $B$, then there exists a leftmost right bound $\overline{B}$
+of the set $[x]$. If the set has a left bound, it has a rightmost left
+bound.
+\end{other}
+The leftmost right bound of the set of points corresponding to the
+numbers $3.$, $3.1$, $3.14$, etc., is the point which corresponds to
+the number $\pi$. In the same manner it follows from the postulate
+that there is a definite point on the line corresponding to any
+decimal with an infinitude of digits.\footnote{%
+  It is not implied here, of course, that it is possible to write a
+  decimal with an infinitude of digits, or to mark the corresponding
+  points. What is meant is that if an infinite sequence of digits is
+  determined, a definite number and a definite point are thereby
+  determined. Thus $\sqrt{2}$ determines an infinite sequence of
+  digits, that is, it furnishes the law whereby the sequence can be
+  extended at will.}
+
+Conversely, given any point on the line, e.g., a point $P$, to the
+right of $0$, there corresponds to it one and only one number.  This
+is evident since, in dividing the line according to a decimal scale,
+either the point in question is one of the division-points, in which
+case the number corresponding to the point is a terminating decimal,
+or in case it is not a division-point we will have an infinite set of
+division\correction{-}{ }points to the left of it, the point in question being the
+leftmost right bound of the set. If now we pick out the rightmost
+point of this left set in every division and note the corresponding
+number, we have a set of numbers whose least upper bound corresponds
+to the point $P$.
+%-----File: 044.png---Folio 32-------
+
+The ordinary analytic geometry furnishes a scheme for setting all
+pairs of real numbers into correspondence with all points of a plane,
+and all triples of real numbers into correspondence with all points in
+space. Indeed, it is upon this correspondence that the analytic
+geometry is based.
+
+It should be noticed that the correspondence between numbers and
+points on the line preserves order, that is, if we have three numbers,
+$a$, $b$, $c$, so that $a < b < c$, then the corresponding points $A$,
+$B$, $C$ are under the ordinary conventions so arranged that $B$ is to
+the right of $A$, and $C$ to the right of $B$.
+
+It will be observed that we have not put this matter of the one-to-one
+correspondence between points and numbers into the form of a
+theorem. Rather than aiming at a rigorous demonstration from a body of
+sharply stated axioms, we have attempted to place the subject-matter
+before the reader in such a manner that he will understand on the one
+hand the necessity, and on the other the grounds, for the hypothesis.
+
+\section{Segments and Intervals. Theorem of Borel.}\hypertarget{chIIsec2}{}%[2]
+
+\begin{definition}\index{Segment}
+A \textit{segment} $\overline{a\ b}$ is the set of all numbers greater
+than $a$ and less than $b$. It does not include its end-points $a$ and
+$b$. An \index{Interval}\textit{interval} $\interval{a}{b}$ is the segment
+$\overline{a\ b}$ together with $a$ and $b$. For a segment plus its
+end point $a$ we use the notation $\linterval{a}{b}$, and when $a$ is
+absent and $b$ present $\rinterval{a}{b}$. All these notations imply
+that $a<b$.\footnote{%
+  The notation $\overline{a\ b}$, $\interval{a}{b}$, $\linterval{a}{b}$, etc.,
+  to denote the presence or absence of end-points is due to
+  \textsc{G.~Peano}, \textit{Analisi Infinitisimali.}  Torino, 1893.}
+Sometimes we denote a segment or interval by a single letter. This is
+done in case it is not important to designate a definite segment or
+interval.
+
+The set of all numbers greater than $a$ is the \index{Segment!infinite}\index{Infinite segment}\textit{infinite
+segment} $\overline{a\ \infty}$, and the set of all numbers less than
+$a$ is the infinite segment $\overline{-\infty\ a}$. The infinite
+segments $\overline{a\ \infty}$ and $\overline{-\infty\ a}$, together
+with the point $a$, are respectively the infinite intervals
+$\linterval{a}{\infty}$ and $\rinterval{-\infty}{a}$.
+%-----File: 045.png---Folio 33-------
+Unless otherwise specified the expressions \emph{segment} and
+\emph{interval} will be understood to refer to segments and intervals
+whose end-points are finite.
+
+By means of the one-to-one correspondence of numbers and points on a
+line we define the length of a segment as follows: The length of a
+segment $\overline{a\ b}$ with respect to the unit segment
+$\overline{0\ 1}$ is the number $|a-b|$. This definition applies
+equally to all segments whether they are commensurable or
+\label{chIIp33}incommensurable with the unit segment.
+\end{definition}
+\begin{definition}\index{Covering of interval or segment}
+A set of segments or intervals $[\sigma]$ \textit{covers} a
+segment or interval $t$ if every point of $t$ is a point of some
+$\sigma$.
+\end{definition}
+
+On the interval $\interval{-1}{1}$ consider the set of points
+$\left[\dfrac{1}{2^n}\right]$. The
+\begin{figure}[!htbp]\label{fig02}\hypertarget{fig02}{}
+\centering
+\setlength{\unitlength}{0.06\textwidth}
+\begin{picture}(16,2)(-8,-1)
+\scriptsize
+\put(-8,0){\line(1,0){16}}
+\put(-8,0){\line(0,1){0.25}}
+\put(0,0){\line(0,1){0.25}}
+\put(1,0){\line(0,1){0.25}}
+\put(2,0){\line(0,1){0.25}}
+\put(4,0){\line(0,1){0.25}}
+\put(8,0){\line(0,1){0.25}}
+\put(-8,0.5){\makebox(0,0)[cc]{$-1$}}
+\put(0,0.5){\makebox(0,0)[cc]{$0$}}
+\put(1,0.5){\makebox(0,0)[cc]{$1/8$}}
+\put(2,0.5){\makebox(0,0)[cc]{$1/4$}}
+\put(4,0.5){\makebox(0,0)[cc]{$1/2$}}
+\put(8,0.5){\makebox(0,0)[cc]{$1$}}
+\normalsize
+\put(0,-0.3){\makebox(0,0)[tc]{\sc Fig.~2}}
+\end{picture}
+\end{figure}
+set of intervals $\interval{-1}{0}$, $\interval{\dfrac12}{1}$,
+$\interval{\dfrac 14}{\dfrac 12}, \ldots$, 
+$\interval{\dfrac{1}{2^n}}{\dfrac{1}{2^{n-1}}},\ldots$ covers the interval 
+$\interval{-1}{1}$,
+because every point of $\interval{-1}{1}$ is a point of one of the
+intervals. On the other hand a set of segments $\overline{-1\ 0}$,
+$\overline{\dfrac12\ 1},\ldots, \overline{\dfrac1{2^n}\
+\dfrac1{2^{n-1}}}$, etc., does not cover the interval because it does
+not include the points $-1$, $1$, $\dfrac 12,\ldots$,
+$\dfrac1{2^n},\ldots,$ or $0$.  In order to obtain a set of segments
+which does cover the interval, it is necessary to adjoin a set of
+segments, no matter how small, such that one includes $-1$, one
+includes $0$, one includes $1$, $\frac12$, $\frac14,\ldots$.
+
+The segment including $0$, no matter how small it is, must include an
+infinitude of the points $\dfrac{1}{2^n}$, and there are only a finite
+number of them which do not lie on that segment. It therefore follows
+that in this enlarged set there is a subset of segments,
+%-----File: 046.png---Folio 34-------
+finite in number, which includes all the points of \correction{$\interval{-1}{1}$}{$-\interval{1}{1}$}. 
+This turns out to be a general theorem, namely, that if any set
+of segments covers an interval, there is a finite subset of it which
+also covers the interval. The example we have just given shows that
+such a theorem is not true of the covering of an interval by a set of
+intervals; furthermore, it is not true of the covering of a segment
+either by a set of segments or by a set of intervals.
+
+\begin{theorem}[10\footnotemark]\footnotetext{%
+  This theorem is due to \textsc{E.~Borel}, Annales de l'{\'E}cole
+  Normale Sup{\'e}rieure, 3d~series, Vol.~12 (1895), p.~51. It is
+  frequently referred to as the \textsc{Heine-Borel} theorem, because
+  it is essentially involved in the proof of the theorem of uniform
+  continuity given by E.~Heine, \textit{Die Elemente der
+  Functionenlehre}, Crelle's Journal, Vol.~74 (1872), page~188.}
+\hypertarget{thm10}{}If an interval $\interval{a}{b}$ is covered by any set $[\sigma]$ of
+segments, it is covered by a finite number of segments
+$\sigma_1,\ldots,\sigma_n$ of $[\sigma]$.
+\end{theorem}
+
+\begin{proof}
+It is evident that at least a part of $\interval{a}{b}$
+is covered by a finite number of $\sigma$'s; for example, if
+$\sigma_0$ is the $\sigma$ or one of the $\sigma$'s which include $a$
+and if $b'$ is any point of $\rinterval{a}{b}$ which lies in
+$\sigma_0$, then $\interval{a}{b'}$ is covered by $\sigma_0$. Let
+$[b']$ be the set of all points of $\rinterval{a}{b}$, such that
+$\interval{a}{b'}$ is covered by a finite number of $\sigma$'s. By
+Theorem~\hyperlink{thm4}{4} $[b']$ has a least upper bound $B$. To complete our proof we
+show (\textit{a}) that $B$ is in $[b']$, and (\textit{b}) that $B=b$.
+\begin{enumerate}
+\item[(\textit{a})] Let $\overline{a''\ b''}$ be a segment of
+$[\sigma]$ including $B$. Since $B$ is the least upper bound of
+$[b']$, there is a point of $[b']$, $b'$, between $a''$ and $B$. But
+if $\sigma_1, \sigma_2, \ldots, \sigma_e,$ be the finite set of
+segments covering the interval $\interval{a}{b'}$, this set together
+with $\overline{a''\ b''}$ will cover $\interval{a}{B}$, which proves
+that $B$ is a point of $[b']$.
+
+\item[(\textit{b})] If $B\neq b$, then $B<b$ and the set $\sigma_1$,
+$\sigma_2$, $\ldots$, $\sigma_e$, together with $\overline{a''\ b''}$,
+would cover an interval $\interval{a}{c}$, where $c$ is a point
+between $B$ and $b''$; $c$ would therefore be a point of $[b']$, which
+is contrary to the hypothesis that $B$ is an upper bound of $[b']$.
+Hence $B=b$ and the theorem is proved.
+\end{enumerate}
+\end{proof}
+%-----File: 047.png---Folio 35-------
+
+An immediate consequence of this theorem is the following, which may
+be called the \index{Theorem of uniformity}\index{Uniformity}\emph{theorem of uniformity}.
+
+\begin{theorem}[11]\hypertarget{thm11}{}
+If an interval $\interval{a}{b}$ is covered by a set of segments
+$[\sigma]$, then $\interval{a}{b}$ may be divided into $N$ equal
+intervals such that each interval is entirely within a $\sigma$.
+\end{theorem}
+
+\begin{proof}
+By Theorem~\hyperlink{thm10}{10} $\interval{a}{b}$ is covered by a finite set of
+$\sigma$'s, $\sigma_1,\sigma_2,\ldots,\sigma_n$. The end points of
+these $\sigma$'s, together with $a$ and $b$, are a finite set of
+points. Let $d$ be the smallest distance between any two distinct
+points of this set. Because of the overlapping of the $\sigma$'s, any
+two points not in the same segment are separated by at least two end
+points. Therefore any two points whose distance apart is less than $d$
+must lie on the same segment of
+$\sigma_1,\sigma_2,\ldots,\sigma_n$. Now let $N$ be such that
+$\dfrac{b-a}{N}<d$, then each interval of length $\dfrac{b-a}{N}$ is
+contained in a $\sigma$.
+\end{proof}
+\begin{figure}[!htbp]\label{fig03}\hypertarget{fig03}{}
+\setlength{\unitlength}{0.01\textwidth}
+\centering
+\begin{picture}(100,11)(-50,-4)
+\scriptsize
+\put(-50,0){\line(1,0){100}}
+\put(-45,-1){\line(0,1){1}}
+\put(-45,-2){\makebox(0,0)[tc]{$a$}}
+\qbezier(-48,0)(-40,8)(-32,0)
+\put(-40,7){\makebox(0,0)[tc]{$\sigma_1$}}
+\qbezier(-36,0)(-27,8)(-18,0)
+\put(-27,7){\makebox(0,0)[tc]{$\sigma_2$}}
+\put(-18,-1){\line(0,1){1}}
+\qbezier(-24,0)(-17,4)(-10,0)
+\put(-17,7){\makebox(0,0)[tc]{$\sigma_3$}}
+\put(-16,-1){\line(0,1){1}}
+\put(-17,-2){\makebox(0,0)[tc]{$d$}}
+\qbezier(-16,0)(-7,6)(2,0)
+\put(-7,7){\makebox(0,0)[tc]{$\sigma_4$}}
+\qbezier(-2,0)(7,9)(16,0)
+\put(7,7){\makebox(0,0)[tc]{$\sigma_5$}}
+\qbezier(12,0)(18,5)(24,0)
+\put(18,7){\makebox(0,0)[tc]{$\sigma_6$}}
+\qbezier(20,0)(27,5)(36,0)
+\put(27,7){\makebox(0,0)[tc]{$\sigma_7$}}
+\qbezier(30,0)(39,5)(48,0)
+\put(39,7){\makebox(0,0)[tc]{$\sigma_8$}}
+\put(45,-1){\line(0,1){1}}
+\put(45,-2){\makebox(0,0)[tc]{$b$}}
+\normalsize
+\put(0,-4){\makebox(0,0)[tc]{\textsc{Fig.~3.}}}
+\end{picture}
+\end{figure}
+
+By this argument we have also proved the following:
+
+\begin{theorem}[12]\hypertarget{thm12}{}
+If an interval $\interval{a}{b}$ is covered by a set of segments, then
+there is a number $d$ such that for any two numbers $x_1$ and $x_2$
+such that $a\leqq x_1<x_2\leqq b$ and $|x_1-x_2|<d$, there is a
+segment $\sigma$ of $[\sigma]$ which contains both $x_1$ and $x_2$. In
+other words, any interval of length $d$ lies entirely within some
+$\sigma$.
+\end{theorem}
+
+The sense in which these are theorems of uniformity is the
+following. Any point $x$ of $\interval{a}{b}$, being within a segment
+$\sigma$, can be regarded as the middle point of an interval $i_x$ of
+length $l_x$ which is entirely within some $\sigma$. The length $l_x$
+is in general different for different points, $x$. Our theorem states
+that a value $l$ can be found which is effective as an $l_x$ for every
+$x$, i.e.,
+%-----File: 048.png---Folio 36-------
+\emph{uniformly over the interval $\interval{a}{b}$}. The distinction
+here drawn is one of the most important in rigorous analysis. It was
+first observed in connection with the theorem of uniform continuity;
+see page~\pageref{t48p89}.
+
+The presence of both end points of $\interval{a}{b}$ is essential, as
+is shown by the following example. $\rinterval{0}{1}$ is covered by
+the segments
+$\overline{\dfrac{1}{2}\ 2}$,
+$\overline{\dfrac{1}{4}\ 1}$,
+$\overline{\dfrac{1}{8}\ \dfrac{1}{2}}$, \ldots,
+$\overline{\dfrac{1}{2^n}\ \dfrac{1}{2^{n-2}}}$, $\ldots$,
+but as we take points nearer to 0, $l_x$ becomes smaller with the
+lower bound $0$, and no $l$ can be found which is effective for all
+points of $\rinterval{0}{1}$.  When the end points are absent it is
+possible, however, to modify the notion of covering, so that our
+theorem remains true. This is sufficiently indicated by the following
+theorem, which is an immediate consequence of Theorem~\hyperlink{thm10}{10}.
+
+\begin{theorem}[13]\hypertarget{thm13}{}
+If on a segment $\overline{a\ b}$ there exists any set $[\sigma]$ of
+segments such that
+\begin{enumerate}
+\item[\textnormal{(1)}]\hypertarget{item1p36}{} $[\sigma]$ includes a segment of which $a$ is an end point
+and a segment of which $b$ is an end point.
+\item[\textnormal{(2)}]\hypertarget{item2p36}{} Every point of the segment $\overline{a\ b}$ lies on one or
+more of the segments of the set $[\sigma]$.
+\end{enumerate}
+Then among the segments of the set $[\sigma]$ there exists a finite
+set of segments $\sigma_1$, $\sigma_2$, $\ldots$, $\sigma_n$ which
+satisfies conditions \hyperlink{item1p36}{\textnormal{(1)}} and \hyperlink{item2p36}{\textnormal{(2)}}.
+\end{theorem}
+
+The theorems which we have just proved can be generalized to space of
+any number of dimensions. A planar generalization of a segment is a
+parallelogram with sides parallel to the coordinate axes, the boundary
+being excluded. The planar generalization of an interval is the same
+with the boundary included.  The theorem of \textsc{Borel} becomes:
+
+\begin{theorem}[14]\hypertarget{thm14}{}
+If every point of the interior or boundary of a parallelogram $P$ is
+interior to at least one parallelogram $p$ of a set of parallelograms
+$[p]$, then every point of $P$ is interior to at least one
+parallelogram of a finite subset $p_1\ldots p_n$ of $[p]$.
+\end{theorem}
+
+\begin{proof}
+Let $x=0$, $x=a>0$, $y=0$, $y=b>0$ determine the boundary of $P$. Let
+$0\leqq y_1\leqq b$. Upon the interval $i$ of the line
+%-----File: 049.png---Folio 37-------
+$y=y_1$, cut off by $P$, those parallelograms of $[p]$ that include
+points of $i$ as interior points determine a set of segments $[\pi]$
+such that every point of $i$ is an interior point of one of these
+segments $\pi$. There is by Theorem~\hyperlink{thm10}{10} a finite subset of $[\pi]$,
+$\pi_1$ $\ldots$ $\pi_n$, including every point of $i$, and therefore
+a finite subset $p_1$ $\ldots$ $p_n$ of $[p]$, including as interior
+points every point of $i$. Moreover, since the number of $p_1$
+$\ldots$ $p_n$ is finite, they include in their interior all the
+points of a definite strip, e.g., the points between the lines
+$y=y_1-e$ and $y=y_1+e$.
+\begin{figure}[!htpb]\label{fig04}\hypertarget{fig04}{}
+\centering
+\setlength{\unitlength}{0.008\textwidth}
+\begin{picture}(120,50)(-60,-25)
+\scriptsize
+\put(-50,-20){\line(0,1){40}} \put(40,-20){\line(0,1){40}} \put(-50,-20){\line(1,0){90}} \put(-50,20){\line(1,0){90}}
+\put(-50,0){\line(1,0){97}}
+\dashline{1}(-50,4)(47,4)
+\dashline{1}(-50,-4)(47,-4)
+\put(-56,-5){\line(0,1){9}} \put(-56,4){\line(1,0){16}}
+  \put(-56,-5){\line(1,0){16}} \put(-40,-5){\line(0,1){9}}
+\put(-44,-7){\line(0,1){17}} \put(-44,10){\line(1,0){22}}
+  \put(-44,-7){\line(1,0){22}} \put(-22,-7){\line(0,1){17}}
+\put(-30,-9){\line(0,1){16}} \put(-30,7){\line(1,0){19}}
+  \put(-30,-9){\line(1,0){19}} \put(-11,-9){\line(0,1){16}}
+\put(-17,-4){\line(0,1){12}} \put(-17,8){\line(1,0){20}}
+  \put(-17,-4){\line(1,0){20}} \put(3,-4){\line(0,1){12}}
+\put(-5,-5){\line(0,1){10}} \put(-5,5){\line(1,0){12}}
+  \put(-5,-5){\line(1,0){12}} \put(7,-5){\line(0,1){10}}
+\put(6,-15){\line(0,1){30}} \put(6,15){\line(1,0){15}}
+  \put(6,-15){\line(1,0){15}} \put(21,-15){\line(0,1){30}}
+\put(20,-9){\line(0,1){18}} \put(20,9){\line(1,0){25}}
+  \put(20,-9){\line(1,0){25}} \put(45,-9){\line(0,1){18}}
+\put(-49,-21){\makebox(0,0)[tl]{$y=0$}}
+\put(-49,19){\makebox(0,0)[tl]{$y=b$}}
+\put(-54,-13){$x=0$}
+\put(36,-13){$x=a$}
+\put(48,4){$y_1+e$}
+\put(48,0){$y_1$}
+\put(48,-4){$y_1-e$}
+\normalsize
+\put(0,-22){\makebox(0,0)[tc]{\textsc{Fig.~4.}}}
+\end{picture}
+\end{figure}
+
+Thus for every $y_1$ $(0\leqq y_1\leqq b)$ we obtain a strip of the
+parallelogram $P$ such that every point of its interior is interior to
+one of a finite number of the parallelograms $[p]$. These strips
+intersect the $y$-axis in a set of segments that include every point
+of the interval $\interval{0}{b}$. There is therefore, by Theorem~\hyperlink{thm10}{10},
+a finite set of strips which includes every point in $P$. Since each
+strip is included by a finite number of parallelograms $p$, the whole
+parallelogram $P$ is included by a finite subset of $[p]$.
+\end{proof}
+
+The generalization of Theorems \hyperlink{thm11}{11} and \hyperlink{thm12}{12} is left to the reader.
+
+\section{Limit Points. Theorem of Weierstrass.}\hypertarget{chIIsec3}{}%[3]
+
+\begin{definition}
+A \emph{neighborhood} or \emph{vicinity}\index{Vicinity} of a point $a$ in a line (or
+simply a line neighborhood of $a$) is a segment of this line such that
+$a$ lies within the segment. We denote a line neighborhood
+%-----File: 050.png---Folio 38-------
+of a point $a$ by $V(a)$\index{Vofa@$V(a)$}. The symbol $V^*(a)$\index{Vstarofa@$V^*(a)$} denotes the set of all
+points of $V(a)$ except $a$ itself. The symbols $V(\infty)$ and
+$V^*(\infty)$ are both used to denote infinite segments $\overline{a\
++\infty}$, and $V(-\infty)$ and $V^*(-\infty)$ to denote infinite
+segments $\overline{-\infty\ a}$.\footnote{%
+This notation is taken
+from \textsc{Pierpont's} \textit{Theory of Functions of Real
+Variables}. It is used here, however, with a meaning slightly
+different from that of \textsc{Pierpont}.}
+
+\index{Neighborhood}A neighborhood of a point in a plane (or a plane neighborhood of a
+point) is the interior of a parallelogram within which the point
+lies. A neighborhood of a point $(a,b)$ is denoted by $V(a,b)$ if
+$(a,b)$ is included and by $V^*(a,b)$ if $(a,b)$ is excluded.  Instead
+of the three linear vicinities $V(a)$, $V(\infty)$, and $V(-\infty)$
+we have the following nine in the case of the plane:
+
+\begin{figure}[!hbtp]\label{fig05}\hypertarget{fig05}{}
+\centering
+\includegraphics{images/fig05}
+%\correction{$V(-\infty,-\infty)$}{$V(-\infty,\infty)$}
+\end{figure}
+%-----File: 051.png---Folio 39-------
+\end{definition}
+
+It follows at once from a consideration of the scheme for setting the
+points on the line into correspondence with all numbers that in every
+neighborhood of a point there is a point whose corresponding number is
+rational.
+
+\begin{definition}\index{Limit!point}
+A point $a$ is said to be a \textit{limit point} of a set if there are
+points of the set, other than $a$, in every neighborhood of $a$. In
+case of a line neighborhood this says that there are points of the set
+in every $V^*(a)$. In the planar case this is equivalent to saying
+that $(a,b)$ is a limit point of the set $[x,y]$, either if for every
+$V^*(a)$ and $V(b)$ there is an $(x,y)$ of which $x$ is in $V^*(a)$
+and $y$ in $V(b)$, or if for every $V(a)$ and $V^*(b)$ there is an
+$(x,y)$ of which $x$ is in $V(a)$ and $y$ in $V^*(b)$.
+\end{definition}
+
+Thus $0$ is a limit point of the set $\left[\tfrac{1}{2^k}\right]$,
+where $k$ takes all positive integral values. In this case the limit
+point is not a point of the set. On the other hand, in the set $1$,
+$1-\frac12$, $1-\frac{1}{2^2}$,\ldots, $1-\frac{1}{2^k}$, $1$ is a
+limit point of the set and also a point of the set. In this case $1$
+is the least upper bound of the set. In case of the set $1$, $2$, $3$,
+the number $3$ is the least upper bound without being a limit
+point. The fundamental theorem about limit points is the following
+(due to \textsc{Weierstrass}):
+
+\begin{theorem}[15]\hypertarget{thm15}{}
+Every infinite bounded set $[p]$ of points on a line has at least one
+limit point.
+\end{theorem}
+
+\begin{proof}
+Since the set $[p]$ is bounded, every one of its points lies on a
+certain interval $\interval{a}{b}$. If the set $[p]$ has no limit
+point, then about every point of the interval $\interval{a}{b}$ there
+is a segment $\sigma$ which contains not more than one point of the
+set $[p]$. By Theorem~\hyperlink{thm10}{10} there is a finite set of the segments
+$[\sigma]$ such that every point of $\interval{a}{b}$ and hence of
+$[p]$ belongs to at least one of them, but each $\sigma$ contains at
+most one point of the set $[p]$, whence $[p]$ is a finite set of
+points. Since this is contrary to the hypothesis, the assumption that
+there is no limit point is not tenable.
+\end{proof}
+%-----File: 052.png---Folio 40-------
+
+It is customary to say that a set which has no finite upper bound has
+the upper bound \index{Infinity as a limit}$+\infty$, and that one which has no finite lower
+bound has the lower bound $-\infty$. In these cases, since the set has
+a point in every $V^*(+\infty)$ or in every $V^*(-\infty)$ $+\infty$
+and $-\infty$ are also called limit points. With these conventions the
+theorem may be stated as follows:
+
+\begin{theorem}[16]\hypertarget{thm16}{}
+Every infinite set of points has a limit point, finite or infinite.
+\end{theorem}
+
+The theorem also generalizes in space of any number of dimensions. In
+the planar case we have:
+
+\begin{theorem}[17]\hypertarget{thm17}{}
+An infinite set of points lying entirely within a parallelogram has at
+least one limit point.
+\end{theorem}
+
+Theorem~\hyperlink{thm17}{17} is a corollary of the stronger theorem that follows:
+
+\begin{theorem}[18]\hypertarget{thm18}{}
+If $[(x,y)]$ is any set of number pairs and if $a$ is a limit point of
+the numbers $[x]$, there is a value of $b$, finite or $+\infty$ or
+$-\infty$, such that for every $V^*(a)$ and $V(b)$ there is an $(x,y)$
+of which $x$ is in $V^*(a)$ and $y$ is in $V(b)$.
+\end{theorem}
+
+\begin{proof}
+Suppose there is no value $b$ finite or $+\infty$ or $-\infty$ such as
+is required by the theorem. Since neither $+\infty$ nor $-\infty$
+possesses the property required of $b$, there is a $\overline{V^*}(a)$
+and a $V(\infty)$ and a $V(-\infty)$ such that for every pair $(x,y)$
+of $[(x,y)]$ whose $x$ lies in $\overline{V^*}(a)$ $y$ fails to lie in
+either $V(\infty)$ or $V(-\infty)$.  This means that there exists a
+pair of numbers $M$ and $m$ such that for every $(x, y)$ whose $x$ is
+in $\overline{V^*}(a)$ the $y$ satisfies the condition
+$m<y<M$. Further, since there exists no $b$ such as is required by the
+theorem, there is for every number $k$ on the interval $\interval{m}{M}$ 
+a $V(k)$ and a $V_k^*(a)$, such that for no $(x,y)$ is $x$ in
+$V_k^*(a)$ and $y$ in $V(k)$. This set of segments $[V(k)]$ covers the
+interval $\interval{m}{M}$, whence by Theorem~\hyperlink{thm10}{10} there is a finite
+subset of $[V(k)]$, $V_1(k)$, $\ldots$, $V_n(k)$ which covers
+$\interval{m}{M}$, and hence a finite set of corresponding
+$V_k^*(a)$'s. Let $\overline{\overline{V^*}}(a)$ be a vicinity of $a$
+contained in every one of the finite set of $V_k^*(a)$'s and in
+$\overline{V^*}(a)$.  Hence if the $x$ of a pair $(x,y)$ is in
+$\overline{\overline{V^*}}(a)$, its $y$ cannot lie in one
+%-----File: 053.png---Folio 41-------
+of the infinite segments $\overline{M\ \infty}$ and
+$\overline{-\infty\ m}$, or in one of the finite segments $V_1(k)$,
+$\ldots$, $V_n(k)$, i.e., no $y$ corresponds to this $x$, which is
+contrary to the hypothesis. This argument covers the cases when $a$ is
+$+\infty$ and when $a$ is $-\infty$.
+\end{proof}
+
+We add the definitions of a few of the technical terms that are used
+in point-set theory.\footnote{%
+  For bibliography and an exposition in English see
+  \textsc{W.~H.~Young} and \textsc{G.~C.~Young}, \textit{The Theory of
+  Sets of Points}.  Cambridge, The University Press.}
+
+\begin{definition}\index{Closed set}
+A set of points which includes all its limit points is called a
+\emph{closed} set.
+
+A set of points every one of which is a limit point of the set is
+called \index{Dense in itself}\emph{dense in itself}.\footnote{%
+  In German ``in sich dicht.''}
+
+A set of points which is both \emph{closed} and \emph{dense in itself}
+is called \label{dp41}\index{Perfect set}\emph{perfect}.
+
+A set having no finite limit point is called \index{Discrete set}\emph{discrete}.
+\end{definition}
+
+A segment not including its end points is an example of a set
+\emph{dense in itself} but not \emph{closed}. If the end points are
+added, the set is \emph{closed} and therefore \emph{perfect}. The set
+of rational numbers is another case of a set \emph{dense in itself}
+but not \emph{closed}. Any set containing only a finite number of
+points is \emph{closed}, according to our definition.
+
+If every point of an interval $\interval{a}{b}$ is a limit point of a
+set $[x]$, then $[x]$ is \index{Everywhere dense}\index{Dense}\emph{everywhere dense} on $\interval{a}{b}$. Such a set has a point between every two points of the
+interval. A set which is everywhere dense on no interval is called
+\index{Nowhere dense}\emph{nowhere dense}. All rational numbers between $0$ and $1$ form an
+\emph{everywhere dense} set.
+
+
+\section{Second Proof of Theorem~\protect\hyperlink{thm15}{15}.}%[4]
+
+To make the reader familiar with a style of argument which is
+frequently used in proving theorems which in this book are made to
+depend upon Theorems \hyperlink{thm10}{10} and \hyperlink{thm14}{14}, we adjoin the following lemma and base
+upon it another proof of Theorem~\hyperlink{thm15}{15}.
+%-----File: 054.png---Folio 42-------
+\begin{lemma}\label{lp42}\emph{Hypothesis:} On a straight line there is an infinite
+set of intervals $\interval{a_1}{b_1}$, $\interval{a_2}{b_2}$,
+$\ldots$, $\interval{a_n}{b_n}$, $\ldots$ conditioned as
+follows:\footnote{%
+  In particular the set of segments assumed in the hypothesis may be
+  obtained by dividing any given segment into a given number of equal
+  segments, then one of these segments into the same number of equal
+  segments and so on indefinitely. To show that the sequential
+  division into a number of equal segments gives a set of segments
+  satisfying the conditions of the hypothesis we have merely to show
+  that such division gives a segment less than any assigned segment
+  $\overline{a_e\ b_e}$. This is equivalent to the statement that for
+  every number $e$ there is an integer $n$, such that $\dfrac{1}{n}<e$
+  a direct consequence of Theorem~\hyperlink{thm3}{3}. This involves the notion that no
+  constant infinitesimal exists. It may appear at first sight that a
+  proof of this statement is superfluous. The fact is, however, as was
+  first proved by \textsc{Veronese}, that the non-existence of
+  constant infinitesimals is not provable without some axiom such as
+  the continuity axiom or the so-called Archimedean Axiom.}
+\begin{enumerate}
+
+\item[\textnormal{(1)}] Interval $\interval{a_2}{b_2}$ lies on interval
+$\interval{a_1}{b_1}$, $\interval{a_3}{b_3}$ on $\interval{a_2}{b_2}$,
+etc.  In general $\interval{a_n}{b_n}$ lies on $\interval{a_{n-1}}{b_{n-1}}$. 
+(This does not exclude the case $a_k=a_{k+1}$.)
+
+\item[\textnormal{(2)}] For every \correction{length}{interval} $e>0$, however small, there is some $n$,
+say $n_e$, such that $|b_{n_e}-a_{n_e}|< e$.
+\end{enumerate}
+\emph{Conclusion:} There is one and only one point $b$ which lies upon
+every interval $\interval{a_n}{b_n}$.
+\end{lemma}
+\begin{proof}
+Since the set of points $a_1\ldots a_n\ldots$ is bounded, we
+have at once, by the postulate of continuity, that this set has a
+leftmost right bound $\overline{B}_a$. Similarly, the set $b_1\ldots
+b_n\ldots$ has a rightmost left bound $\underline{B}_b$. It follows at
+once that $\overline{B}_a=\underline{B}_b$, for if not, we get either
+an $a$ point to the right of $\overline{B}_a$, or a $b$ point to the
+left of $\underline{B}_b$ when $n_e$ is so chosen that
+$|b_{n_e}-a_{n_e}|< \overline{B}_a-\underline{B}_b$.
+\end{proof}
+
+We now give another proof for Theorem~\hyperlink{thm11}{11}. Divide the interval
+$\interval{a}{b}$ on which all points of $[p]$ lie into two equal
+intervals.  Then there is an infinite number of points $[p]$ on at
+least one of these intervals which we call $\interval{a_1}{b_1}$. Divide this interval
+%-----File: 055.png---Folio 43-------
+into two equal parts and so on indefinitely, always selecting for
+division an interval which contains an infinite number of points of
+the set $[p]$. We thus obtain an infinite sequence of intervals
+$\interval{a_1}{b_1}$, $\interval{a_2}{b_2}$, $\ldots$,
+$\interval{a_n}{b_n} \ldots$ which satisfies the hypothesis of the
+lemma. There is therefore a point $B$ which belongs to every one of
+the intervals $\interval{a_1}{b_1}$, $\interval{a_2}{b_2}$, $\ldots$,
+$\interval{a_n}{b_n} \ldots$, and therefore there is a point of the
+set $[p]$ in every neighborhood of $B$.
+
+It should be noticed that the intervals in this sequence may be such
+that all intervals after a certain one will have, say, the right
+extremities in common. In this case the right extremity is the point
+$B$. Such is the sequence, obtained by decimal division, representing
+the number $2=1.99999 \ldots$.
+%-----File: 056.png---Folio 44-------
+
+
+\chapter{FUNCTIONS IN GENERAL\@. SPECIAL CLASSES OF FUNCTIONS.}\hypertarget{chapIII}{}%[III]
+
+\section{Definition of a Function.}\hypertarget{chIIIsec1}{}%[1]
+\index{Function}
+\begin{definition}\index{Constant}
+A \emph{variable} is a symbol which represents any one of a set of
+numbers. A \emph{constant} is a special case of a variable where the
+set consists of but one number.
+\end{definition}
+
+\begin{definition}
+A variable $y$ is said to be a \index{Single-valued functions}\emph{single-valued function} of
+another variable $x$ if to every value of $x$ there corresponds one
+and only one value of $y$. The letter $x$ is called the
+\emph{independent}\index{Variable!independent}\index{Independent variable} variable and $y$ the \emph{dependent}\index{Variable!dependent}\index{Dependent variable}
+variable.\footnote{%
+  \protect\hypertarget{fn}{}This definition of function is the culmination of a long development
+  of the use of the word. The idea of function arose in connection
+  with coordinate geometry, \textsc{Ren\'e Descartes} using the word
+  as early as 1637. From this time to that of \textsc{Leibnitz}
+  ``function'' was used synonymously with the word ``power,'' such as
+  $x^2$, $x^3$, etc.
+
+  \textsc{G.~W.~Leibnitz} regarded ``function'' as ``any expression
+  standing for certain lengths connected with a curve, such as
+  coordinates, tangents, radii of curvature, normals, etc.''
+
+  \textsc{Johann Bernoulli} (1718) defined ``function'' as ``an
+  expression made up of one variable and any constants whatever.''
+
+  \textsc{Leonard Euler} (1734) called the expression described by
+  \textsc{Bernoulli} an analytic function and introduced the notation
+  $f(x)$. \textsc{Euler} also distinguished between algebraic and
+  transcendental functions. He wrote the first treatise on ``The
+  Theory of Functions.''
+
+  The problem of vibrating strings led to the consideration of
+  trigonometric series. \textsc{J.~B.~Fourier} set the problem of
+  determining what kind of relations can be expressed by trigonometric
+  series. The possibility then under consideration that any relation
+  might be so expressed led \textsc{Lejeune Dirichlet} to state his
+  celebrated definition, which is the one given above. See the
+  Encyclop\"adie der mathematischen Wissenschaften, II~A.~1, pp.~3--5;
+  also \textsc{Ball}'s History of Mathematics, p.~378.}
+\end{definition}
+
+\begin{definition}\index{Many-valued function}
+A variable $y$ is said to be a many-valued function or multiple-valued
+function of another variable $x$ if to every value of $x$ there
+correspond one or more values of $y$.  The class of multiple-valued
+functions thus includes the class of single-valued
+functions.\hyperlink{fn}{\footnotemark[1]}
+\end{definition}
+%-----File: 057.png---Folio 45-------
+
+It is sometimes convenient to think of special values taken by these
+two variables as arranged in two tables, one table containing values
+of the independent variable and the other containing the corresponding
+values of the dependent variable.
+\begin{center}
+\begin{tabular}{r|l}
+  Independent Variable & Dependent Variable\\
+  \hline
+  $x_1$ & $y_1$\\
+  $x_2$ & $y_2$\\
+  $\,\cdot\,$ & $\,\cdot\,$ \\
+  $\,\cdot\,$ & $\,\cdot\,$\\
+  $\,\cdot\,$ & $\,\cdot\,$\\
+  $x_n$ & $y_n$
+\end{tabular}
+\end{center}
+
+If $y$ is a single-valued function of $x$, one and only one value of
+$y$ will appear in the table for each $x$. It is evident that
+functionality is a reciprocal relation; that is, if $y$ is a function
+of $x$, then $x$ is a function of $y$. It does not follow, however,
+that if $y$ is a single-valued function of $x$, then $x$ is a
+single-valued function of $y$, e.g., $y=x^2$. It is also to be noticed
+that such tables cannot exhibit the functional relation completely
+when the independent variable takes all values of the continuum, since
+no table contains all such values.
+
+\begin{definition}\label{dp45}
+That $y$ is a function of $x$ (and hence that $x$ is a function of
+$y$) is expressed by the equation $y=f(x)$ or by $x=f^{-1}(y)$. If $y$
+and $x$ are connected by the equation $y=f(x)$, \index{Function!inverse}\index{Inverse function}$f^{-1}(y)$ is called
+the inverse function of $f(x)$.
+\end{definition}
+
+Thus $y=x^2$ has the inverse function $x=\pm\sqrt y$. In this case,
+while the first function $y=x^2$ is defined for all real values of
+$x$, the inverse function $x = \pm\sqrt y$ is defined only for
+positive values of $y$.
+
+The independent variable may or may not take all values between any
+two of its values. Thus $n!$ is a function of $n$ where $n$ takes only
+integral values. $S_n$, the sum of the first
+%-----File: 058.png---Folio 46-------
+$n$ terms of a series, is a function of $n$ where $n$ takes only
+integral values. Again, the amount of food consumed in a city is a
+function of the number of people in the city, where the independent
+variable takes on only integral values. Or the independent variable
+may take on all values between any two of its values, as in the
+formula for the distance fallen from rest by a body in time $t$,
+$s=\dfrac{gt^2}{2}$.
+
+It follows from the correspondence between pairs of numbers and points
+in a plane that the functional relation between two variables may be
+represented by a set of points in a plane.  The points are so taken
+that while one of the two numbers which correspond to a point is a
+value of the independent variable, the other number is the
+corresponding value, or one of the corresponding values, of the
+dependent variable. Such representations are called \index{Function!graph of}\index{Graph of a function}graphs of the
+function. Cases in point where the function is single-valued are: the
+hyperbola referred to its asymptotes as axes
+$\left(y=\dfrac{1}{x}\right)$; a straight line not parallel to the $y$
+axis $(y=ax+b)$; or a broken line such that no line parallel to the
+$y$ axis contains more than one of its points.  In general, the graph
+of a single-valued function with a single-valued inverse is a set of
+points $[(x, y)]$ such that no two points have the same $x$ or the
+same $y$.
+
+Following is a graph of a function where the independent variable does
+not take all values between any two of its values.  Consider $S_n$,
+the sum of the first $n$ terms as a function of $n$ in the series
+\[
+  S = 1+\frac12+\frac{1}{2^2}+\ldots+\frac{1}{2^{n-1}}+\ldots.
+\]
+
+The numbers on the $x$ axis are the values taken by the independent
+variable, while the functional relation is represented by the points
+within the small circles. Thus it is seen that the graph of this
+function consists of a discrete set of points. (Fig.~\hyperlink{fig06}{6}.)
+%-----File: 059.png---Folio 47-------
+
+The definition of a function here given is very general. It will
+permit, for instance, a function such that for all rational values of
+the independent variable the value of the function is unity, and for
+irrational values of the independent variable the value of the
+function is zero.
+
+\begin{figure}[!hbtp]\label{fig06}\hypertarget{fig06}{}
+\centering
+\setlength{\unitlength}{0.08\textwidth}
+\begin{picture}(10,6)(0,-0.5)
+\thicklines
+\put(0,0){\line(1,0){10}}
+\put(0,0){\line(0,1){5.5}}
+\thinlines
+\multiput(1,0)(1,0){5}{\line(0,1){3}}
+\put(1,1.5){\circle{0.2}}
+\put(2,2.25){\circle{0.2}}
+\put(3,2.625){\circle{0.2}}
+\put(4,2.8125){\circle{0.2}}
+\put(5,2.90625){\circle{0.2}}
+\dashline{0.1}(0,3)(6,3)
+\put(10,0.25){\makebox(0,0)[br]{$x$}}
+\put(0.25,5.5){\makebox(0,0)[tl]{$y$}}
+\put(5,-0.25){\makebox(0,0)[tc]{\textsc{Fig.~6.}}}
+\end{picture}
+\end{figure}
+
+\section{Bounded Functions.}\hypertarget{chIIIsec2}{}%[2]
+
+Since the definition of function is so general there are few theorems
+that apply to all functions. If the restriction that $f(x)$ shall be
+bounded is introduced, we have at once a very important theorem.
+
+\begin{definition}\index{Bounds!upper and lower}
+\index{Function!upper and lower bound of}\index{Upper bound!of a function}\index{Lower bound!of a function}A function, $f(x)$, has an \textit{upper bound for a set of values
+$[x]$} of the independent variable if there exists a finite number $M$
+such that $f(x)<M$ for every value of $x$ in the set $[x]$. The
+function has a lower bound $m$ if $f(x)>m$ for every value of $x$ in
+$[x]$. A function which for a given set of values of $x$ has no \index{Infinity as a limit}finite
+upper bound is said to be \index{Function!unbounded}\index{Unbounded function}unbounded on that set, or to have an upper
+bound $+\infty$ on that set, and if it has
+%-----File: 060.png---Folio 48-------
+no lower bound on the set the function is said to have the lower bound
+$-\infty$ on the set.
+\end{definition}
+
+\begin{theorem}[19]\hypertarget{thm19}{}
+If on an interval $\interval{a}{b}$ a function has an upper bound $M$,
+then it has a least upper bound $\overline{B}$, and there is at least
+one value of $x$, $x_1$ on $\interval{a}{b}$ such that the least upper
+bound of the function on every neighborhood of $x_1$ contained in
+$\interval{a}{b}$ is $\overline{B}$.
+\end{theorem}
+
+\begin{proof}
+(1) The set of values of the function $f(x)$ form a bounded set of
+numbers. By Theorem~\hyperlink{thm4}{4} the set has a least upper bound $\overline{B}$.
+
+(2) Suppose there were no point $x_1$ on $\interval{a}{b}$ such that
+the least upper bound on every neighborhood of $x_1$ contained in
+\correction{$\interval{a}{b}$}{$\interval{a\text{---}}{!b}$} is
+$\overline{B}$. Then for every $x$ of $\interval{a}{b}$ there would be
+a segment $\sigma_x$ containing $x$ such that the least upper bound of
+$f(x)$ for values of $x$ common to $\sigma_x$ and $\interval{a}{b}$ is
+less than $\overline{B}$. The set $[\sigma_x]$ is infinite, but by
+Theorem~\hyperlink{thm10}{10} there exists a finite subset $[\sigma_n]$ of the set
+$[\sigma_x]$ covering $\interval{a}{b}$. Therefore, since the upper
+bound of $f(x)$ is less than $\overline{B}$ on that part of every one
+of these segments of $[\sigma_n]$ which lies on $\interval{a}{b}$, it
+follows that the least upper bound of $f(x)$ on $\interval{a}{b}$ is
+less than $\overline{B}$. Hence the hypothesis that no point $x_1$
+exists is not tenable, and there is a point $x_1$ such that the least
+upper bound of the function on every one of its neighborhoods which
+lies in $\interval{a}{b}$ is $\overline{B}$.
+\end{proof}
+
+This argument applies to multiple-valued as well as to single-valued
+functions.
+
+As an exercise the reader may repeat the above argument to prove the
+following:
+
+\begin{corollary}
+If on an interval $\interval{a}{b}$ a function has an upper bound
+$+\infty$, then there is at least one value of $x$, $x_1$ on
+$\interval{a}{b}$ such that in every neighborhood of $x_1$ the upper
+bound of the function is $+\infty$.
+\end{corollary}
+%-----File: 061.png---Folio 49-------
+\section{Monotonic Functions; Inverse Functions.}\hypertarget{chIIIsec3}{}%[3]
+
+\begin{definitions}\index{Decreasing function}\index{Function!monotonic!increasing}\index{Increasing function}\index{Function!monotonic!decreasing}\index{Monotonic function}
+If a single-valued function $f(x)$ on an interval $\interval{a}{b}$ is
+such that $f(x_1)<f(x_2)$ whenever $x_1<x_2$, the function is said to
+be \emph{monotonic increasing} on that interval. If $f(x_1)> f(x_2)$
+whenever $x_1<x_2$, the function is said to be \emph{monotonic
+decreasing}.
+\begin{figure}[!htbp]\label{fig07}\hypertarget{fig07}{}
+\centering
+\setlength{\unitlength}{0.025\textwidth}
+\begin{picture}(40,30)(0,-5)
+\put(0,0){\line(1,0){40}}
+\put(0,0){\line(0,1){25}}
+\put(2,10){\line(3,4){10}}
+\path(12,10)(17,14)(22,14)(27,24)
+\path(23,10)(27,16)(29,14)(34,18)(35,12)(38,22)
+\dashline{0.75}(29,14)(29,0)
+\dashline{0.75}(35,12)(35,0)
+\dashline{0.75}(31,15.6)(31,0)
+\put(0,-1){\makebox(0,0)[tl]{$0$}}
+\put(29,-1){\makebox(0,0)[tc]{$x_1$}}
+\put(31,-1){\makebox(0,0)[tc]{$x_2$}}
+\put(35,-1){\makebox(0,0)[tc]{$x_3$}}
+\put(40,1){\makebox(0,0)[bc]{$x$}}
+\put(1,25){\makebox(0,0)[tl]{$y$}}
+\put(20,-2){\makebox(0,0)[tc]{\sc Fig.~7.}}
+\end{picture}
+\end{figure}
+
+If there exist three values of $x$ on the interval $\interval{a}{b}$,
+$x_1$, $x_2$, and $x_3$ such that $f(x_2)>f(x_1)$ and $f(x_2)>f(x_3)$
+while $x_1<x_2<x_3$ or $f(x_2)<f(x_1)$ and $f(x_2)<f(x_3)$, while
+$x_1<x_2<x_3$, the function is said to be \index{Oscillation of a function}\index{Function!oscillating}\textit{oscillating} on that
+interval. A function which is \index{Non-oscillating function}not oscillating on an interval is called
+\index{Function!non-oscillating}\textit{non-oscillating}. It should be noticed that a function is not
+necessarily oscillating even if it is not monotonic. That is, it may
+be constant on some parts of the interval.
+\end{definitions}
+The terms monotonic and oscillating are not convenient of application
+to multiple-valued functions. Hence we restrict their use to
+single-valued functions.
+
+\begin{definition}
+A function $f(x)$ is said to have a finite number of oscillations on
+an interval $\interval{a}{b}$ if there exists a finite
+%-----File: 062.png---Folio 50-------
+number of points $a=x_0$, $x_1$, $\ldots$, $x_n=b$, such that on each
+interval $\interval{x_{k-1}}{x_k}$ ($k=1, 2, 3, \ldots, n$) $f(x)$ is
+non-oscillating. It is evident that if a function has only a finite
+number of oscillations on an interval $\interval{a}{b}$ and if there
+is no subinterval of $\interval{a}{b}$ on which the function is
+constant, then the interval $\interval{a}{b}$ may be subdivided into a
+finite set of intervals on each of which the function is
+monotonic. \index{Function!partitively monotonic}\index{Partitively monotonic}Such a function may be called \textit{partitively
+monotonic} (Abteilungsweise monoton).
+\end{definition}
+
+\begin{figure}[!hbtp]\label{fig08}\hypertarget{fig08}{}
+\centering
+\includegraphics{images/fig08}
+\end{figure}
+
+The function $f(x) = \sin\dfrac{1}{x}$, for $x\neq0$, and $f(x)=0$,
+for $x=0$, is an example of a function with an infinite number of
+oscillations on
+%-----File: 063.png---Folio 51-------
+every neighborhood of a point. $f(x) =x \sin\dfrac{1}{x}$, for $x\neq
+0$, $f(0)=0$, and $f(x) =x^2 \sin\dfrac{1}{x}$, for $x\neq 0$,
+$f(0)=0$ have the above property and also are continuous (see
+page~\pageref{dp61} for meaning of the term continuous function).
+
+\begin{figure}[!hbtp]\label{fig09}\hypertarget{fig09}{}
+\centering
+\includegraphics{images/fig09}
+\end{figure}
+
+\label{oscillp51}There exist continuous functions which have an infinite
+number of oscillations on every neighborhood of every point.  The
+first function of this type is probably the one discovered by
+Weierstrass,\footnote{%
+    According to F.~Klein, this function was discovered by Weierstrass
+    in 1851. See \textsc{Klein}, \textit{Anwendung der Differential-
+    und Integralrechnung auf Geometrie}, p.~83 et seq. The function
+    was first published in a paper entitled \textit{Abhandlungen aus
+    der Functionenlehre}, \textsc{Du Bois Reymond}, \textit{Crelle's
+    Journal}, Vol.~79, p.~29 (1874).}
+which is continuous over an interval and does not possess a derivative
+at any point on this interval (see page~\pageref{nowherediffp150}).
+%-----File: 064.png---Folio 52-------
+Other functions of this type have been published by \textsc{Peano},
+\textsc{Moore}, and others.\footnote{%
+    \textsc{G.~Peano}, \textit{Sur une courbe, qui remplit toute une
+    aire plane, Mathematische Annalen}, Vol.~36, pp.~157--160
+    (1890). \textsc{Cesaro}, \textit{Sur la repr\'esentation
+    analytique des r\'egions et des courbes qui les remplisent},
+    \textit{Bulletin des Sciences Math\'ematiques}, 2d~Ser., Vol.~21,
+    pp.~257--267. \textsc{E.~H.~Moore}, \textit{On Certain Crinkly
+    Curves}. \textit{Transactions of the American Mathematical
+    Society,} Vol.~1, pp.~73--90 (1899). See also \textsc{Steinitz},
+    \textit{Mathematische Annalen}, Vol.~52, pp.~58--69 (1899).}
+These latter investigators have obtained the function in question in
+connection with space-filling curves.
+
+\begin{theorem}[20]\hypertarget{thm20}{}
+If $y$ is a monotonic function of $x$ on the interval $\interval{a}{b}$, with bounds $A$ and $B$, then in turn $x$ is a single-valued
+monotonic function of $y$ on $\interval{A}{B}$, whose upper and lower
+bounds are $b$ and $a$.
+\end{theorem}
+
+\begin{figure}[!hbtp]\label{fig10}\hypertarget{fig10}{}
+\centering
+\includegraphics{images/fig10}
+\end{figure}
+
+\begin{proof}
+It follows from the monotonic character of $y$ as a function of $x$
+that for no two values of $x$ does $y$ have the same value. Hence for
+every value of $y$ on $\interval{A}{B}$ there exists one and
+%-----File: 065.png---Folio 53-------
+only one value of $x$. That is, $x$ is a single-valued function of
+$y$.\footnote{%
+  It is clear that the independent variable $y$ of the inverse
+  function may not take on all values of a continuum even if $x$ does
+  take on all such values.}
+Moreover, it is clear that for any three values of $y$, $y_1$, $y_2$,
+$y_3$, such that $y_2$ is between $y_1$ and $y_3$, the corresponding
+values of $x$, $x_1$, $x_2$, $x_3$, are such that $x_2$ is between
+$x_1$ and $x_3$, i.e., $x$ is a monotonic function of $y$, which
+completes the proof of the theorem.
+\end{proof}
+\begin{corollary}
+If a function $f(x)$ has a finite number $k$ of oscillations and is
+constant on no interval, then its inverse is at most
+$(k+1)$-valued. For example, the inverse of $y=x^2$ is double-valued.
+\end{corollary}
+
+\section[Rational, Exponential, and Logarithmic Functions.]{Rational, Exponential, and Logarithmic Functions.}\hypertarget{chIIIsec4}{}%[4]
+\label{s4p53}
+
+\begin{definitions}
+The symbol $a^m$, where $m$ is a positive integer and $a$ any real
+number whatever, means the product of $m$ factors $a$. This definition
+gives a meaning to the symbol
+\[
+  y=a_mx^m + a_{m-1}x^{m-1} + \ldots + a_1x + a_0,
+\]
+where $a_0 \ldots a_m$ are any real numbers and $m$ any positive
+integer.  In this case $y$ is called a \index{Rational!integral functions}\index{Function!rational integral}rational integral function of
+$x$ or a \index{Polynomial}polynomial in $x$.\footnote{%
+  The notion of polynomial finds its natural generalization in that of
+  a power series
+\[
+  y=c_0+c\cdot x+c_2\cdot x^2+ \ldots + c_nx^n+ \ldots
+\]
+
+  For conditions under which a series defines $y$ as a function of $x$
+  see Chapter~\hyperlink{chapIV}{IV}, \hyperlink{chIVsec3}{\S~3}.}
+
+In case
+\[
+  y = \frac{a_mx^m + a_{m-1}x^{m-1} + \ldots + a_1\cdot x + a_0}
+           {b_nx^n + b_{n-1}x^{n-1} + \ldots + b_1\cdot x + b_0},
+\]
+$m$ and $n$ being positive integers and $a_k$ ($k=0,\ldots m$) and
+$b_l\ (l=0,\ldots n)$ being real numbers, $y$ is called a \index{Rational!functions}\index{Function!rational}rational
+function of $x$.
+
+If\index{Algebraic!functions}\index{Function!algebraic}
+\[
+  y^n + y^{n-1}R_1(x) + y^{n-2}R_2(x) + \ldots
+  + yR_{n-1}(x) + R_n(x) = 0,
+\]
+where $R_1(x) \ldots R_n(x)$ are rational functions of $x$, then $y$
+is said to
+%-----File: 066.png---Folio 54-------
+be an algebraic function of $x$. Any function which is not algebraic
+is \index{Transcendental!functions}\index{Function!transcendental}transcendental.
+\end{definitions}
+The symbol $a^x$, where $x=\dfrac{m}{n}$, $m$ and $n$ being positive
+integers and $a$ any positive real number, is defined to be the $n$th
+root of the $m$th power of $a$. By elementary algebra it is easily
+shown that
+\[
+  a^{x_1} \cdot a^{x_2} = a^{x_1+x_2} \quad\text{and}\quad
+  (a^{x_1})^{x_2} = a^{x_1 \cdot x_2}.
+\]
+
+If
+\[
+  y=a^x,
+\]
+then $y$ is an \index{Exponential function}\index{Function!exponential}\textit{exponential} function of $x$. At present this
+function is defined only for rational values of $x$.
+
+\begin{figure}[!hbtp]\label{fig11}\hypertarget{fig11}{}
+\centering
+\includegraphics{images/fig11}
+\end{figure}
+
+\begin{theorem}[21]\hypertarget{thm21}{}
+The function $a^x$ for $x$ on the set $\left[ \dfrac{m}{n} \right]$ is
+a monotonic increasing function if $1<a$, and a monotonic decreasing
+function if $0<a<1$.
+\end{theorem}
+
+\begin{proof}\begin{enumerate}
+\item[(\textit{a})]For integral values of $x$ the theorem is obvious.
+\item[(\textit{b})] If $x_1=\dfrac{m_1}{n_1}$ and
+$x_2=\dfrac{m_2}{n_1}$, where $\dfrac{m_2}{n_1} > \dfrac{m_1}{n_1}$,
+then
+%-----File: 067.png---Folio 55-------
+$a^{x_1}<a^{x_2}$ if $a>1$ and $a^{x_1}>a^{x_2}$ if $a<1$. The proof
+of this follows at once from case ($a$), since
+$a^\frac{m_1}{n_1}=\left(a^\frac{1}{n_1}\right)^{m_1}$ (by definition
+and elementary algebra) and
+$a^\frac{m_2}{n_1}=\left({a^\frac{1}{n_1}}\right)^{m_2}$.
+\item[(\textit{c})] If $x_1=\dfrac{m_1}{n_1}$ and
+$x_2=\dfrac{m_2}{n_2}$, where $\dfrac{m_1}{n_1}<\dfrac{m_2}{n_2}$, we
+have $a^\frac{m_1}{n_1}=a^\frac{m_1{\cdot}n_2}{n_1{\cdot}n_2}$ and
+$a^\frac{m_2}{n_2}=a^\frac{m_2{\cdot}n_1}{n_2{\cdot}n_1}$, where
+$m_1{\cdot}n_2\text{\correction{$<$}{$>$}}m_2{\cdot}n_1$, which reduces case (\emph{c}) to case
+(\emph{b}).\qedhere
+\end{enumerate}
+\end{proof}
+
+This theorem makes it natural to define $a^x$, where $a>1$ and $x$ is
+a positive irrational number, as the least upper bound of all numbers
+of the form $\left[a^\frac mn\right]$, where \correction{$\left[\dfrac{m}{n}\right]$}{$\dfrac{m}{n}$} is the set
+of all positive rational numbers less than $x$, i.e., $a^x =
+\overline{B}\left[a^\frac mn\right]$. It is, however, equally natural
+to define $a^x$ as $\underline{B}\left[a^\frac pq\right]$, where
+$\left[\dfrac{p}{q}\right]$ is the set of all rational numbers greater
+than $x$. We shall prove that the two definitions are equivalent.
+
+\begin{lemma}
+If $[x]$ is the set of all positive rational numbers, then
+\begin{align*}
+  \underline{B}[a^x]&=1 \qquad \text{if } a>1\\
+\intertext{and}
+  \overline{B}[a^x]&=1 \qquad \text{if } a<1.
+\end{align*}
+\end{lemma}
+\begin{proof}
+We prove the lemma only for the case $a>1$, the argument in the other
+case being similar. If $x$ is any positive rational number,
+$\dfrac{m}{n}$, then the number $\dfrac{1}{n}$ is less than or equal
+to $x$, and since $a^x$ is a monotonic function, $a^\frac1n
+\qqle a^\frac mn$. But $\left[\dfrac{1}{n}\right]$ is a
+subset of $\left[\dfrac{1}{n}\right]$. Hence
+\[
+  \underline{B}[a^x]=\underline{B}\left[a^\frac1n\right],
+\]
+where $[n]$ is the set of all positive integers.
+\end{proof}
+%-----File: 068.png---Folio 56-------
+
+If $\underline{B}\left[ a^{\frac1n} \right]$ were less than $1$, then
+there would be a value, $n_1$, of $n$ such that
+$a^{\frac{1}{n_1}}<1$. This implies that $a<1$, which is contrary to
+the hypothesis. On the other hand, if
+$\underline{B}\left[a^{\frac1n}\right] > 1$, there is a number of the
+form $1+e$, where $e>0$, such that $1+e<a^{\frac1n}$ for every
+$n$. Hence $(1 +e)^n<a$ for every $n$, but by the binomial theorem for
+integral exponents
+\[
+  (1+e)^n>1+ne,
+\]
+and the latter expression is clearly greater than $a$ if
+\[
+  n>\frac ae.
+\]
+
+Since $\underline{B}\left[a^{\frac1n}\right]$ cannot be either greater
+or less than $1$,
+\[
+  \underline{B}\left[a^{\frac1n}\right] = 1.
+\]
+
+\begin{theorem}[22]\hypertarget{thm22}{}
+If $x$ is any real number, and $\left[ \dfrac{m}{n} \right]$ the set
+of all rational numbers less than $x$, and $\left[\dfrac{p}{q}\right]$
+the set of all rational numbers greater than $x$, then
+\begin{align*}
+    \overline{B}\left[a^{\frac{m\vphantom{p}}{n}}\right]
+  &= \underline{B}\left[a^{\frac pq}\right]
+  &&\text{if $a>1$,}
+\\
+   \underline{B}\left[a^{\frac{m\vphantom{p}}{n}}\right]
+  &= \overline{B}\left[a^{\frac pq}\right]
+  &&\text{if $0<a<1$.}
+\end{align*}
+\end{theorem}
+
+\begin{proof}
+We give the detailed proof only in the case $a>1$, the other case
+being similar. By the lemma, since
+$\underline{B}\left[\frac{p}{q}-\frac{m}{n}\right]$ is zero,
+\[
+  \underline{B}\left[ a^{\frac pq}-\text{\correction{$a^{\frac mn}$}{$a^m_n$}}\right]
+  = \underline{B}\left[
+  a^\frac pq \left(1-a^{\frac mn-\frac pq} \right) \right]
+\]
+is also zero. Now if
+\[
+  \overline{B}\left[a^{\frac{m\vphantom{p}}{n}}\right] \neq
+  \underline{B}\left[a^{\frac pq}\right],
+\]
+%-----File: 069.png---Folio 57-------
+since $a^{\frac pq}$ is always greater than $a^\frac mn$,
+\[
+  \underline{B}\left[a^\frac pq\right]-
+  \overline{B}\left[a^\frac{m\vphantom{p}}{n}\right] = \varepsilon > 0.
+\]
+
+But from this it would follow that
+\[
+  a^\frac pq-a^\frac mn
+\]
+is at least as great as $\varepsilon$, whereas we have proved that
+\[
+  \underline{B}\left[a^\frac pq-a^\frac mn \right] = 0.\\
+\]
+Hence
+\[
+  \overline{B}\left[a^\frac{m\vphantom{p}}{n}\right] =
+  \underline{B}\left[a^\frac pq\right]
+\]
+if $a>1$.
+\end{proof}
+
+\begin{definition}\label{dp57}In case $x$ is a positive irrational number,
+and $\left[\dfrac{p}{q}\right]$ is the set of all rational numbers
+greater than $x$, and $\left[\dfrac{m}{n}\right]$ is the set of all
+rational numbers less than $x$, then
+\begin{alignat*}{2}
+  a^x &= \underline{B}\left[a^\frac pq\right] =
+  \overline{B}\left[a^\frac{m\vphantom{p}}{n}\right]&\qquad&\text{if $a> 1$}\\
+\intertext{and}
+   a^x &= \overline{B}\left[a^\frac pq\right] =
+  \underline{B}\left[a^\frac{m\vphantom{p}}{n}\right]&&\text{if $0<a<1$.}
+\end{alignat*}
+Further, if $x$ is any negative real number, then
+\[
+  a^x = \frac{1}{a^{-x}} \quad\text{and}\quad a^0=1.
+\]
+\end{definition}
+
+\begin{theorem}[23]\hypertarget{thm23}{}
+The function $a^x$ is a monotonic increasing function of $x$ if $a>1$,
+and a monotonic decreasing function if $0<a<1$.  In both cases its
+upper bound is $+\infty$ and its lower bound is zero, the function
+taking all values between these bounds; further,
+\[
+  a^{x_1}\cdot a^{x_2}=a^{x_1+x_2} \quad\text{and}\quad
+ (a^{x_1})^{x_2}=a^{x_1\cdot x_2}.
+\]
+\end{theorem}
+
+The proof of this theorem is left as an exercise for the reader.  The
+proof is partly contained in the preceding theorems and
+%-----File: 070.png---Folio 58-------
+involves the same kind of argument about upper and lower bounds that
+is used in proving them.
+
+\begin{definition}\index{Logarithms}
+The \textit{logarithm} of $x$ ($x>0$) to the \textit{base} $a$ ($a>0$)
+is a number $y$ such that $a^y=x$, or $a^{\log_a x}=x$. That is, the
+function $\log_a x$ is the inverse of $a^x$. The identity
+\begin{align*}
+  a^{x_1} \cdot a^{x_2} &= a^{x_1 + x_2}\\
+\intertext{gives at once}
+\log_a x_1 + \log_a x_2 &= \log_a (x_1 \cdot x_2),
+\end{align*}
+and
+\[
+  (a^{x_1})^{x_2}=a^{x_1 \cdot x_2}\quad\text{gives}\quad
+  x_1\cdot \log_a x_2 = \log_a x_2^{x_1}.
+\]
+\end{definition}
+
+By means of Theorem~\hyperlink{thm20}{20}, the logarithm $\log_a x$, being the inverse of
+a monotonic function, is also a monotonic function, increasing if $1 <
+a$ and decreasing if $0<a<1$. Further, the function has the upper
+bound $+\infty$ and the lower bound $-\infty$, and takes on all real
+values as $x$ varies from $0$ to $+\infty$. Thus it follows that for
+$x<a$, $1<b$,
+\[
+  \overline{B}(\log_b x) = \log_b a = \log_b(\overline{B}x).
+\]
+By means of this relation it is easy to show that the function
+\[
+  x^a,\quad (x>0)
+\]
+is monotonic increasing for all values of $a$, $a>0$, that its lower
+bound is zero and its upper bound is $+\infty$, and that it takes on
+all values between these bounds.
+
+The proof of these statements is left to the reader. The general type
+of the argument required is exemplified in the following, by means of
+which we infer some of the properties of the function $x^x$.
+
+If $x_1<x_2$, then
+\begin{align*}
+  \log_2x_1&<\log_2x_2,\\
+\intertext{and}
+  x_1 \cdot \log_2 x_1 &< x_2 \cdot \log_2 x_2,\\
+\intertext{and}
+  \log_2 x_1^{x_1} &< \log_2 x_2^{x_2}.\\
+\therefore x_1^{x_1} &< x_2^{x_2}.
+\end{align*}
+%-----File: 071.png---Folio 59-------
+
+Hence $x^x$, $(x>0)$ is a monotonic increasing function of $x$.  Since
+the upper bound of $x\cdot\log_2x=\log_2x^x$ is $+\infty$, the upper
+bound of $x^x$ is $+\infty$. The lower bound of $x^x$ is not negative,
+since $x>0$, and must not be greater than the lower bound of $2^x$,
+since if $x<2$, $x^x<2^x$; since the lower bound of $2^x$ is
+zero\footnote{%
+    The lower bound of $a^x$ is zero by Theorem~\hyperlink{thm23}{23}.}
+the lower bound of $x^x$ must also be zero.
+
+Further theorems about these functions are to be found on pages
+\pageref{logp64}, \pageref{logp81}, \pageref{s4p97}, \pageref{p123},
+and \pageref{t101p160}.
+%-----File: 072.png---Folio 60-------
+
+
+
+\chapter{THEORY OF LIMITS.}\hypertarget{chapIV}{}%[IV]
+
+\section{Definitions. Limits of Monotonic Functions.}\hypertarget{chIVsec1}{}%[1]
+
+\begin{definition}
+If a point $a$ is a limit point of a set of values taken by a variable
+$x$, the variable is said \emph{to approach $a$ upon} the set; we
+denote this by the symbol $x\doteq a$. $a$ may be finite or $+\infty$
+or $-\infty$.
+\end{definition}
+
+In particular the variable may approach $a$ from the left or from the
+right, or in the case where $a$ is finite, the variable may take
+values on each side of the limit point. Even when the variable takes
+all values in some neighborhood on each side of the limit point it may
+be important to consider it first as taking the values on one side and
+then those on the other.
+
+\begin{definition}\index{Function!limit of}\index{Limit!of a function}
+A value $b$ ($b$ may be \index{Infinity as a limit}$+\infty$ or $-\infty$ or a finite number) is
+a \emph{value approached}\index{Value approached by!a function}\index{Function!value approached by} by $f(x)$ as $x$ approaches\index{Value approached by!the independent variable} $a$ if for every
+$V^*(a)$ and $V(b)$ there is at least one value of $x$ such that $x$
+is in $V^*(a)$ and $f(x)$ in $V(b)$. Under these conditions $f(x)$ is
+also said to approach $b$ as $x$ approaches $a$.
+\end{definition}
+
+\begin{definition}\index{Convergence!to a limit}
+If $b$ is the only value approached as $x$ approaches $a$, then $b$ is
+called \emph{the limit of $f(x)$} as $x$ approaches $a$.  This is also
+indicated by the phrase ``\emph{$f(x)$ converges to a unique limit
+$b$} as $x$ approaches $a$,'' or \index{Approach to a limit}``\emph{$f(x)$ approaches $b$ as a
+limit},'' or by the notation
+\[
+  \mathop{L}_{\text{\correction{$x\doteq a$}{$x=a$}}} f(x)=b.
+\]
+\end{definition}
+
+The function $f(x)$ is sometimes referred to as the \index{Limitand function}\emph{limitand}.
+The set of values taken by $x$ is sometimes indicated by the symbol
+for a limit, as, for example,
+%-----File: 073.png---Folio 61-------
+\begin{align*}
+  \mathop{L}_{\substack{x>a \\x\doteq a}} f(x)=b &&\text{or}
+&&\mathop{L}_{\substack{x<a \\x\doteq a}} f(x)=b &&\text{or}
+&&\mathop{L}_{\substack{x|[x]\\x\doteq a}} f(x)=b. &
+\end{align*}
+The first means that $x$ approaches $a$ from the right, the second
+that $x$ approaches $a$ from the left, and the third indicates that
+the approach is over some set $[x]$ otherwise defined.
+
+\begin{definition}\label{dp61}\index{Function!continuity of!at a point}\index{Continuity!at a point}\index{Continuous!function}
+If $f(x)$ is single-valued and converges to a finite limit as $x$
+approaches $a$ and
+\[
+  \mathop{L}_{x\doteq a} f(x)=f(a),
+\]
+then $f(x)$ is said to be \emph{continuous} at $x=a$.
+\end{definition}
+
+By reference to \hyperlink{chIIsec3}{\S~3}, Chapter~\hyperlink{chapII}{II}, the reader will see that if $b$ is a
+value approached by $f(x)$ as $x$ approaches $a$, then $(a, b)$ is a
+limit point of the set of points $(x, f(x))$. Theorem~\hyperlink{thm18}{18} therefore
+translates into the following important statement:
+
+\begin{theorem}[24]\hypertarget{thm24}{}
+If $f(x)$ is any function defined for any set $[x]$ of which $a$ is a
+(finite or $+\infty$ or $-\infty$) limit point, then there is at least
+one value (finite or $+\infty$ or $-\infty$) approached by $f(x)$ as
+$x$ approaches $a$.
+\end{theorem}
+
+\begin{corollary}
+If $f(x)$ is a bounded function, the values approached by $f(x)$ are
+all finite.
+\end{corollary}
+
+In the light of this theorem we see that the existence of
+\[
+  \mathop{L}_{x\doteq a} f(x)
+\]
+simply means that $f(x)$ approaches only one value, while the
+non-existence of
+\[
+  \mathop{L}_{x\doteq a} f(x)
+\]
+means that $f(x)$ approaches at least two values as $x$ approaches
+$a$.
+
+In case $f(x)$ is monotonic (and hence single-valued), or more
+generally if $f(x)$ is a non-oscillating function, these ideas are
+particularly simple. We have in fact the theorem:
+
+\begin{theorem}[25]\hypertarget{thm25}{}
+If $f(x)$ is a non-oscillating function for a set of values $[x] < a$,
+$a$ being a limit point of $[x]$, then as $x$ approaches $a$
+%-----File: 074.png---Folio 62-------
+from the left on the set $[x]$, $f(x)$ approaches one and only one
+value $b$, and if $f(x)$ is an increasing function,
+\[
+  b=\overline{B}f(x)
+\]
+for $x$ on $[x]$, whereas if $f(x)$ is a decreasing function,
+\[
+  b=\underline{B}f(x)
+\]
+for $x$ on $[x]$.
+\end{theorem}
+
+\begin{proof}
+Consider an increasing non-oscillating function and let
+\[
+  b=\overline{B}f(x)
+\]
+for $x$ on $[x]$.
+
+In view of the preceding theorem we need to prove only that no value
+$b'\neq b$ can be a value approached. Suppose $b'>b$; then since
+$\overline{B}f(x) =b$, there would be no value of $f(x)$ between $b$
+and $b'$, that is, there would be a $V(b')$ which could contain no
+value of $f(x)$, whence $b'>b$ is not a value approached. Suppose
+$b'<b$. Then take $b'<b''<b$, and since $\overline{B}f(x)=b$, there
+would be a value $x_1$ of $[x]$ such that $f(x_1)>b''$. If $x_1<x<a$,
+then $b''<f(x_1)\leqq f(x)$, because $f(x)$ cannot decrease as $x$
+increases.  This defines a $V^*(a)$ and a $V(b')$ such that if $x$ is
+in $V^*(a)$, $f(x)$ cannot be in $V(b')$. Hence $b'<b$ is not a value
+approached. A like argument applies if $f(x)$ is a decreasing
+function, and of course the same theorem holds if $x$ approaches $a$
+from the right.
+\end{proof}
+
+It does not follow that\index{Discontinuity}
+\[
+  \mathop{L}_{\substack{x<a\\x\doteq a}} f(x)
+= \mathop{L}_{\substack{x>a\\x\doteq a}} f(x),
+\]
+nor that either of these limits is equal to $f(a)$. A case in point is
+the following: Let the temperature of a cooling body of water be the
+independent variable, and the amount of heat given out in cooling from
+a certain fixed temperature be the dependent variable. When the water
+reaches the freezing-point
+%-----File: 075.png---Folio 63-------
+a great amount of heat is given off without any change in
+temperature. If the zero temperature is approached from below, the
+function approaches a definite limit point $k$, and if the temperature
+approaches zero from above, the function
+\begin{figure}[!htbp]\label{fig12}\hypertarget{fig12}{}
+\centering
+\setlength{\unitlength}{0.05\textwidth}
+\begin{picture}(10,10)(0,-1)
+\put(0,0){\line(1,0){10}}
+\put(0,0){\line(0,1){9}}
+\path(1,2)(4,5)(4,7)(8,9)
+\put(5,-0.5){\makebox(0,0)[tc]{\sc Fig.~12}}
+\put(0,9){\makebox(0,0)[tl]{Heat}}
+\put(8,0){\makebox(0,0)[bc]{Temp.}}
+\end{picture}
+\end{figure}
+approaches an entirely different point $k'$. \index{Discontinuity}This function, however,
+is multiple-valued at the zero point. A case where the limit fails to
+exist is the following: The function $y=\sin\ 1/x$; (see Fig.~\hyperlink{fig08}{8},
+page~\pageref{fig08}) approaches an infinite number of values as x approaches zero. The value of the function will be alternately $1$ and $-1$, as
+$x=\dfrac{2}{\pi}$, $\dfrac{2}{3\pi}$, $\dfrac{2}{5\pi}$ etc., and for
+all values of $x$ between any two of these the function will take all
+values between $1$ and $-1$. Clearly every value between $1$ and $-1$
+is a value approached as $x$ approaches zero. In like manner
+%-----File: 076.png---Folio 64-------
+$y = \dfrac{1}{x}\sin\dfrac{1}{x}$ approaches all values between and
+including $+\infty$ and $-\infty$, cf.\ Fig.~\hyperlink{fig13}{13}.
+
+\begin{figure}[!hbtp]\label{fig13}\hypertarget{fig13}{}
+\centering
+\includegraphics{images/fig13}
+\end{figure}
+
+The functions $a^x$, $\log_a x$\label{logp64}, $x^a$ defined in \hyperlink{chIIIsec4}{\S~4}
+of the \hyperlink{chapIII}{last chapter} are all monotonic and all satisfy the condition
+that
+\[
+  \mathop{L}_{\substack{x>a\\x\doteq a}}f(x)
+  = f(a) = \mathop{L}_{\substack{x<a\\x\doteq a}}f(x)
+\]
+at all points where the functions are defined. These functions are
+therefore all continuous.
+%-----File: 077.png---Folio 65-------
+
+
+\section{The Existence of Limits.}\hypertarget{chIVsec2}{}%[2]
+
+\begin{theorem}[26]\hypertarget{thm26}{}
+A \index{Necessary and sufficient condition}necessary and sufficient condition\footnote{%
+    This means:
+    \begin{enumerate}
+    \item[(\textit{a})] If $\displaystyle\mathop{L}_{x \doteq a} f(x) = b$, then for
+      every $V(b)$ there exists a $V^*(a)$, as specified by the theorem.
+    \item[(\textit{b})] If for every $V(b)$ there exists a $V^*(a)$ as specified,
+      then $\displaystyle\mathop{L}_{x \doteq a} f(x) = b$.
+    \end{enumerate}
+
+    A condition is necessary for a certain conclusion if it can be deduced
+    from that conclusion; a condition sufficient for a conclusion is one
+    from which the conclusion can be deduced. A man sufficient for a task
+    is a man who can perform the task, while a man necessary for the task
+    is such that the task cannot be performed without him.}
+that $f(x)$ shall converge to a unique limit $b$ as $x$ approaches
+$a$, i.e., that
+\[
+  \mathop{L}_{x \doteq a} f(x) = b,
+\]
+is that for every $V(b)$ there shall exist a $V^*(a)$ such that for
+every $x$ in $V^*(a)$, $f(x)$ is in $V(b)$.
+\end{theorem}
+
+\begin{proof}
+(1) \textit{The condition is necessary.} It is to be proved that if
+$\displaystyle\mathop{L}_{x \doteq a} f(x) = b$, then for every $V(b)$ there exists
+a $V^*(a)$ such that for every $x$ in $V^*(a)$ the corresponding
+$f(x)$ is in $V(b)$. If this conclusion did not follow, then for some
+$V(b)$ every $V^*(a)$ would contain at least one $x'$ such that
+$f(x')$ is not in $V(b)$.  There is thus defined a set of points
+$[x']$ of which $a$ is a limit point. By Theorem~\hyperlink{thm20}{20} $f(x)$ would
+approach at least one value $b'$ as $x$ approaches $a$ on the set
+$[x']$. But by the definition of $[x']$, $b'$ is distinct from
+$b$. Hence the hypothesis would be contradicted.
+
+(2) \textit{The condition is sufficient.} We need only to show that if
+for every $V(b)$ there exists a $V^*(a)$ such that for every $x$ in
+$V^*(a)$ the corresponding $f(x)$ is in $V(b)$, then $f(x)$ can
+approach no other value than $b$. If $b' \ne b$, then there exists a
+$\overline V(b')$ and a $\overline V(b)$ which have no point in
+common. Now if $\overline V^*(a)$ is such that for every $x$ of
+$\overline V^*(a)$, $f(x)$ is in $\overline V(b)$, then
+%-----File: 078.png---Folio 66-------
+for no such $x$ is $f(x)$ in $\overline{V}(b')$ and hence $b'$ is not
+a value approached.
+\end{proof}
+
+The reader should observe that this proof applies also to
+multiple-valued functions, although worded to fit the single-valued
+case. It is worthy of note that in case $b$ is a finite number, our
+theorem becomes:
+
+\textit{A necessary and sufficient condition that}
+\[
+  \mathop{L}_{x \doteq a} f(x) = b
+\]
+\textit{is that for every $\varepsilon>0$ there exists a
+${V_\varepsilon}^*(a)$ such that for every $x$ in
+${V_\varepsilon}^*(a)$, $|f(x)-b|< \varepsilon$.}
+
+In case $a$ also is finite, the condition may be stated in a form
+which is frequently used as the definition of a limit, namely:
+
+\textit{$\displaystyle\mathop{L}_{x \doteq a} f(x) =b$ means that for
+every $\varepsilon>0$ there exists a $\delta_\varepsilon >0$ such that
+if $|x-a|< \delta_\varepsilon$ and $x\neq a$, then $|f(x)-b|<
+\varepsilon$.}\footnote{%
+  The $\varepsilon$ subscript to $\delta_\varepsilon$ or to
+  ${V_\varepsilon}^*(a)$ denotes that $\delta_\varepsilon$ or
+  ${V_\varepsilon}^*(a)$ is a function of $\varepsilon$. It is to be
+  noted that inasmuch as any number less than $\delta_\varepsilon$ is
+  effective as $\delta_\varepsilon$, $\delta_\varepsilon$ is a
+  multiple-valued function of $\varepsilon$.}
+
+\begin{theorem}[27]\hypertarget{thm27}{}
+A necessary and sufficient condition that $f(x)$ shall converge to a
+finite limit as $x$ approaches $a$ is that for every $\varepsilon>0$
+there shall exist a ${V_\varepsilon}^*(a)$ such that if $x_1$ and
+$x_2$ are any two values of $x$ in ${V_\varepsilon}^*(a)$, then
+\[
+  |f(x_1)-f(x_2)|< \varepsilon.
+\]
+\end{theorem}
+
+\begin{proof}
+(1) \textit{The condition is necessary.} If $\displaystyle\mathop{L}_{x\doteq a}
+f(x)=b$ and $b$ is finite, then by the preceding theorem for every
+$\frac{\varepsilon}{2}>0$ there exists a $V^*(a)$ such that if $x_1$
+and $x_2$ are in $V^*(a)$, then
+\begin{align*}
+  |f(x_1)-b|&< \frac\varepsilon2 \\
+\intertext{and}
+  |f(x_2)-b|&< \frac\varepsilon2,
+\end{align*}
+from which it follows that
+\[
+  |f(x_1)-f(x_2)|< \varepsilon.
+\]
+%-----File: 079.png---Folio 67-------
+
+(2) \textit{The condition is sufficient.} If the condition is
+satisfied, there exists a $\overline{V^*}(a)$ upon which the
+function $f(x)$ is bounded.
+For let $\overline{\varepsilon}$ be some fixed number. By
+hypothesis there exists a $\overline{V^*}(a)$ such that if $x$ and
+$x_0$ are on $\overline{V^*}(a)$, then
+\[
+  |f(x)-f(x_0)|< \overline{\varepsilon}.
+\]
+Taking $x_0$ as a fixed number, we have that
+\[
+  f(x_0)-\overline{\varepsilon} < f(x) < f(x_0)
+  + \overline{\varepsilon}
+\]
+for every $x$ on $\overline{V^*}(a)$. Hence there is at least one
+\textit{finite} value, $b$, approached by $f(x)$. Now for every
+$\varepsilon>0$ there exists a $V_\varepsilon^*(a)$ such that if $x_1$
+and $x_2$ are any two \correction{values}{valves} of $x$ in
+$V_\varepsilon^*(a)$, $|f(x_1)-f(x_2)|< \varepsilon$. Hence by the
+definition of value approached there is an $x_\varepsilon$ of
+$V_\varepsilon^*(a)$ for which
+\begin{align*}
+  |f(x_\varepsilon)-b|&< \varepsilon\tag{\textit{a}}\\
+\intertext{and}
+  |f(x_\varepsilon)-f(x)|&< \varepsilon\tag{\textit{b}}
+\end{align*}
+for every $x$ of $V_\varepsilon^*(a)$. Hence, combining (\textit{a})
+and (\textit{b}), for every $x$ of $V_\varepsilon^*(a)$ we have
+\[
+  |f(x)-b|< 2\varepsilon,
+\]
+and hence by the preceding theorem we have
+\[
+  \mathop{L}_{x \doteq a} f(x)=b.\qedhere
+\]
+\end{proof}
+
+In case $a$ as well as $b$ is finite, Theorem~\hyperlink{thm27}{27} becomes:
+
+\textit{A necessary and sufficient condition that
+\[
+  \mathop{L}_{x\doteq a}f(x)
+\]
+shall exist and be finite is that for every $\varepsilon>0$ there
+exists a $\delta_\varepsilon > 0$ such that
+\[
+  |f(x_1)-f(x_2)|<\varepsilon
+\]
+%-----File: 080.png---Folio 68-------
+for every $x_1$ and $x_2$ such that
+\[
+  x_1 \neq a,\quad x_2\neq a,\quad
+  |x_1-a|< \delta_\varepsilon,\quad
+  |x_2-a|< \delta_\varepsilon.
+\]}
+
+In case $a$ is $+\infty$ the condition becomes:
+
+\textit{For every $\varepsilon >0$ there exists a $N_\varepsilon>0$ such
+that}
+\[
+  |f(x_1)-f(x_2)|<\varepsilon
+\]
+\textit{for every $x_1$ and $x_2$ such that $x_1>N_\varepsilon$,
+$x_2>N_\varepsilon$.}
+
+The necessary and sufficient conditions just derived have the
+following evident corollaries:
+
+\begin{ncorollary}[1]\hypertarget{cor1th27}{}
+The expression
+\[
+  \mathop{L}_{x \doteq a}f(x)=b,
+\]
+where $b$ is finite, is equivalent to the expression
+\[
+  \mathop{L}_{x \doteq a}(f(x)-b)=0,
+\]
+and whether $b$ is finite or infinite
+\[
+  \mathop{L}_{x \doteq a} f(x) =b \text{ is equivalent to }
+  \mathop{L}_{x \doteq a} (-f(x)) =-b.
+\]
+\end{ncorollary}
+\begin{ncorollary}[2]\hypertarget{cor2th27}
+The expressions
+\[
+  \mathop{L}_{x \doteq a} f(x) = 0 \text{ and }
+  \mathop{L}_{x \doteq a} |f(x)|= 0
+\]
+are equivalent.
+\end{ncorollary}
+\begin{ncorollary}[3]
+The expression
+\[
+  \mathop{L}_{x \doteq a} f(x)=b
+\]
+is equivalent to
+\[
+  \mathop{L}_{y \doteq 0} f(y+a)=b,
+\]
+where $y+a=x$.
+\end{ncorollary}
+%-----File: 081.png---Folio 69-------
+
+\begin{ncorollary}[4]
+The expression
+\[
+  \mathop{L}_{\stackrel{x < a}{x \doteq a}} f(x)=b
+\]
+is equivalent to
+\[
+  \mathop{L}_{z \doteq + \infty} f \left({a + \frac1z}\right) = b,
+\]
+where $z = \frac{1}{x-a}$.
+\end{ncorollary}
+The reader should verify these corollaries by writing down the
+necessary and sufficient condition for the existence of each
+limit. The following less obvious statement is proved in detail for
+the case when $b$ is finite, the case when $b$ is $+ \infty$ or
+$-\infty$ being left to the reader.
+
+\begin{ncorollary}[5]
+If
+\[
+  \mathop{L}_{x \doteq a} f(x) = b,
+\]
+then
+\[
+  \mathop{L}_{x \doteq a} |f(x)| = |b|.
+\]
+\end{ncorollary}
+\begin{proof}
+By the necessary condition of Theorem~\hyperlink{thm26}{26} for every $\varepsilon$ there
+exists a $V_{\varepsilon}^*(a)$ such that for every $x_1$ of
+$V_{\varepsilon}^*(a)$
+\[
+  |f(x_1)-b|< \varepsilon.
+\]
+If $f(x_1)$ and $b$ are of the same sign, then
+\[
+  \bigl||f(x_1)|-|b|\bigr|
+= |f(x_1)-b|< \varepsilon,
+\]
+and if $f(x_1)$ and $b$ are of opposite sign, then
+\[
+  \bigl||f(x_1)|-|b|\bigr|
+  < |f(x_1)-b|< \varepsilon.
+\]
+Hence, by the sufficient condition of Theorem~\hyperlink{thm26}{26},
+\[
+  \mathop{L}_{x \doteq a} |f(x)|
+\]
+exists and is equal to $|b|$.
+\end{proof}
+%-----File: 082.png---Folio 70-------
+
+\begin{ncorollary}[6]
+If a function $f(x)$ is continuous at $x=a$, then $|f(x)|$ is
+continuous at $x=a$.
+\end{ncorollary}
+It should be noticed that
+\begin{align*}
+  \mathop{L}_{x \doteq a} |f(x)|&= |b|\\
+  \intertext{is \textit{not equivalent} to}
+  \mathop{L}_{x \doteq a} f(x)&=b.
+\end{align*}
+Suppose $f(x) = +1$ for all rational values of $x$ and $f(x) =-1$ for
+all irrational values of $x$. Then $\displaystyle\mathop{L}_{x \doteq
+a} |f(x)|= +1$, but $\displaystyle\mathop{L}_{x \doteq a} f(x)$ does
+not exist, since both $+1$ and $-1$ are values approached by $f(x)$ as
+$x$ approaches any value whatever.
+
+\begin{definition}\index{Numbers!sequence of}\index{Sequence of numbers}
+Any set of numbers which may be written $[x_n]$, where
+\begin{align*}
+  n &= 0, 1, 2, \ldots, \kappa, \\
+  \text{or } \qquad n &= 0, 1, 2, \ldots, \kappa, \ldots,
+\end{align*}
+is called a \textit{sequence}.
+\end{definition}
+
+To the corollaries of this section may be added a corollary related to
+the definition of a limit.
+
+\begin{ncorollary}[7]
+If for every sequence of numbers $[x_n]$ having $a$ as a limit point,
+\[
+  \mathop{L}_{\substack{x|[x_n] \\ x \doteq a}} f(x)=b,
+  \quad\text{then}\quad \mathop{L}_{x \doteq a} f(x)=b.
+\]
+\end{ncorollary}
+\begin{proof}
+In case two values $b$ and $b_1$ were approached by $f(x)$ as $x$
+approaches $a$, then, as in the first part of the proof of Theorem~\hyperlink{thm26}{26},
+two sequences could be chosen upon one of which $f(x)$ approached $b$
+and upon the other of which $f(x)$ approached $b_1$.
+\end{proof}
+
+\section{Application to Infinite Series.}\hypertarget{chIVsec3}{}%[3]
+\index{Convergence!of infinite series}\index{Infinite series}\index{Series!infinite}
+The theory of limits has important applications to infinite series. An
+\textit{infinite series} is defined as an expression of the form
+%-----File: 083.png---Folio 71-------
+\[
+  \sum_{k=1}^\infty a_k = a_1 + a_2 + a_3 + \ldots + a_n + \ldots.
+\]
+If $S_n$ is defined as
+\[
+  a_1 + \ldots + a_n = \sum_{k=1}^n a_k,
+\]
+$n$ being any positive integer, then the sum of the series is
+defined\label{dp71} as
+\[
+  \mathop{L}_{n=\infty} S_n = S
+\]
+if this limit exists.
+
+If the limit exists and is finite, the series is said to be
+\index{Infinite series!convergence and divergence of}\index{Series!infinite!convergence and divergence of}\textit{convergent}.  If $S$ is infinite or if $S_n$ approaches more
+than one value as $n$ approaches infinity, then the series is
+\index{Divergence}\textit{divergent}. For example, $S$ is infinite if
+\[
+  \sum_{k=1}^\infty a_k = 1 + 1 + 1 + 1 \ldots,
+\]
+and $S_n$ has more than one value approached if
+\[
+  \sum_{k=1}^\infty a_k = 1-1 + 1-1 + 1 \ldots.
+\]
+It is customary to write
+\[
+  R_n=S-S_n.
+\]
+
+A necessary and sufficient condition for the convergence of an
+infinite series is obtained from Theorem~\hyperlink{thm27}{27}.
+
+(1) \textit{For every $\varepsilon > 0$ there exists an integer
+  $N_{\varepsilon}$, such that if $n > N_{\varepsilon}$ and $n' >
+  N_{\varepsilon}$ then}
+\[
+  |S_n-S_{n'}|< \varepsilon.
+\]
+
+This condition immediately translates into the following form:
+%-----File: 084.png---Folio 72-------
+
+(2) \textit{For every $\varepsilon>0$ there exists an integer
+  $N_\varepsilon$, such that if $n>N_\varepsilon$, then for every $k$}
+\[
+  |a_n + a_{n+1} + \ldots + a_{n+k}|< \varepsilon.
+\]
+
+\begin{corollary}\label{cp72}
+If $\sum\limits_{k=1}^\infty a_k$ is a convergent series, then
+$\displaystyle\mathop{L}_{k \doteq \infty} a_k=0$.
+\end{corollary}
+
+\begin{definition}\index{Absolute convergence of infinite series}
+A series
+\[
+  \sum_{k=0}^\infty a_k=a_0+a_1+ \ldots+a_n+ \ldots
+\]
+is said to be \textit{absolutely convergent} if
+\[
+  |a_0|+ |a_1|+ \ldots + |a_n| + \ldots
+\]
+is convergent.
+\end{definition}
+
+Since
+\[
+  |a_n + a_{n+1} +\ldots +a_{n+k}|
+  < |a_n|+ |a_{n+1}|+ \ldots|a_{n+k}|,
+\]
+the above criteria give
+
+\begin{theorem}[28]\hypertarget{thm28}{}
+A series is convergent if it is absolutely convergent.
+\end{theorem}
+
+\begin{theorem}[29]\hypertarget{thm29}{}
+If $\sum\limits_{k=0}^\infty b_k$ is a convergent series all of whose
+terms are positive and $\sum\limits_{k=0}^\infty a_k$ is a series such
+that for every $k$, $|a_k|\leqq b_k$, then
+\[
+  \sum_{k=0}^\infty a_k
+\]
+is absolutely convergent.
+\end{theorem}
+
+\begin{proof}
+By hypothesis
+\[
+  \sum_{k=0}^n|a_k|\leqq \sum_{k=0}^n b_k.
+\]
+%-----File: 085.png---Folio 73-------
+
+Hence
+\[
+  \sum_{k=0}^n|a_k|
+\]
+is bounded, and being an increasing function of $n$, the series is
+convergent according to Theorem~\hyperlink{thm25}{25}.
+\end{proof}
+
+This theorem gives a useful method of determining the convergence or
+divergence of a series, namely, by comparison with a known
+series. Such a known series is the \index{Geometric series}\index{Series!geometric}geometric series
+\[
+  a+ar+ar^2 + \ldots +ar^n+ \ldots,
+\]
+where $0 < r < 1$ and $a > 0$. In this series
+\[
+  \sum_{k=0}^n ar^k = a\frac{1-r^{n+1}}{1-r} < \frac{a}{1-r},
+\]
+which shows that the series is convergent. Moreover, it can easily be
+seen to have the sum $\dfrac{a}{1-r}$.
+
+If $r \qqge  1$, the geometric series is evidently
+divergent. This result can be used to prove the ``ratio-test'' for
+convergence.
+
+\begin{theorem}[30]\index{Ratio test for convergence of infinite series}
+\hypertarget{thm30}{}If there exists a number, $r$, $0<r<1$, such that
+\[
+  \left|\frac{a_n}{a_{n-1}} \right|< r
+\]
+for every integral value of $n$, then the series
+\hypertarget{eq1p73}{\[
+\tag{1}
+a_1 + a_2 + \ldots + a_n + \ldots
+\]}
+is absolutely convergent. If $\left|\frac{a_n}{a_{n-1}}
+\right|\qqge 1$ for every $n$, the series is divergent.
+\end{theorem}
+
+\begin{proof}
+The series \hyperlink{eq1p73}{(1)} may be written
+\[
+  a_1 +
+  a_1\frac{a_2}{a_1} +
+  a_1\frac{a_2}{a_1} \cdot \frac{a_3}{a_2} +
+  \ldots +
+  a_1\frac{a_2}{a_1} \ldots \frac{a_n}{a_{n-1}}
+\tag{2}
+\]
+%-----File: 086.png---Folio 74-------
+$\left|\dfrac{a_n}{a_{n-1}}\right|<r$, this is numerically less term
+by term than
+\[
+\tag{3}
+  a_1 + a_1r + a_1r^2 + \ldots + a_1r^n + \ldots
+\]
+and therefore converges absolutely.  If $\left|\dfrac{a_n}{a_{n-1}}
+\right|\geqq 1$, $a_n \geqq a_1$ for every $n$; hence, by the
+corollary, page~\pageref{cp72}, \hyperlink{eq1p73}{(1)} is divergent.
+\end{proof}
+
+Nothing is said about the case when
+\[
+  \left\vert \frac{a_n}{a_{n-1}} \right|< 1,
+\quad\text{but}\quad
+  \mathop{L}_{n \doteq \infty}
+  \left\vert \frac{a_n}{a_{n-1}} \right|= 1.
+\]
+It is evident that the ratio test need be applied only to terms beyond
+some fixed term $a_n$, since the sum of the first $n$ terms
+\[
+  a_1 + a_2 + \ldots + a_n
+\]
+may be regarded as a finite number $S_n$ and the whole series as
+\[
+  S_n + a_{n+1} + a_{n+2} + \ldots,
+\]
+i.e., a finite number plus the infinite series
+\[
+  a_{n+1} + a_{n+2} + \ldots.
+\]
+
+\section{Infinitesimals. Computation of Limits.}\hypertarget{chIVsec4}{}%[4]
+
+\begin{theorem}[31]\hypertarget{thm31}{}
+A necessary and sufficient condition that
+\[
+  \mathop{L}_{x \doteq a} f(x) = b
+\]
+is that for the function $\varepsilon(x)$ defined by the equation
+$f(x)=b + \varepsilon(x)$
+\[
+  \mathop{L}_{x \doteq a} \varepsilon(x) =0.
+\]
+\end{theorem}
+%-----File: 087.png---Folio 75-------
+\begin{proof}
+Take $\varepsilon(x)=f(x)-b$ and apply Theorem~\hyperlink{thm26}{26}. A special case of
+this theorem is: \textit{A necessary and sufficient condition for the
+convergence of a series to a finite value $b$ is that for every
+$\varepsilon>0$ there exists an integer $N_\varepsilon$, such that if
+$n>N_\varepsilon$ then $|R_n|< \varepsilon$.}
+\end{proof}
+
+\begin{definition}\label{dp75}A function $f(x)$ such that
+\[
+  \mathop{L}_{x \doteq a} f(x)=0
+\]
+is called an \index{Infinitesimals}\textit{infinitesimal} as $x$ approaches $a$.\footnote{%
+  No constant, however small if not zero, is an infinitesimal, the
+  essence of the latter being that it varies so as to approach zero as
+  a limit. Cf.\ Goursat, Cours d'Analyse, tome~I, p.~21, etc.}
+\end{definition}
+
+\begin{theorem}[32]\hypertarget{thm32}{}
+The sum, difference, or product of two infinitesimals is an
+infinitesimal.
+\end{theorem}
+
+\begin{proof}
+Let the two infinitesimals be $f_1(x)$ and $f_2(x)$. For every
+$\varepsilon$, $1> \varepsilon >0$, there exists a $V_1^*(a)$ for
+every $x$ of which
+\[
+  |f_1(x)|< \frac\varepsilon2,
+\]
+and a $V_2^*(a)$ for every $x$ of which
+\[
+  |f_2(x)|< \frac\varepsilon2.
+\]
+Hence in any $V^*(a)$ common to $V_1^*(a)$ and $V_2^*(a)$
+\begin{align*}
+& |f_1(x) + f_2(x)|\leqq
+  |f_1(x)|+ |f_2(x)|< \varepsilon,
+\\
+& |f_1(x)-f_2(x)|\leqq
+  |f_1(x)|+ |f_2(x)|< \varepsilon,
+\\
+& |f_1(x) \cdot f_2(x)|=
+  |f_1(x)|\cdot|f_2(x)|< \varepsilon.
+\end{align*}
+From these inequalities and Theorem~\hyperlink{thm26}{26} the conclusion follows.
+\end{proof}
+
+\begin{theorem}[33]\hypertarget{thm33}{}
+If $f(x)$ is bounded on a certain $\overline{V^*}(a)$ and
+$\varepsilon(x)$ is an infinitesimal as $x$ approaches $a$, then
+$\varepsilon(x)\cdot f(x)$ is also an infinitesimal as $x$ approaches
+$a$.
+\end{theorem}
+%-----File: 088.png---Folio 76-------
+
+\begin{proof}
+By hypothesis there are two numbers $m$ and $M$, such that $M>f(x)>m$
+for every $x$ on $\overline{V^*}(a)$. Let $k$ be the larger of $|m|$
+and $|M|$. Also by hypothesis there exists for every $\varepsilon$ a
+${V_\varepsilon}^*(a)$ within $\overline{V^*}(a)$ such that if $x$ is
+in ${V_\varepsilon}^*(a)$, then
+\begin{align*}
+  |\varepsilon(x)|&< \frac{\varepsilon}{k} \\
+\intertext{or}
+   k|\varepsilon(x)|&< \varepsilon.
+\end{align*}
+But for such values of $x$
+\[
+  |f(x)\cdot\varepsilon(x)|
+< k\cdot|\varepsilon(x)|< \varepsilon,
+\]
+and hence for every $\varepsilon$ there is a ${V_\varepsilon}^*(a)$
+such that for $x$ an ${V_\varepsilon}^*(a)$
+\[
+  |f(x)\cdot\varepsilon(x)|< \varepsilon.\qedhere
+\]
+\end{proof}
+
+\begin{corollary}
+If $f(x)$ is an infinitesimal and $c$ any constant, then $c \cdot
+f(x)$ is an infinitesimal.
+\end{corollary}
+
+\begin{theorem}[34]\hypertarget{thm34}{}
+If $\displaystyle \mathop{L}_{x \doteq a} f_1(x)=b_1$ and
+$\displaystyle \mathop{L}_{x \doteq a} f_2(x)=b_2$, $b_1$ and $b_2$
+being finite, then
+\begin{align*}
+\tag{$\alpha$}
+  &\mathop{L}_{x\doteq a} \{f_1(x) \pm f_2(x)\} = b_1 \pm b_2,
+\\
+\tag{$\beta$}
+  &\mathop{L}_{x\doteq a} \{f_1(x) \cdot f_2(x)\} = b_1 \cdot b_2;
+\\
+\intertext{and if $b_2\neq 0$,}
+\tag{$\gamma$}
+    &\mathop{L}_{x\doteq a} \frac{f_1(x)}{f_2(x)} = \frac{b_1}{b_2}
+\end{align*}
+\end{theorem}
+
+\begin{proof}
+According to Theorem~\hyperlink{thm31}{31}, we write
+\begin{align*}
+f_1(x) &= b_1 + \varepsilon_1(x),\\
+f_2(x) &= b_2 + \varepsilon_2(x),
+\end{align*}
+%-----File: 089.png---Folio 77-------
+where $\varepsilon_1(x)$ and $\varepsilon_2(x)$ are
+infinitesimals. Hence
+\begin{gather*}
+\tag{$\alpha'$}
+  f_1(x) + f_2(x) = b_1 + b_2 + \varepsilon_1(x) + \varepsilon_2(x),
+\\
+\tag{$\beta'$}
+  f_1(x)\cdot f_2(x) = b_1 \cdot b_2
++ b_1 \cdot \varepsilon_2(x) + b_2 \cdot \varepsilon_1(x)
++ \varepsilon_1(x) \cdot \varepsilon_2(x).
+\end{gather*}
+But by the preceding theorem the terms of $(\alpha')$ and $(\beta')$
+which involve $\varepsilon_1(x)$ and $\varepsilon_2(x)$ are
+infinitesimals, and hence the conclusions $(\alpha)$ and $(\beta)$ are
+established.
+
+To establish ($\gamma$), observe that by Theorem~\hyperlink{thm26}{26} there exists a
+$V^*(a)$ for every $x$ of which $|f_2(x)-b_2|< |b_2|$ and hence upon
+which $f_2(x)\neq 0$. Hence
+\[
+  \frac{f_1(x)}{f_2(x)}
+= \frac{b_1 + \varepsilon_1(x)}{b_2 + \varepsilon_2(x)}
+= \frac{b_1}{b_2} + \frac{b_2 \varepsilon_1(x)-b_1 \varepsilon_2(x)}
+                         {b_2 \{ b_2 + \varepsilon_2(x) \}},
+\]
+the second term of which is infinitesimal according to Theorems \hyperlink{thm32}{32} and
+\hyperlink{thm33}{33}.
+\end{proof}
+
+Some of the cases in which $b_1$ and $b_2$ are $\pm\infty$ are covered
+by the following theorems. The other cases ($\infty-\infty$,
+$\dfrac{\infty}{\infty}$, $\dfrac{0}{0}$, etc.), are treated in
+Chapter~\hyperlink{chapVI}{VI}.
+
+\begin{theorem}[35]\hypertarget{thm35}{}
+If $f_2(x)$ has a lower bound on some $V^*(a)$, and if
+\[
+  \mathop{L}_{x \doteq 0} f_1(x) = +\infty,
+\]
+then
+\[
+  \mathop{L}_{x \doteq 0} \{f_2(x) + f_1(x)\} = +\infty.
+\]
+\end{theorem}
+
+\begin{proof}
+Let $M$ be the lower bound of $f_2(x)$. By hypothesis, for every
+number $E$ there exists a $V_E^*(a)$ such that for $x$ on $V_E^*(a)$
+\[
+  f_1(x) > E-M.
+\]
+Since
+\[
+  f_2(x) > M,\\
+\]
+this gives
+\[
+  f_1(x) + f_2(x) > E,
+\]
+which means that $ f_1(x) + f_2(x)$ approaches the limit $+\infty$.
+\end{proof}
+%-----File: 090.png---Folio 78-------
+
+\begin{theorem}[36]\hypertarget{thm36}{}
+If $\displaystyle \mathop{L}_{x \doteq a} f_1(x) = + \infty$ or
+$-\infty$, and if $f_2(x)$ is such that for a
+$\overline{V^*}(a)$\correction{,}{} $f_2(x)$ has a lower bound greater
+than zero or an upper bound less than zero, then $\displaystyle
+\mathop{L}_{x \doteq a} \{ f_1(x) \cdot f_2(x)\}$ is definitely
+infinite; i.e., if $f_2(x)$ has a lower bound greater than zero and
+$\displaystyle \mathop{L}_{x \doteq a} f_1(x) = +\infty$, then
+$\displaystyle\mathop{L}_{x\doteq a}\{f_1(x)\cdot f_2(x)\} = +\infty$,
+etc.
+\end{theorem}
+
+\begin{proof}
+Suppose $f_2(x)$ has a lower bound greater than zero, say $M$, and
+that $\displaystyle \mathop{L}_{x \doteq a} f_1(x) = +\infty$. Then
+for every $E$ there exists a $V_E^*(a)$ within $\overline{V^*}(a)$
+such that for every $x_1$ of $V_E^*(a)$, $f_1(x_1) > \dfrac{E}{M}$,
+and therefore $f_1(x_1)\cdot f_2(x_1)\qqge  f_1(x_1)\cdot
+M>E$. Hence by the definition of limit of a function
+$\displaystyle\mathop{L}_{x\doteq a}\{f_1(x)\cdot f_2(x)\} =
++\infty$. If we consider the case where $f_2(x)$ has an upper bound
+less than zero, we have in the same manner $L \{f_1(x)\cdot f_2(x)\}
+=-\infty$. Similar statements hold for the cases in which
+$\displaystyle \mathop{L}_{x \doteq a} f_1(x) =-\infty$.
+\end{proof}
+
+\begin{corollary}
+If $f_2(x)$ is positive and has a finite upper bound and
+$\displaystyle \mathop{L}_{x \doteq a} f_1(x) = +\infty$, then
+\[
+  \mathop{L}_{x \doteq a} \frac{f_1(x)}{f_2(x)} = +\infty.
+\]
+\end{corollary}
+
+\begin{theorem}[37]\hypertarget{thm37}{}
+If $\displaystyle \mathop{L}_{x \doteq a} f(x)= +\infty$, then
+$\displaystyle \mathop{L}_{x \doteq a} \frac{1}{f(x)} = 0$, and there
+is a vicinity $V^*(a)$ upon which $f(x)>0$. Conversely, if
+$\displaystyle \mathop{L}_{x \doteq a} f(x) =0$ and there is a
+$V^*(a)$ upon which $f(x) > 0$, then $\displaystyle \mathop{L}_{x
+\doteq a} \frac{1}{f(x)} = +\infty$.
+\end{theorem}
+
+\begin{proof}
+If $\displaystyle \mathop{L}_{x \doteq a} f(x) = +\infty$, then for
+every $\varepsilon$ there exists a ${V_\varepsilon}^*(a)$ such that if
+$x$ is in ${V_\varepsilon}^*(a)$, then
+\[
+  f(x) > \frac{1}{\varepsilon}
+\]
+%-----File: 091.png---Folio 79-------
+and
+\[
+  \frac{1}{f(x)} < \varepsilon.
+\]
+\[
+  \therefore \mathop{L}_{x \doteq a} \frac{1}{f(x)} = 0,
+\]
+since both $f(x)$ and $\dfrac{1}{f(x)}$ are positive.
+
+Again, if $\displaystyle \mathop{L}_{x \doteq a} f(x) =0$, then for
+every $\varepsilon$ there is a $\overline{V_\varepsilon^*}(a)$ such
+that for $x$ in $\overline{V_\varepsilon^*}(a)$, $|f(x)|<\varepsilon$
+or $\dfrac{1}{f(x)}>\dfrac{1}{\varepsilon}$ ($f(x)$ being positive).
+Hence
+\[
+  \mathop{L}_{x \doteq a} \frac{1}{f(x)} = + \infty.\qedhere
+\]
+\end{proof}
+
+\begin{ncorollary}[1]
+If $f_1(x)$ has finite upper and lower bounds on some $V^*(a)$ and
+$\displaystyle \mathop{L}_{x \doteq a} f_2(x) = +\infty$ or $-\infty$,
+then
+\[
+  \mathop{L}_{x \doteq a} \frac{f_1(x)}{f_2(x)} = 0.
+\]
+\end{ncorollary}
+\begin{ncorollary}[2]
+If $f_2(x)$ is positive and $f_1(x)$ has a positive lower bound on
+some $V^*(a)$ and $\displaystyle \mathop{L}_{x \doteq a} f_2(x)=0$,
+then
+\[
+  \mathop{L}_{x \doteq a} \frac{f_1(x)}{f_2(x)} = +\infty.
+\]
+\end{ncorollary}
+\begin{theorem}[38]\index{Change of variable}\emph{(change of variable).}\hypertarget{thm38}{} If
+\begin{enumerate}
+\item[\textnormal{(1)}]
+$\displaystyle\mathop{L}_{x \doteq a } f_1(x) = b_1$
+and
+$\displaystyle\mathop{L}_{x \doteq b_1} f_2(y) = b_2$
+when $y$ takes all valves of $f_1(x)$ corresponding to values of $x$ on
+some $\overline{V^*}(a)$, and if
+\item[\textnormal{(2)}]\hypertarget{item2p79}
+$\displaystyle f_1(x) \neq b_1 \text{ for } x \text{ on } \overline{V^*}(a)$,
+\end{enumerate}
+then
+\[
+ \mathop{L}_{x \doteq a} f_2(f_1(x)) = b_2.
+\]
+\end{theorem}
+%-----File: 092.png---Folio 80-------
+
+\begin{proof}
+($\alpha$) Since $\displaystyle \mathop{L}_{y \doteq b_1} f_2(y)
+=b_2$, for every $V(b_2)$ there exists a $V^*(b_1)$ such that if $y$
+is in $V^*(b_1)$, $f_2(y)$ is in $V(b_2)$. Since $\displaystyle
+\mathop{L}_{x \doteq a}f_1(x) =b_1$, for every $V(b_1)$ there exists a
+$V^*(a)$ in $\overline{V^*}(a)$ such that if $x$ is in $V^*(a)$,
+$f_1(x)$ is in $V(b_1)$. But by \hyperlink{item2p79}{(2)} if $x$ is in $V^*(a)$, $f_1(x)\neq
+b$. Hence ($\beta$) for every $V^*(b_1)$ there exists a $V^*(a)$ such
+that for every $x$ in $V^*(a)$, $f_1(x)$ is in $V^*(b_1)$.
+
+Combining statements ($\alpha$) and ($\beta$): for every $V(b_2)$
+there exists a $V^*(a)$ such that for every $x$ in $V^*(a)$ $f_1(x)$
+is in $V^*(b_1)$, and hence $f_2(f(x))$ is in $V(b_2)$. This means,
+according to Theorem~\hyperlink{thm26}{26}, that
+\[
+  \mathop{L}_{x \doteq a} f_2(f_1(x)) = b_2.\qedhere
+\]
+\end{proof}
+
+\begin{theorem}[39]\hypertarget{thm39}{}
+If $\displaystyle \mathop{L}_{x \doteq a} f_1(x) =b$ and
+$\displaystyle \mathop{L}_{y \doteq b} f_2(y) =f_2(b)$, where $y$
+takes all values taken by $f_1(x)$ for $x$ on some
+$\overline{V^*}(a)$, then
+\[
+  \mathop{L}_{x \doteq a} f_2(f_1(x)) = f_2(b).
+\]
+\end{theorem}
+
+\begin{proof}
+The proof of the theorem is similar to that of Theorem~\hyperlink{thm38}{38}.  In this
+case the notation $f_2(b)$ implies that $b$ is a finite number. Thus
+for every $\varepsilon_1$ there exists a ${V_{\varepsilon_1}}^*(a)$
+entirely within $\overline{V^*}(a)$ such that if $x$ is in
+${V_{\varepsilon_1}}^*(a)$,
+\[
+  |f_1(x)-b|< \varepsilon_1.
+\]
+
+Furthermore, for every $\varepsilon_2$ there exists a
+$\delta_{\varepsilon_2}$ such that for every $y$, $y \neq b$,
+$|y-b|<\delta_{\varepsilon_2}$,
+\[
+  |f_2(y)-f_2(b)|< \varepsilon_2.
+\]
+But since $|f_2(y)-f_2(b)|= 0$ when $y = b$, this means that for all
+values of $y$ (equal or unequal to $b$) such that $|y-b|<
+\delta_{\varepsilon_2}$, $|f_2(y)-f_2(b)|< \varepsilon_2$. Now let
+$\varepsilon_1 = \delta_{\varepsilon_2}$; then, if $x$ is in
+${V_{\varepsilon_1}}^*(a)$, it follows that $|f_1(x)-b|<
+\delta_{\varepsilon_2}$ and therefore that
+\[
+  |f_2(f_1(x))-f_2(b)|< \varepsilon_2.
+\]
+Hence
+\[
+  \mathop{L}_{x \doteq a} f_2(f_1(x)) = f_2(b).\qedhere
+\]
+\end{proof}
+%-----File: 093.png---Folio 81-------
+\begin{ncorollary}[1]\hypertarget{cor1p81}{}
+If $f_1(x)$ is continuous at $x=a$, and $f_2(y)$ is continuous at
+$y=f_1(a)$, then $f_2(f_1(x))$ is continuous at $x=a$.
+\end{ncorollary}
+
+\begin{ncorollary}[2]\hypertarget{cor2p81}{}
+If $k \neq 0$, $f(x) \geqq 0$, and $\displaystyle \mathop{L}_{x \doteq
+a} f(x) =b$, then
+\[
+  \mathop{L}_{x \doteq a} (f(x))^k = b^k,
+\]
+under the convention that $\infty^k = \infty$ if $k>0$ and
+$\infty^k=0$ if $k<0$.
+\end{ncorollary}
+
+\begin{ncorollary}[3]\label{logp81}
+If $c>0$ and $f(x)>0$ and $b>0$ and $\displaystyle \mathop{L}_{x
+\doteq a} f(x)=b$, then
+\[
+  \mathop{L}_{x \doteq a} \log_c f(x) = \log_c b,
+\]
+under the convention that $\log_c (+\infty) = +\infty$ and $\log_c 0
+=-\infty$.
+\end{ncorollary}
+
+The conclusions of the last two corollaries may also be expressed by
+the equations
+\[
+   \mathop{L}_{x \doteq a} (f(x))^k
+= (\mathop{L}_{x \doteq a} f(x))^k
+\]
+and
+\[
+  \log_c \mathop{L}_{x \doteq a} f(x)
+= \mathop{L}_{x \doteq a} \log_c f(x).
+\]
+
+\begin{ncorollary}[4]
+If $\displaystyle \mathop{L}_{x \doteq a} (f(x))^k$ or $\displaystyle
+\mathop{L}_{x \doteq a} \log f(x)$ fails to exist, then $\displaystyle
+\mathop{L}_{x \doteq a} f(x)$ does not exist.
+\end{ncorollary}
+
+
+\section{Further Theorems on Limits.}\hypertarget{chIVsec5}{}%[5]
+
+\begin{theorem}[40]\hypertarget{thm40}{}
+If $f(x) \leqq b$ for all values of a set $[x]$ on a certain $V^*(a)$,
+then every value approached by $f(x)$ as $x$ approaches $a$ is less
+than or equal to $b$. Similarly if $f(x) \geqq b$ for all values of a
+set $[x]$ on a certain $V^*(a)$, then every value approached by $f(x)$
+as $x$ approaches $a$ is greater than or equal to $b$.
+\end{theorem}
+
+\begin{proof}
+If $f(x) \leqq b$ on $V^*(a)$, then if $b'$ is any value greater than
+$b$, and $V(b')$ any vicinity of $b'$ which does not include $b$,
+there is no value of $x$ on $V^*(a)$ for which $f(x)$ is in
+$V(b')$. Hence $b'$ is not a value approached. A similar argument
+holds for the case where $f(x) \geqq b$.
+\end{proof}
+%-----File: 094.png---Folio 82-------
+
+\begin{ncorollary}[1]\hypertarget{cor1p82}{}
+If $f(x)\geqq 0$ in the neighborhood of $x=a$, then if
+\[
+  \mathop{L}_{x\doteq a} f(x)\text{ exist, }
+  \mathop{L}_{x\doteq a} f(x) \geqq 0.
+\]
+\end{ncorollary}
+\begin{ncorollary}[2]\hypertarget{cor2p82}{}
+If $f_1(x)\geqq f_2(x)$ in the neighborhood of $x=a$, then
+\[
+  \mathop{L}_{x\doteq a} f_1(x) \geqq
+  \mathop{L}_{x\doteq a} f_2(x)
+\]
+if both these limits exist.
+\end{ncorollary}
+\begin{proof}
+Apply Corollary~\hyperlink{cor1p82}{1} to $f_1(x)-f_2(x)$.
+\end{proof}
+
+\begin{ncorollary}[3]\hypertarget{cor3p82}{}
+If $f_1(x)\geqq f_2(x)$ in the neighborhood of $x=a$, then the largest
+value approached by $f_1(x)$ is greater than or equal to the largest
+value approached by $f_2(x)$.
+\end{ncorollary}
+\begin{ncorollary}[4]\hypertarget{cor4p82}{}
+If $f_1(x)$ and $f_2(x)$ are both positive in the neighborhood of
+$x=a$, and if $f_1(x)\geqq f_2(x)$, then if
+$\displaystyle\mathop{L}_{x\doteq a} f_1(x)=0$, it follows that
+\[
+  \mathop{L}_{x\doteq a} f_2(x)=0.
+\]
+\end{ncorollary}
+\begin{theorem}[41]\hypertarget{thm41}{}
+If $[x']$ is a subset of $[x]$, $a$ being a limit point of $[x']$, and
+if $\displaystyle\mathop{L}_{x\doteq a} f(x)$ exists, then
+$\displaystyle\mathop{L}_{\text{\correction{$x'$}{$x$}}\doteq a} f(x')$ exists and
+\[
+  \mathop{L}_{x\doteq a} f(x)= \mathop{L}_{x'\doteq a} f(x').%
+\footnote{The notation $f(x')$ is used to indicate that $x$ takes
+  the values of the set $[x']$.}
+\]
+\end{theorem}
+
+\begin{proof}
+By hypothesis there exists for every $V(b)$ a $V^*(a)$ such that for
+every $x$ of the set $[x]$ which is in $V^*(a)$, $f(x)$ is in
+$V(b)$. Since $[x']$ is a subset of $[x]$, the same $V^*(a)$ is
+evidently efficient for $x$ on $[x']$.
+\end{proof}
+
+In the statement of necessary and sufficient conditions for the
+existence of a limit we have made use of a certain positive
+multiple-valued function of $\varepsilon$ denoted by
+$\delta_\varepsilon$. If a given value is effective as a
+$\delta_\varepsilon$, then every positive value smaller than this is
+also effective.
+
+\begin{theorem}[42]\hypertarget{thm42}{}
+For every $\varepsilon$ for which the set of values of
+$\delta_\varepsilon$ has an upper bound there is a greatest
+$\delta_\varepsilon$.
+\end{theorem}
+%-----File: 095.png---Folio 83-------
+
+\begin{proof}
+Let $\overline{B}[\delta_\varepsilon]$ be the least upper bound of the
+set of values of $\delta_\varepsilon$, for a particular
+$\varepsilon$. If $x$ is such that $|x-a|<
+\overline{B}[\delta_\varepsilon]$, then there is a
+$\delta_\varepsilon$ such that $|x-a|< \delta_\varepsilon$. But if
+$|x-a|< \delta_\varepsilon$, $|f(x)-b|< \varepsilon$. Hence, if
+$|x-a|< \overline{B}[\delta_\varepsilon]$, $|f(x)-b|< \varepsilon$.
+\end{proof}
+
+\begin{theorem}[43]\hypertarget{thm43}{}
+The limit of the least upper bound of a function $f(x)$ on a variable
+segment $\overline{a\ x}$, $a < x$, as the end point approaches $a$,
+is the least upper bound of the values approached by the function as
+$x$ approaches $a$ from the right.
+\end{theorem}
+
+\begin{proof}
+Let $l$ be the least upper bound of the values approached by the
+function as $x$ approaches $a$ from the right, and let $b(x)$
+represent the upper bound of $f(x)$ for all values of $x$ on
+$\overline{a\ x}$. Since $\overline{B}f(x)$ on the segment
+$\overline{a\ x_1}$ is not greater than $\overline{B}f(x)$ on a
+segment $\overline{a\ x_2}$ if $x_1$ lies on $\overline{a\ x_2}$,
+$b(x)$ is a non-oscillating function decreasing as $x$
+decreases. Hence $\displaystyle \mathop{L}_{x \doteq a} b(x)$ exists
+by Theorem~\hyperlink{thm21}{21}; and by Corollary~\hyperlink{cor3p82}{3}, Theorem~\hyperlink{thm40}{40}, $\displaystyle
+\mathop{L}_{x \doteq a} b(x) \geqq l$. If $\displaystyle \mathop{L}_{x
+\doteq a} b(x) = k > l$, then there are two vicinities of $k$,
+$V_1(k)$ contained in $V_2(k)$ and $V_2(k)$ not containing $l$. By
+Theorem~\hyperlink{thm26}{26} a $V_1^*(a)$ exists such that if $x$ is in $V_1^*(a)$,
+$b(x)$ is in $V_1(k)$. Furthermore, by the definition of $b(x)$, if
+$x_1$ is an arbitrary value of $x$ on $V_1^*(a)$, then there is a
+value of $x$ in $\overline{a\ x_1}$ such that $f(x)$ is in
+$V(k)$. Hence $k$ would be a value approached by $f(x)$ contrary to
+the hypothesis $k>l$.
+\end{proof}
+
+\section{Bounds of Indetermination. Oscillation.}\hypertarget{chIVsec6}{}%[6]
+
+It is a corollary of Theorem~\hyperlink{thm43}{43} that in the approach to any point $a$
+from the right or from the left the least upper \correction{bounds}{bound} and the greatest
+lower bounds of the values approached by $f(x)$ are themselves values
+approached by $f(x)$. The four numbers thus indicated may be denoted
+by
+\[\label{limp84}
+  \overline{f(a+0)} =
+   \mathop{\overline{L}}_{x \doteq a+0} f(x)
+  = \stackrel{\leftarrow}{\mathop{L}_{x \doteq a}} f(x),
+\]
+%-----File: 096.png---Folio 84-------
+the least upper bound of the values approached from the right:
+\[
+  \overline{f(a-0)} =
+   \mathop{\overline{L}}_{x \doteq a-0}
+   f(x) =
+  \mathop{\stackrel{\rightarrow}{L}}_{x \doteq a} f(x),
+\]
+the least upper bound of the values approached from the left:
+\[
+  \underline{f(a+0)} =
+   \mathop{\underline{L}}_{x \doteq a + 0}
+   f(x) =
+  \mathop{L}_{\stackrel{\leftarrow}{x \doteq a}} f(x),
+\]
+the greatest lower bound of the values approached from the right:
+\[
+  \underline{f(a-0)} =
+   \mathop{\underline{L}}_{x \doteq a-0}
+   f(x) =
+  \mathop{L}_{\stackrel{\rightarrow}{x \doteq a}} f(x),
+\]
+the greatest lower bound of the values approached from the left.
+
+If all four of these values coincide, there is only one value
+approached and $\displaystyle \mathop{L}_{x \doteq a} f(x)$ exists.
+If $\overline{f(a+0)}$ and $\underline{f(a+0)}$ coincide, this value
+is denoted by $f(a+0)$ and is the same as $\displaystyle
+\mathop{L}_{\stackrel{x>a}{x \doteq a}} f(x)$.  Similarly if
+$\overline{f(a-0)}$ and $\underline{f(a-0)}$ coincide, their common
+value, $\displaystyle \mathop{L}_{\stackrel{x<a}{x \doteq a}} f(x)$,
+is denoted by $f(a-0)$. The \index{Bounds!of indetermination}larger of $\overline{f(a+0)}$ and
+$\overline{f(a-0)}$ is denoted by $\displaystyle
+\mathop{\overline{L}}_{x \doteq a} f(x)$, and is called the \index{Limit!upper}upper\index{Upper!limit}
+limit of $f(x)$ as $x$ approaches $a$. Similarly $\displaystyle
+\mathop{\underline{L}}_{x \doteq a} f(x)$, the \index{Limit!lower}lower limit of $f(x)$,
+is the smaller of $\underline{f(a+0)}$ and $\underline{f(a-0)}$.
+$\displaystyle \mathop{\overline{L}}_{x \doteq a} f(x)$ and
+$\displaystyle \mathop{\underline{L}}_{x \doteq a} f(x)$ are called
+the bounds of indetermination of $f(x)$ at $x=a$
+(Unbestimmtheitsgrenzen).  See the Encyclop\"adie der mathematischen
+Wissenschaften, II~41.
+
+In order that a function shall be continuous at a point $a$ it is
+necessary and sufficient that
+\[
+\tag{a}
+f(a) =
+  \overline{f(a+0)} =
+  \underline{f(a+0)} =
+  \overline{f(a-0)} =
+  \underline{f(a\text{\correction{$-$}{$+$}}0)}.
+\]
+
+The difference between the greatest and the least of these
+%-----File: 097.png---Folio 85-------
+values is called the \index{Oscillation of a function!at a point}\emph{oscillation} of the function at the point
+$a$.  It is denoted by $O_af(x)$, and according to the theorem above
+is equivalent to the lower bound of all values of $Of(x)$, where
+\[
+  Of(x) = \overline{B}f(x)-\underline{B}f(x) \text{ for a segment } V(a).
+\]
+$O_a^bf(x)$ is used for the \label{chIVp85}oscillation of $f(x)$ on the segment
+$\overline{a\ b}$. It is sometimes also used for the oscillation of
+$f(x)$ on the interval $\interval{a}{b}$. The word oscillation may
+also be applied to the difference between the upper and lower bounds
+of the function on a $V^*(a)$.  Denote this by $O_{{V^*}(a)}f(x)$. The
+lower bound of these values may be denoted by $O_a^*f(x)$ and is the
+difference between the greatest and the least of the four values
+$\overline{f(a+0)}$, $\overline{f(a-0)}$, $\underline{f(a+0)}$,
+$\underline{f(a-0)}$.
+
+The reader will find it a useful exercise to construct examples and to
+enumerate the different ways in which a function may be discontinuous\index{Discontinuity!of the first and second kind},
+according as $f(a+0)$ or $f(a-0)$ exist or do not exist, and according
+as $f(a)$ does or does not coincide with any of the values approached
+by $f(x)$. (Compare the reference to the E.~d.~m.~W. given above.) The
+principal classification used is into \emph{discontinuities of the
+first kind}, where $f(a+0)$ and $f(a-0)$ both exist, and
+\emph{discontinuities of the second kind}, where not both $f(a+0)$ and
+$f(a-0)$ exist.
+
+\begin{theorem}[44]\hypertarget{thm44}{}
+If $a$ is a limit point of $[x]$, then a necessary and sufficient
+condition that $b_2$ and $b_1$ shall be the upper and lower bounds of
+indetermination of $f(x)$, as $x\doteq a$, is that for every set of
+four numbers $a_1$, $a_2$, $c_1$, $c_2$, such that\footnote{%
+    If $b_1 =-\infty$, $a_1=b_1$ replaces $a_1<b_1$. If $b_2 =
+    +\infty$, $a_2=b_2$ replaces $b_2<a_2$.}
+\[
+  a_1<b_1<c_1<c_2<b_2<a_2,
+\]
+there exists a $V^*(a)$ such that for every $x$ on $V^*(a)$
+\[
+  a_1<f(x)<a_2,
+\]
+and for some $x'$, $x''$ on $V^*(a)$
+\[
+  f(x')>c_2 \text{ and } f(x'')<c_1.
+\]
+\end{theorem}
+%-----File: 098.png---Folio 86-------
+
+\begin{proof}
+I. \textit{The condition is necessary.} It is to be proved that if
+$b_2$ and $b_1$ are the upper and lower bounds of indetermination of
+$f(x)$, as $x\doteq a$ on $[x]$, then for every four numbers
+$a_1<b_1<c_1<c_2<b_2<a_2$ there exists a $V^*(a)$ such that:---
+
+(1) For all values of $x$ on $V^*(a)$, $a_1<f(x)<a_2$. If this
+conclusion does not follow, then for a particular pair of numbers
+$a_1$, $a_2$, there are values of $f(x)$ greater than $a_2$ or less
+than $a_1$ for $x$ on any $V^*(a)$, and by Theorems \hyperlink{thm24}{24} and \hyperlink{thm40}{40} there is
+at least one value approached greater than $b_2$ or less than
+$b_1$. This would contradict the hypothesis, and there is therefore a
+$V^*(a)$ such that for all values of $x$ on $V^*(a)$, $a_1<f(x)<a_2$.
+
+(2) For some $x'$, $x''$ on $V^*(a)$, $f(x')>c_2$ and $f(x'')<c_1$. If
+this conclusion should not follow, then for some $V^*(a)$ there would
+be no $x'$ such that $f(x') >c_2$, or no $x''$ such that $f(x'')<c_1$,
+and therefore $b_1$ and $b_2$ could not both be values approached.
+
+II. \textit{The condition is sufficient.} It is to be proved that
+$b_2$ and $b_1$ are the upper and lower bounds of the values
+approached.  If the condition is satisfied, then for every four
+numbers $a_1$, $a_2$, $c_1$, $c_2$, such that
+$a_1<b_1<c_1<c_2<b_2<a_2$ there is a $V^*(a)$ such that for all $x$'s
+on $V^*(a)$ $a_1<f(x)<a_2$, and for some $x'$, $x'',$ $f(x')>c_2$ and
+$f(x'')<c_1$. By Theorem~\hyperlink{thm24}{24} there are values approached, and hence we
+need only to show that $b_2$ is the least upper and $b_1$ the greatest
+lower bound of the values approached.  Suppose some $B>b_2$ is the
+least upper bound of the values approached; $a_2$ may then be so
+chosen that $b_2 < a_2 < B$, so that by hypothesis for $x$ on $V^*(a)$
+$B$ cannot be a value approached.  Again, suppose $B<b_2$ to be the
+least upper bound; $c$ may then be chosen so that $B<c_2$, and hence
+for some value $x'$ on each $V^*(a)$, $f(x')<c_2$. By the set of
+values $f(x')$ there is at least one value approached. This value is
+greater than $c_2>B$.  Therefore $B$ cannot be the least upper
+bound. Since the least upper bound may not be either less than $b_2$
+or greater than $b_2$, it must be equal to $b_2$. A similar argument
+will prove $b_1$ to be the greatest lower bound of the values
+approached.
+\end{proof}
+%-----File: 099.png---Folio 87-------
+
+
+\chapter{CONTINUOUS FUNCTIONS.}\hypertarget{chapV}{}%[V]
+
+\section{Continuity at a Point.}\hypertarget{chVsec1}{}%[1]
+
+The notion of continuous functions will in this chapter, as in the
+definition on page~\pageref{dp61}, be confined to single-valued
+functions.  It has been shown in Theorem~\hyperlink{thm34}{34} that if $f_1(x)$ and
+$f_2(x)$ are continuous at a point $x=a$, then
+\[
+  f_1(x) \pm f_2(x), \quad
+  f_1(x) \cdot f_2(x), \quad
+  f_1(x)/f_2(x), \quad
+ (f_2(x) \neq 0)
+\]
+are also continuous at this point. Corollary~\hyperlink{cor1p81}{1} of Theorem~\hyperlink{thm39}{39} states
+that a continuous function of a continuous function is continuous.
+
+The definition of continuity at $x=a$, namely,
+\[
+\mathop{L}_{x \doteq a} f(x) = f(a),
+\]
+is by Theorem~\hyperlink{thm26}{26} equivalent to the following proposition:
+
+\emph{For every $\varepsilon>0$ there exists a $\delta_\varepsilon>0$
+such that if $|x-a|< \delta_\varepsilon$, then $|f(x)-f(a)|<
+\varepsilon$.}
+
+It should be noted that the restriction $x \neq a$ which appears in
+the general form of Theorem~\hyperlink{thm26}{26} is of no significance here, since for
+$x=a$, $|f(x)-f(a)|= 0 < \varepsilon$. In other words, we may deal
+with vicinities of the type $V(a)$ instead of $V^*(a)$.
+
+The difference of the least upper and the greatest lower bound of a
+function on an interval $\interval{a}{b}$ has been called in
+Chapter~\hyperlink{chapIV}{IV}, page~\pageref{chIVp85}, the oscillation of $f(x)$ on that interval, and
+denoted by $O_a^b(x)$. The definition of continuity and Theorem~\hyperlink{thm27}{27},
+Chapter~\hyperlink{chapIII}{III}, give the following necessary and sufficient condition for
+the continuity of a function $f(x)$ at the
+%-----File: 100.png---Folio 88-------
+\textit{For every $\varepsilon>0$ there exists a
+$\delta_\varepsilon>0$ such that if $|x_1-a|< \delta_\varepsilon$, and
+$|x_2-a|< \delta_\varepsilon$ then $|f(x_1)-f(x_2)|<
+\dfrac{\varepsilon}{2}$. This means that for all values of $x_1$ and
+$x_2$ on the segment $\overline{(a-\delta_\varepsilon)\ (a +
+\delta_\varepsilon)}$}
+\[
+  \overline{B} |f(x_1)-f(x_2)|\leqq \frac\varepsilon2 < \varepsilon,
+\]
+and this means
+\[
+\overline{B}f(x)-\underline{B}f(x) < \varepsilon,
+\]
+or
+\[
+  O^{a + \delta_\varepsilon}_{a-\delta_\varepsilon} f(x)
+  < \varepsilon.
+\]
+Then we have
+
+\begin{theorem}[45]\hypertarget{thm45}{}
+If $f(x)$ is continuous for $x=a$, then for every $\varepsilon>0$
+there exists a $V_\varepsilon(a)$ such that on $V_\varepsilon(a)$ the
+oscillation of $f(x)$ is less than $\varepsilon$.
+\end{theorem}
+
+\begin{theorem}[46]\hypertarget{thm46}{}
+If $f(x)$ is continuous at a point $x=a$ and if $f(a)$ is positive,
+then there is a neighborhood of $x=a$ upon which the function is
+positive.
+\end{theorem}
+
+\begin{proof}
+If there were values of $x$, $[x']$ within every neighborhood of $x=a$
+for which the function is equal to or less than zero, then by
+Theorem~\hyperlink{thm24}{24} there would be a value approached by $f(x')$ as $x'$
+approaches $a$ on the set $[x']$. That is, by Theorem~\hyperlink{thm40}{40}, there would
+be a negative or zero value approached by $f(x)$, which would
+contradict the hypothesis.
+\end{proof}
+
+\section{Continuity of a Function on an Interval.}\hypertarget{chVsec2}{}%[2]
+
+\begin{definition}\index{Continuity!over an interval}\index{Function!continuity of!over an interval}
+A function is said to be continuous on an interval $\interval{a}{b}$
+if it is continuous at every point on the interval.
+\end{definition}
+
+\begin{theorem}[47]\hypertarget{thm47}{}
+If $f(x)$ is continuous on a finite interval $\interval{a}{b}$, then
+for every $\varepsilon > 0$, $\interval{a}{b}$ can be divided into a
+finite number of equal intervals upon each of which the oscillation of
+$f(x)$ is less than $\varepsilon$.\footnote{%
+    The importance of this theorem in proving the properties of
+    continuous functions seems first to have been recognized by
+    \textsc{Goursat}. See his \textit{Cours d'Analyse}, Vol.~1,
+    page~161.}
+\end{theorem}
+%-----File: 101.png---Folio 89-------
+
+\begin{proof}
+By Theorem~\hyperlink{thm45}{45} there is about every point of $\interval{a}{b}$ a
+segment $\sigma$ upon which the oscillation is less than
+$\varepsilon$. This set of segments $[\sigma]$ covers $\interval{a}{b}$, and by Theorem~\hyperlink{thm11}{11} $\interval{a}{b}$ can be divided into a finite
+number of equal intervals each of which is interior to a $\sigma$;
+this gives the conclusion of our theorem.
+\end{proof}
+
+\begin{theorem}[48]\hypertarget{thm48}{} (Uniform continuity.)\label{t48p89}\index{Uniform continuity}\index{Continuity!uniform}\index{Function!uniform continuity of}
+If a function is continuous on a finite interval $\interval{a}{b}$,
+then for every $\varepsilon>0$ there exists a $\delta_{\varepsilon}>0$
+such that for any two values of $x$, $x_1$, and $x_2$, on
+$\interval{a}{b}$ where $|x_1-x_2|< \delta_{\varepsilon}$,
+$|f(x_1)-f(x_2)|< \varepsilon$.
+\end{theorem}
+
+\begin{proof}
+This theorem may be inferred in an obvious way from the preceding
+theorem, or it may be proved directly as follows:
+
+By Theorem~\hyperlink{thm27}{27}, for every $\varepsilon$ there exists a neighborhood
+$V_{\varepsilon}(x')$ of every $x'$ of $\interval{a}{b}$ such that if
+$x_1$ and $x_2$ are on \correction{$V_\varepsilon(x')$}{$V (x')$},
+then $|f(x_1)-f(x_2)|< \varepsilon$. The $V_{\varepsilon}(x)$'s
+constitute a set of segments which cover $\interval{a}{b}$. Hence, by
+Theorem~\hyperlink{thm12}{12}, there is a $\delta_{\varepsilon}$ such that if $|x_1-x_2|\text{\correction{$<$}{$>$}}
+\delta_{\varepsilon}$, $x_1$ and $x_2$ are on the same
+\correction{$V_\varepsilon(x')$}{$V (x')$} and
+consequently $|f(x_1)-f(x_2)|< \varepsilon$.
+\end{proof}
+
+The uniform continuity theorem is due to \textsc{E.~Heine}.\footnote{%
+    \textsc{E.~Heine:} \textit{Die Elemente der Functionenlehre},
+    Crelle, Vol.~74 (1872), p.~188.}
+The proof given by him is essentially that given above.
+
+In 1873 \textsc{L\"uroth}\footnote{%
+   \textsc{L\"uroth:} \textit{Bemerkung \"uber Gleichm\"assige
+    Stetigkeit}, Mathematische Annalen, Vol.~6, p.~319.}
+gave another proof of the theorem which is based on the following
+definition of continuity:
+
+A single-valued function is continuous at a point $x=a'$ if for every
+positive $\varepsilon$ there exists a $\delta_{\varepsilon}$, such
+that for every $x_1$ and $x_2$ on the interval
+$\interval{a-\delta_{\varepsilon}}{a + \delta_{\varepsilon}}$,
+$|f(x_1)-f(x_2)|< \varepsilon$ (Theorem~\hyperlink{thm45}{45}).
+
+By Theorem~\hyperlink{thm42}{42} there exists a greatest $\delta$ for a given point and
+for a given $\varepsilon$. Denote this by
+$\Delta_{\varepsilon}(x)$. If the function is continuous at every
+point of $\interval{a}{b}$, then for every $\varepsilon$ there will be
+a value of $\Delta_{\varepsilon}(x)$ for every point of the interval,
+i.e., $\Delta_{\varepsilon}(x)$, for any particular $\varepsilon$,
+will be a single-valued function of $x$.
+%-----File: 102.png---Folio 90-------
+
+The essential part of \textsc{L\"uroth's} proof consists in
+establishing the following fact: If $f(x)$ is continuous at every
+point of its interval, then for any particular value of $\varepsilon$
+the function $\Delta_\varepsilon(x)$ is also a continuous function of
+$x$. From this it follows by Theorem~\hyperlink{thm50}{50} that the function
+$\Delta_\varepsilon(x)$ will actually reach its greatest lower bound,
+that is, will have a minimum value; and this minimum value, like all
+other values of $\delta_\varepsilon$, will be positive.\footnote{%
+  It is interesting to note that this proof will not hold if the
+  condition of Theorem~\hyperlink{thm26}{26} is used as a definition of continuity. On
+  this point see \textsc{N.~J.  Lennes}: The Annals of Mathematics,
+  second series, Vol.~6, p.~86.}
+This minimum value of \correction{$\Delta_\varepsilon(x)$}{$\Delta$}
+on the interval under consideration will be effective as a
+$\delta_\varepsilon$, independent of $x$.
+
+The property of a continuous function exhibited above is called
+uniform continuity, and Theorem~\hyperlink{thm48}{48} may be briefly stated in the form:
+\emph{Every function continuous on an interval is uniformly continuous
+on that interval.}\footnote{%
+  It should be noticed that this theorem does not hold if ``segment''
+  is substituted for ``interval,'' as is shown by the function
+  $\dfrac1x$ on the segment $\overline{0\ 1}$, which is continuous but
+  not uniformly continuous. The function is defined and continuous for
+  every value of $x$ on this \textit{segment}, but not for every value
+  of $x$ on the \emph{interval} $\interval{0}{1}$.}
+
+This theorem is used, for example, in proving the integrability of
+continuous functions. See page~\pageref{t98p157}.
+
+\begin{theorem}[49]\hypertarget{thm49}{}
+If a function is continuous on an interval $\interval{a}{b}$, it is
+bounded on that interval.
+\end{theorem}
+
+\begin{proof}
+By Theorem~\hyperlink{thm46}{46} the interval $\interval{a}{b}$ can be divided into a
+finite number of intervals, such that the oscillation on each interval
+is less than a given positive number $\varepsilon$. If the number of
+intervals is $n$, then the oscillation on the interval $\interval{a}{b}$ is less than $n\varepsilon$. Since the function is defined at all
+points of the interval, its value being $f(x_1)$ at some point $x_1$,
+it follows that every value of $f(x)$ on $\interval{a}{b}$ is less
+than $f(x_1) +n\varepsilon$ and greater than $f(x_1)-n\varepsilon$;
+which proves the theorem.
+\end{proof}
+
+\begin{theorem}[50]\hypertarget{thm50}{}
+If a function $f(x)$ is continuous on an interval
+%-----File: 103.png---Folio 91-------
+$\interval{a}{b}$, then the function assumes as values its least upper
+and its greatest lower bound.
+\end{theorem}
+
+\begin{proof}
+By the preceding theorem the function is bounded and hence the least
+upper and greatest lower bounds are finite.  By Theorem~\hyperlink{thm19}{19} there is a
+point $k$ on the interval $\interval{a}{b}$ such that the least upper
+bound of the function on every neighborhood of $x=k$ is the same as
+the least upper bound on the interval $\interval{a}{b}$.  Denote the
+least upper bound of $f(x)$ on $\interval{a}{b}$ by $B$. It follows
+from Theorem~\hyperlink{thm43}{43} that $B$ is a value approached by $f(x)$ as $x$
+approaches $k$. But since $\displaystyle \mathop{L}_{x\doteq k} f(x)
+=f(k)$, the function being continuous at $x=k$, we have that $f(k) =
+B$. In the same manner we can prove that the function reaches its
+greatest lower bound.
+\end{proof}
+
+\begin{corollary}
+If $k$ is a value not assumed by a continuous function on an interval
+$\interval{a}{b}$, then $f(x)-k$ or $k-f(x)$ is a continuous function
+of $x$ and assumes its least upper and greatest lower bounds. That is,
+there is a definite number $\Delta$ which is the least difference
+between $k$ and the set of values of $f(x)$ on the interval
+$\interval{a}{b}$.
+\end{corollary}
+
+\begin{theorem}[51]\hypertarget{thm51}{}
+If a function is continuous on an interval $\interval{a}{b}$, then the
+function takes on all values between its least upper and its greatest
+lower bound.
+\end{theorem}
+
+\begin{proof}
+If there is a value $k$ between these bounds which is not assumed by a
+continuous function $f(x)$, then by the corollary of the preceding
+theorem there is a value $\Delta$ such that no values of $f(x)$ are
+between $k-\Delta$ and $k+\Delta$. With $\varepsilon$ less than
+$\Delta$ divide the interval $\interval{a}{b}$ into subintervals
+according to Theorem~\hyperlink{thm47}{47}, such that the oscillation on every interval
+is less than $\varepsilon$. No interval of this set can contain values
+of $f(x)$ both greater and less than $k$, and no two consecutive
+intervals can contain such values. Suppose the values of $f(x)$ on the
+first interval of this set are all greater than $k$, then the same is
+%-----File: 104.png---Folio 92-------
+true of the second interval of the set, and so on. Hence it follows
+that all values of $f(x)$ on $\interval{a}{b}$ are either greater than
+or less than $k$, which is contrary to the hypothesis that $k$ lies
+between the least upper and the greatest lower bounds of the function
+on $\interval{a}{b}$. Hence the hypothesis that $f(x)$ does not assume
+the value $k$ is untenable.
+\end{proof}
+
+By the aid of Theorem~\hyperlink{thm51}{51} we are enabled to prove the following:
+
+\begin{theorem}[51a]\hypertarget{thm51a}{}
+If $f_1(x)$ is continuous at every point of an interval $\interval{a'}{b'}$ except at a certain point $a$, and if
+\[
+  \mathop{L}_{x \doteq a} f_1(x) = +\infty \text{ \textit{and} }
+  \mathop{L}_{x \doteq a} f_2(x) =-\infty,
+\]
+then for every $b$, finite or $+\infty$ or $-\infty$, there exist two
+sequences of points, $[x_i]$ and $[x'_i]$ ($i=0, 1, 2, \ldots$), each
+sequence having a as a limit point, such that
+\[
+  \mathop{L}_{i \doteq \infty} \{ f_1(x_i) + f_2(x'_i) \} = b.
+\]
+\end{theorem}
+
+\begin{proof}
+Let $[x'_i]$ be any sequence whatever on $\interval{a'}{b'}$ having
+$a$ as a limit point, and let $x_0$ be an arbitrary point of
+$\interval{a'}{b'}$. Since $f_1(x)$ assumes all values between
+$f_1(x_0)$ and $+\infty$, and since $\displaystyle\mathop{L}_{x \doteq
+a} f_2(x) =-\infty$, it follows, in case $b$ is finite, that for every
+$i$ greater than some fixed value there exists an $x_i$ such that
+\[
+  f_1(x_i) + f_2(x'_i) = b.
+\]
+In case $b = +\infty$, $x_i$ is chosen so that
+\[
+  f_1(x_i) + f_2(x'_i) > i.\qedhere
+\]
+\end{proof}
+
+\begin{corollary}
+Whether $f_1(x)$ and $f_2(x)$ are continuous or not, if
+$\displaystyle\mathop{L}_{x \doteq a} f_1(x) = +\infty$ and
+$\displaystyle\mathop{L}_{x \doteq a} f_2(x) =-\infty$, there exists a
+pair of
+%-----File: 105.png---Folio 93-------
+sequences $[x_i]$ and $[x_i']$ such that
+\[
+  \mathop{L}_{i\doteq\infty} \{f_1(x_i)+f_2(x_i)\}
+\]
+is $+\infty$ or $-\infty$.
+\end{corollary}
+
+\begin{theorem}[52]\hypertarget{thm52}{}
+If $y$ is a function, $f(x)$, of $x$, monotonic and continuous on an
+interval $\interval{a}{b}$, then $x=f^{-1}(y)$ is a function of $y$
+which is monotonic and continuous on the interval $\interval{f(a)}{f(b)}$.
+\end{theorem}
+
+\begin{proof}
+By Theorem~\hyperlink{thm20}{20} the function $f^{-1}(y)$ is monotonic and has as upper
+and lower bounds $a$ and $b$. By Theorems~50 and 51 the function is
+defined for every value of $y$ between and including $f(a)$ and $f(b)$
+and for no other values. We prove the function continuous on the
+interval $\interval{f(a)}{f(b)}$ by showing that it is continuous at
+any point $y=y_1$ on this interval. As $y$ approaches $y_1$ on the
+interval $\interval{f(a)}{{y_1}}$, $f^{-1}(y)$ approaches a definite
+limit $g$ by Theorem~\hyperlink{thm25}{25}, and by Theorem~\hyperlink{thm40}{40} $a<g\leqq f^{-1}(y_1)\leqq
+b$. If $g<f^{-1}(y_1)$, then for values of $x$ on the interval
+$\interval{g}{f(y_1)}$ there is no corresponding value of $y$,
+contrary to the hypothesis that $f(x)$ is defined at every point of
+the interval $\interval{a}{b}$. Hence $g=f^{-1}(y_1)$, and by similar
+reasoning we show that $f^{-1}(y)$ approaches $f^{-1}(y_1)$ as $y$
+approaches $y_1$ on the interval, $\interval{y_1}{f^{-1}(b)}$.
+\end{proof}
+
+\begin{theorem}[53]\hypertarget{thm53}{}
+If $f(x)$ is single-valued and continuous with $A$, $B$ as lower and
+upper bounds, on an interval $\interval{a}{b}$ and has a single-valued
+inverse on the interval, $\interval{A}{B}$ then $f(x)$ is monotonic on
+$\interval{a}{b}$.
+\end{theorem}
+
+\begin{proof}
+If $f(x)$ is not monotonic, then there must be three values of $x$,
+\[
+  x_1<x_2<x_3,
+\]
+such that either
+\[
+  f(x_1)\leqq f(x_2)\geqq f(x_3)
+\]
+or
+\[
+  f(x_1)\geqq f(x_2)\leqq f(x_3).
+\]
+In either case, if one of the equality signs holds, the hypothesis
+that $f(x)$ has a single-valued inverse is contradicted. If there
+%-----File: 106.png---Folio 94-------
+are no equality signs, it follows by Theorem~\hyperlink{thm51}{51} that there are two
+values of $x$, $x_4$ and $x_5$, such that
+\[
+  x_1 < x_4 < x_2 < x_5 < x_3,
+\]
+and $f(x_4) =f(x_5)$, in contradiction with the hypothesis that $f(x)$
+has a single-valued inverse.
+\end{proof}
+
+\begin{corollary}
+If $f(x)$ is single-valued, continuous, and has a single-valued
+inverse on an interval $\interval{a}{b}$, then the inverse function is
+monotonic on $\interval{A}{B}$.
+\end{corollary}
+
+\section{Functions Continuous on an Everywhere Dense Set.}\hypertarget{chVsec3}{}%[3]
+
+\begin{theorem}[54]\hypertarget{thm54}{}
+If the functions $f_1(x)$ and $f_2(x)$ are continuous on the interval
+$\interval{a}{b}$, and if $f_1(x)=f_2(x)$ on a set everywhere dense,
+then $f_1(x) =f_2(x)$ on the whole interval.\footnote{%
+  I.e., if a function $f(x)$, continuous on an interval $\interval{a}{  b}$, is known on an everywhere dense set on that interval, it is
+  known for every point on that interval.}
+\end{theorem}
+
+\begin{proof}
+Let $[x']$ be the set everywhere dense on $\interval{a}{b}$ for which,
+by hypothesis, $f_1(x) = f_2(x)$. Let $x''$ be any point of the
+interval not of the set $[x']$. By hypothesis $x''$ is a limit point
+of the set $[x']$, and further $f_1(x)$ and $f_2(x)$ are continuous at
+$x=x''$.  Hence
+\begin{align*}
+  \mathop{L}_{x \doteq x''} f_1(x) &= f_1(x'')
+\\
+\intertext{and }
+  \mathop{L}_{x \doteq x''} f_2(x) &= f_2(x'').
+\\
+\intertext{But by Theorem~\hyperlink{thm41}{41} }
+  \mathop{L}_{x'\doteq x''} f_1(x')
+  &= \mathop{L}_{x \doteq x''} f_1(x ),
+\\
+\intertext{and by Theorem~\hyperlink{thm41}{41} }
+ \mathop{L}_{x'\doteq x''} f_2(x')
+ &= \mathop{L}_{x \doteq x''} f_2(x ).
+\\
+\intertext{Therefore }
+f_1(x'') &= f_2(x'').\qedhere
+\end{align*}
+\end{proof}
+%-----File: 107.png---Folio 95-------
+
+\begin{definition}
+On an interval $\interval{a}{b}$ a function $f(x')$ is
+\textit{uniformly continuous} over a set $[x']$ if for every
+$\varepsilon >0$ there exists a $ \delta_\varepsilon > 0$ such that
+for any two values of $x'$, $x_1'$, and $x_2'$ an $\interval{a}{b}$,
+for which $ |x_1'-x_2'| < \delta_\varepsilon$, $|f(x_1')-f(x_2')| <
+\varepsilon$.
+\end{definition}
+
+\begin{theorem}[55]\hypertarget{thm55}{}
+If a function $f(x')$ is defined on a set everywhere dense on the
+interval $\interval{a}{b}$ and is uniformly continuous over that set,
+then there exists one and only one function $f(x)$ defined on the full
+interval $\interval{a}{b}$ such that:
+\begin{enumerate}
+\item[\textnormal{(1)}] $f(x)$ is identical with $f(x')$ where $f(x')$ is defined.
+
+\item[\textnormal{(2)}] $f(x)$ is continuous on the interval $\interval{a}{b}$.
+\end{enumerate}
+\end{theorem}
+
+\begin{proof}
+Let $x''$ be any point on the interval $\interval{a}{b}$, but not of
+the set $[x']$. We first prove that
+\[
+  \mathop{L}_{x' \doteq x''} f(x')
+\]
+exists and is finite. By the definition of uniform continuity, for
+every $\varepsilon$ there exists a $\delta_\varepsilon$ such that for
+any two values of $x'$, $x_1'$, and $x_2'$, where $|x_1'-x_2'| <
+\delta_\varepsilon$, $|f(x_1')-f(x_2)| < \varepsilon $. Hence we have
+for every pair of values $x_1'$ and $x_2'$ where $|x_1'-x''|< \dfrac
+{\delta_\varepsilon}{2}$ and $|x_2'-x''| <
+\dfrac{\delta_\varepsilon}{2}$ that $|f(x_1')-f(x_2')|<
+\varepsilon$. By Theorem~\hyperlink{thm23}{23} this is a sufficient condition that
+\[
+  \mathop{L}_{x' \doteq x''}f(x')
+\]
+shall exist and be finite.
+
+Let $f(x)$ denote a function identical with $f(x')$ on the set $[x']$
+and equal to
+\[
+  \mathop{L}_{x' \doteq x''} f(x')
+\]
+at all points $x''$. This function is defined upon the continuum,
+%-----File: 108.png---Folio 96-------
+since all points $x''$ on $\interval{a}{b}$ are limit points of the
+set $[x']$. Hence the function has the property that $\displaystyle
+\mathop{L}_{x_1\doteq x} f(x')=f(x)$ for every $x$ of $\interval{a}{b}$.
+
+We next prove that $f(x)$ is continuous at every point on the interval
+$\interval{a}{b}$, in other words that $f(x)$ cannot approach a value
+$b$ different from $f(x_1)$ as $x$ approaches $x_1$. We already know
+that $f(x)$ approaches $f(x_1)$ on the set $[x']$. If $b$ is another
+value approached, then for every positive $\varepsilon$ and $\delta$
+there is an $x_{\varepsilon\delta}$ such that
+\hypertarget{eq1p97}{\[
+  |x_{\varepsilon\delta}-x_1|<\delta,
+  \qquad|f(x_{\varepsilon\delta})-b|<\varepsilon.\tag{1}
+\]}
+Since $f(x_{\varepsilon\delta}) =\displaystyle\mathop{L}_{x'\doteq
+x_{\varepsilon\delta}} f(x')$ we have that for every $\varepsilon>0$
+there exists a $\delta_\varepsilon>0$ such that for every $x'$ for
+which $|x'-x_{\varepsilon\delta}|<\delta_\varepsilon$,
+\hypertarget{eq2p97}{\[
+  |f(x')-f(x_{\varepsilon\delta})|<\varepsilon. \tag{2}
+\]}
+From \hyperlink{eq1p97}{(1)} and \hyperlink{eq2p97}{(2)} we have
+\hypertarget{eq3p97}{\[
+  |f(x')-b|<2\varepsilon. \tag{3}
+\]}
+Since the $\delta$ of \hyperlink{eq1p97}{(1)} is any positive number, there is an
+$x_{\varepsilon\delta}$ on every neighborhood of $x_1$ and hence by
+\hyperlink{eq2p97}{(2)} and \hyperlink{eq3p97}{(3)} an $x'$ on every neighborhood of $x_1$ such that
+$|f(x')-b| <2\varepsilon$, $\varepsilon$ being arbitrary and $b$ a
+constant different from $f(x_1'')$. But this is contrary to the fact
+proved above, that $\displaystyle \mathop{L}_{x'\doteq x_1}f(x')$
+exists and is equal to $f(x_1)$. Hence the function is continuous at
+every point of the interval $\interval{a}{b}$. The uniqueness of the
+function follows directly from Theorem~\hyperlink{thm54}{54}.
+\end{proof}
+
+This theorem can be applied, for example, to give an elegant
+definition of the exponential function (see Chap.~\hyperlink{chapIII}{III}). We first show
+that the function $a^\frac mn$ is uniformly continuous on the set of
+all rational values between $x_1$ and $x_2$, and then define
+%-----File: 109.png---Folio 97-------
+$a^x$ on the continuum as that continuous function which coincides
+with $a^\frac mn$ for the rational values $\dfrac mn$. The properties
+of the function then follow very easily. It will be an excellent
+exercise for the reader to carry out this development in detail.
+
+
+\section{The Exponential Function.}\hypertarget{chVsec4}{}%[4]
+\label{s4p97}
+Consider the function defined by the infinite series
+\[
+1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\ldots+\frac{x^n}{n!}+\ldots.
+\tag{1}
+\]
+Applying the ratio test for the convergence of infinite series we have
+\[
+  \frac{x^n}{n!}\div\frac{x^{n-1}}{(n-1)!}=\frac xn.
+\]
+If $n'$ is a fixed integer larger than $x$, this ratio is always less
+than $\dfrac x{n'}<1$. The series~(1) therefore converges absolutely
+for every value of $x$, and we may denote its sum by
+\[
+  e(x).
+\]
+
+From Chap.~\hyperlink{chapI}{I}, page~\pageref{t7p17}, we have that
+\[
+  e(1)= \mathop{L}_{n\doteq\infty} \left(1+\frac1n\right)^n =e.
+\]
+
+\begin{theorem}[56]\hypertarget{thm56}{}
+\[
+  \mathop{L}_{n\doteq\infty} \left(1+\frac xn\right)^n,
+\]
+where $[n]$ is the set of all positive integers, exists and is equal
+to $e(x)$ for all values of $x$.
+\end{theorem}
+%-----File: 110.png---Folio 98-------
+\begin{proof}
+Let
+\[
+E_n(x) = \sum_{k=0}^n\frac{x^k}{k!}
+\]
+(where $0! = 1$).
+Then, since
+\[
+  \left(1+\frac xn\right)^n
+  = 1 + \frac{n!}{(n-1)!} \cdot\frac xn
+    + \frac{n!}{(n-2)!\cdot 2!} \left(\frac xn\right)^2 + \ldots
+    + \frac{n!}{n!} \left(\frac xn\right)^n,
+\]
+it follows that
+\begin{align*}
+  \left|E_n(x)-\left(1+\frac xn\right)^n\right|
+  &= \left|\sum_{k=2}^n\left(\frac1{k!}-
+  \frac{n!}{(n-k)!\cdot k!\,n^k} \right)x^k\right|
+\\
+  &\leqq \sum_{k=2}^n\left(\frac1{k!}-
+  \frac{n(n-1)\ldots(n-k+1)}{k!\,n^k} \right)\cdot|x^k|
+\\
+  &<\sum_{k=2}^n \frac{n^k-(n-k+1)^k}{k!\,n^k}\cdot|x^k|.
+\end{align*}
+Now, since
+\begin{multline*}
+  n^k-(n-k+1)^k = (k-1)\{n^{k-1}+n^{k-2}\cdot(n-k+1)+\ldots
+\\
+  +(n-k+1)^{k-1}\} < (k-1)k\cdot n^{k-1},
+\end{multline*}
+it follows that
+\[
+  \left|E_n(x)-\left(1+\frac xn\right)^n\right|
+  < \sum_{k=2}^n \frac{|x|^k}{(k-2)!\cdot n}
+  < \frac{x^2\cdot e(|x|)}{n}.
+\]
+For a fixed value of $x$, therefore, we have
+\[
+  \left(1+\frac xn\right)^n = E_n(x)+\varepsilon_1(n),
+\]
+where $\varepsilon_1(n)$ is an infinitesimal as $n\doteq\infty$.
+
+At the same time
+\[
+  e(x) = E_n(x) + \varepsilon_2(n),
+\]
+where $\varepsilon_2(n)$ is an infinitesimal as $n\doteq\infty$.
+Hence
+\[
+  \mathop{L}_{n\doteq\infty} \left(1+\frac xn\right)^n = e(x).\qedhere
+\]
+\end{proof}
+%-----File: 111.png---Folio 99-------
+\begin{theorem}[57]\hypertarget{thm57}{}
+\[
+ \mathop{L}_{z\doteq \infty} \left(1+\frac xz\right),
+\]
+where $[z]$ is the set of all real numbers, exists and is equal to
+$e(x)$.
+\end{theorem}
+
+\begin{proof}
+If $z$ is any number greater than $1$, let $n_z$ be the integer such
+that
+\[
+  n_z\leqq z<n_z+1.
+\]
+Hence, if $x>0$,
+\[
+  1+\frac x{n_z}\geqq1+\frac xz >1+\frac x{n_z+1}.
+\tag{1}
+\]
+Hence
+\[
+  \left(1+\frac x{n_z}\right)^{n_z+1}\geqq
+  \left(1+\frac xz\right)^z >
+  \left(1+\frac x{n_z+1}\right)^{n_z},
+\tag{2}
+\]
+or
+\[
+  \left(1+\frac x{n_z}\right)
+  \left(1+\frac x{n_z}\right)^{n_z} \geqq
+  \left(1+\frac xz\right)^z >
+  \left(1+\frac x{n_z+1}\right)^{n_z+1}\cdot
+  \frac{1}{1+\frac{x}{n_z+1}}.
+\tag{3}
+\]
+Since
+\begin{alignat*}{2}
+  \mathop{L}_{z\doteq\infty} \left(1+\frac x{\text{\correction{$n_z$}{$n$}}}\right)
+  &=1,
+  &\text{ and }
+  \mathop{L}_{z\doteq\infty}\left(1+\frac x{n_z+1}\right)
+  &=1,\\
+\intertext{and}
+  \mathop{L}_{z\doteq\infty}\left(1+\frac x{n_z}\right)^{n_z}
+  &=e(x),
+  &\text{and}
+  \mathop{L}_{z\doteq\infty} \left(1+\frac x{n_z+1}\right)^{n_z+1}&=e(x),
+\end{alignat*}
+the inequality~(3), together with Corollary~\hyperlink{cor3p82}{3}, Theorem~\hyperlink{thm40}{40}, leads to
+the result:
+\[
+  \mathop{L}_{z\doteq\infty} \left(1+\frac xz\right)^z=e(x).
+\]
+
+The argument is similar if $x<0$.
+\end{proof}
+
+\begin{corollary}
+\[
+  \mathop{L}_{z\doteq\infty} \left(1+\frac xz\right)^z=e(x),
+\]
+where $[z]$ is any set of numbers with limit point $+\infty$.
+\end{corollary}
+
+\begin{theorem}[58]\hypertarget{thm58}{}\label{t58p99}
+The function $e(x)$ is the same as $e^x$ where
+\[
+  e=1+1+\frac1{2!}+\frac1{3!}+\ldots
+\]
+\end{theorem}
+%-----File: 112.png---Folio 100------
+
+\begin{proof}
+By the continuity of $z^x$ as a function of $z$ (see Corollary~\hyperlink{cor2p81}{2} of
+Theorem~\hyperlink{thm39}{39}), it follows that, since
+\begin{align*}
+  \mathop{L}_{n\doteq \infty}\left(1+\frac1n\right)^n &= e,\\
+  \mathop{L}_{n\doteq \infty}\left(1+\frac1n\right)^{nx} &= e^x.
+\end{align*}
+But
+\[
+  \left(1+\frac1n\right)^{nx}
+  = \left(1+\frac x{nx}\right)^{nx}
+  = \left(1+\frac xz\right)^z,
+\]
+where $z=nx$. Hence by Theorem~\hyperlink{thm39}{39}
+\[
+  e^x = \mathop{L}_{z\doteq \infty}\left(1+\frac xz\right)^z,
+\]
+and by the corollary of Theorem~\hyperlink{thm57}{57} the latter expression is equal to
+$e(x)$. Hence we have
+\hypertarget{eq1p100}{\[
+  e^x = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\ldots.\tag{1}
+\]}
+\hyperlink{eq1p100}{(1)} is frequently used as the definition of $e^x$, $a^x$ being defined
+as $e^x\cdot\log_e a$.
+\end{proof}
+%-----File: 113.png---Folio 101------
+
+
+
+\chapter{INFINITESIMALS AND INFINITES.}\hypertarget{chapVI}{}%[VI]
+
+\section{The Order of a Function at a Point.}\hypertarget{chVIsec1}{}%[1]
+
+An infinitesimal has been defined (page~\pageref{dp75}) as a function
+$f(x)$ such that
+\[
+  \mathop{L}_{x \doteq a} f(x)=0.
+\]
+
+A function which is unbounded in every vicinity of $x=a$ is said to
+have an \index{Function!infinite at a point}\textit{infinity} at $a$, to be or become \index{Infinite}infinite at $x=a$,
+or to have an \index{Singularity}\textit{infinite singularity} at $x=a$.\footnote{%
+  It is perfectly compatible with these statements to say that while
+  $f(x)$ has an infinite singularity at $x=a$, $f(a)=0$ or any other
+  finite number.  For example, a function which is $\dfrac{1}{x}$ for
+  all values of $x$ except $x=0$ is left undefined for $x=0$ and hence
+  at this point the function may be defined as zero or any other
+  number. This function illustrates very well how a function which has
+  a finite value at every point may nevertheless have infinite
+  singularities.}
+The reciprocal of an infinitesimal at $x=a$ is infinite at this point.
+
+A function may be infinite at a point in a variety of ways:
+\begin{enumerate}
+\item[(\textit{a})] It may be monotonic and approach $+\infty$ or
+$-\infty$ as $x \doteq a$; for example, $\dfrac{1}{x}$ as $x$
+approaches zero from the positive side.
+\item[(\textit{b})] It may oscillate on every neighborhood of $x=a$
+and still approach $+\infty$ or $-\infty$ as a unique limit; for
+example,
+\[
+  \frac{\sin\dfrac{1}{x}+2}{x}
+\]
+as $x$ approaches zero.
+%-----File: 114.png---Folio 102------
+\item[(\textit{c})] It may approach any set of real numbers or the set
+of all real numbers; an example of the latter is
+\[
+  \frac{\sin\dfrac1x}{x}
+\]
+as $x$ approaches zero. See Fig.~\hyperlink{fig13}{13}, page~\pageref{fig13}.
+\item[(\textit{d})] $+\infty$ and $-\infty$ may both be approached
+while no other number is approached; for example,
+$\frac1x$ as $x$ approaches zero from both sides.
+\end{enumerate}
+\begin{defnorder}\index{Order of function}
+If $f(x)$ and $\phi(x)$ are two functions such that in some
+neighborhood $V^*(a)$ neither of them changes sign or is zero, and if
+\[
+  \mathop{L}_{x \doteq a} \frac{f(x)}{\phi(x)} = k,
+\]
+where $k$ is finite and not zero, then $f(x)$ and $\phi(x)$ are said
+to be of the \textit{same order} at $x=a$. If
+\[
+  \mathop{L}_{x \doteq a} \frac{f(x)}{\phi(x)} = 0,
+\]
+then $f(x)$ is said to be \textit{infinitesimal with respect to}
+$\phi(x)$, and $\phi(x)$ is said to be \index{Infinite}\textit{infinite with respect
+to} $f(x)$. If
+\[
+  \mathop{L}_{x \doteq a} \frac{f(x)}{\phi(x)} = +\infty \text{ or }-\infty,
+\]
+then, by Theorem~\hyperlink{thm37}{37}, $\phi(x)$ is infinitesimal with respect to
+$f(x)$, and $f(x)$ infinite with respect to $\phi(x)$. If $f(x)$ and
+$\phi(x)$ are both infinitesimal at $x=a$, and $f(x)$ is infinitesimal
+with respect to $\phi(x)$, then $f(x)$ is infinitesimal of a
+\textit{higher order} than $\phi(x)$, and $\phi(x)$ of \textit{lower
+order} than $f(x)$. If $\phi(x)$ and $f(x)$ are both infinite at
+$x=a$, and $f(x)$ is infinite with respect to $\phi(x)$, then $f(x)$
+is
+%-----File: 115.png---Folio 103------
+infinite of higher order than $\phi(x)$, and $\phi(x)$ is infinite of
+lower order than $f(x)$.\footnote{%
+  This definition of order is by no means as general as it might possibly
+  be made. The restriction to functions which are not zero and do not change
+  sign may be partly removed. The existence of
+  \[
+    \underset{x\doteq a}L\frac{f(x)}{\phi(x)}
+  \]
+  is dispensed with for
+  some cases in \hyperlink{chVIsec4}{\S~4} on Rank of Infinitesimals and Infinites.  For an
+  account of still further generalizations (due mainly to
+  \textsc{Cauchy}) see \textsc{E.~Borel}, \textit{S\'eries \correction{\`a}{a} Termes
+  Positifs}, Chapters III and IV, Paris, 1902. An excellent treatment
+  of the material of this section together with extensions of the
+  concept of order of infinity is due to \label{borlottip103}\textsc{E.~Borlotti}, {\it
+  Calcolo degli Infinitesimi}, Modena, 1905 (62 pages).}
+\end{defnorder}
+
+The independent variable $x$ is usually said to be an infinitesimal of
+the first order as $x$ approaches zero, $x^2$ of the second order,
+etc. Any constant $\neq 0$ is said to be infinite of zero order,
+$\dfrac{1}{x}$ is of the first order, $\dfrac{1}{x^2}$ of the second
+order, etc. This usage, however, is best confined to analytic
+functions. In the general case there are no two infinitesimals of
+consecutive order. Evidently there are as many different orders of
+infinitesimals between $x$ and $x^2$ as there are numbers between $1$
+and $2$; i.e., $x^{1+k}$ is of higher order than $x$ for every
+positive value of $k$.
+
+Since $\displaystyle\mathop{L}_{x\doteq
+a}\frac{f_1(x)}{f_2(x)}=\frac1k$ whenever
+$\displaystyle\mathop{L}_{x\doteq a}\frac{f_2(x)}{f_1(x)}=k$, we have
+\begin{theorem}[59]\hypertarget{thm59}{}
+If $f_1(x)$ is of the same order as $f_2(x)$, then $f_2(x)$ is of the
+same order as $f_1(x)$.
+\end{theorem}
+\begin{theorem}[60]\hypertarget{thm60}{}
+The function $cf(x)$ is of the same order as $f(x)$, $c$ being any
+constant not zero.
+\end{theorem}
+\begin{proof}
+By Theorem~\hyperlink{thm34}{34}, $\displaystyle\mathop{L}_{x\doteq
+a}\frac{cf(x)}{f(x)}=c$.
+\end{proof}
+\begin{theorem}[61]\hypertarget{thm61}{}
+If $f_1(x)$ is of the same order as $f_2(x)$, and $f_2(x)$ is of the
+same order as $f_3(x)$, then $f_1(x)$ and $f_3(x)$ are of the same
+order.
+\end{theorem}
+%-----File: 116.png---Folio 104------
+
+\begin{proof}
+By hypothesis
+$\displaystyle\mathop{L}_{x \doteq a} \frac{f_1(x)}{f_2(x)}=k_1$ and
+$\displaystyle\mathop{L}_{x \doteq a} \frac{f_2(x)}{f_3(x)}=k_2$.
+By Theorem~\hyperlink{thm34}{34},
+\[
+ \displaystyle\mathop{L}_{x \doteq a} \frac{f_1(x)}{f_2(x)} \cdot
+ \mathop{L}_{x \doteq a} \frac{f_2(x)}{f_3(x)} =
+ \mathop{L}_{x \doteq a} \frac{f_1(x)}{f_3(x)}.
+\]
+(By definition, $f_2(x) \neq 0$ and $f_3(x)\neq 0$ for some
+neighborhood of $x=a$.) Hence
+\[
+  \mathop{L}_{x \doteq a} \frac{f_1(x)}{f_3(x)} = k_1 \cdot k_2.\qedhere
+\]
+\end{proof}
+
+\begin{theorem}[62]\hypertarget{thm62}{}
+If $f_1(x)$ and $f_2(x)$ are infinitesimal (infinite) and neither is
+zero or changes sign on some $V^*(a)$, then $f_1(x)\cdot f_2(x)$ is
+infinitesimal (infinite) of a higher order than either.
+\end{theorem}
+
+\begin{proof}
+\[
+  \mathop{L}_{x \doteq a} \frac{f_1(x) \cdot f_2(x)}{f_2(x)} =
+  \mathop{L}_{x \doteq a} f_1(x) = 0.\ (\pm \infty.)\qedhere
+\]
+\end{proof}
+
+\begin{theorem}[63]\hypertarget{thm63}{}
+If $f_1(x)$, $\ldots$, $f_n(x)$ have the same sign on some $V^*(a)$
+and if $f_2(x)$, $\ldots$, $f_n(x)$ are infinitesimal (infinite) of
+the same or higher (lower) order than $f_1(x)$, then
+\[
+  f_1(x) + f_2(x) + f_3(x) + \ldots + f_n(x)
+\]
+is of the same order as $f_1(x)$, and if $f_2(x)$, $f_3(x)$, $\ldots$,
+$f_n(x)$ are of higher (lower) order than $f_1(x)$, then $f_1(x) \pm
+f_2(x) \pm f_3(x) \pm \ldots \pm f_n(x)$ is of the same order as
+$f_1(x)$.
+\end{theorem}
+
+\begin{proof}
+We are to show that
+\[
+  \mathop{L}_{x \doteq a} \frac{f_1(x) + f_2(x) + \ldots +
+  f_n(x)}{f_1(x)} = k \neq 0.
+\]
+By hypothesis,
+\[
+  \mathop{L}_{x \doteq a} \frac{f_2(x)}{f_1(x)} = k_2, \;
+  \mathop{L}_{x \doteq a} \frac{f_3(x)}{f_1(x)} = k_3, \;
+  \ldots, \;
+  \mathop{L}_{x \doteq a} \frac{f_n(x)}{f_1(x)} = k_n,
+\]
+and
+\[
+  \mathop{L}_{x \doteq a} \frac{f_1(x)}{f_1(x)} = 1.
+\]
+%-----File: 117.png---Folio 105------
+Hence, by Theorem~\hyperlink{thm30}{30},
+\[
+  \mathop{L}_{x \doteq a} \left\{%
+  \frac{f_1(x)}{f_1(x)} +
+  \frac{f_2(x)}{f_1(x)} +
+  \frac{f_3(x)}{f_1(x)} +
+  \ldots +
+  \frac{f_n(x)}{f_1(x)} \right\} =
+  1 + k_2 \text{\correction{$+\ldots+$}{$\ldots$}} k_n = k \neq 0,
+\]
+since all the $k$'s are positive or zero.
+
+Similarly, under the second hypothesis,
+\begin{align*}
+  \mathop{L}_{x \doteq a} \frac{f_1(x) \pm f_2(x) \pm \ldots \pm
+  f_n(x)}{f_1(x)}
+  & = \mathop{L}_{x \doteq a} \left\{%
+  \frac{f_1(x)}{f_1(x)} \pm \frac{f_2(x)}{f_1(x)} \pm \ldots \pm
+  \frac{f_n(x)}{f_1(x)} \right\}\\
+  & = 1 + 0 + \ldots + 0 = 1.\qedhere
+\end{align*}
+\end{proof}
+
+\begin{theorem}[64]\hypertarget{thm64}{}
+If $f_3(x)$ and $f_4(x)$ are infinitesimals with respect to $f_1(x)$
+and $f_2(x)$, then
+\[
+  \mathop{L}_{x \doteq a}
+  \frac{\{f_1(x) + f_3(x)\} \cdot \{f_2(x) + f_4(x)\}}{f_1(x)\cdot f_2(x)}=1.
+\]
+\end{theorem}
+
+\begin{proof}
+\begin{align*}
+  &\mathop{L}_{x \doteq a} \frac{\{f_1(x) + f_3(x)\} \cdot \{f_2(x) +
+  f_4(x)\}}{f_1(x)\cdot f_2(x)} \\
+  =&\mathop{L}_{x \doteq a} \frac{f_1(x)\cdot f_2(x) + f_1(x)\cdot
+  f_4(x) + f_3(x)\cdot f_2(x) + f_3(x)\cdot f_4(x)}{f_1(x)\cdot
+  f_2(x)} \\
+  =&\mathop{L}_{x \doteq a} \frac{f_1(x)\cdot f_2(x)}{f_1(x)\cdot f_2(x)} +
+  \mathop{L}_{x \doteq a} \frac{f_1(x)\cdot f_4(x)}{f_1(x)\cdot
+  f_2(x)} + \mathop{L}_{x \doteq a} \frac{f_3(x) \cdot f_2(x)}
+  {f_1(x)\cdot f_2(x)} + \mathop{L}_{x \doteq a} \frac{f_3(x) \cdot
+  f_4(x)} {f_1(x)\cdot f_2(x)} = 1.\qedhere
+\end{align*}
+\end{proof}
+
+
+\section{The Limit of a Quotient.}\hypertarget{chVIsec2}{}%[2]
+
+\begin{theorem}[65]\hypertarget{thm65}{}
+If as $x \doteq a$, $\varepsilon_1(x)$ is an infinitesimal with
+respect to $f_1(x)$ and $\varepsilon_2(x)$ with respect to $f_2(x)$,
+then the values approached by
+\[
+  \frac{f_1(x) + \varepsilon_1(x)}{f_2(x) + \varepsilon_2(x)}
+  \quad \text{and} \quad
+  \frac{f_1(x)}{f_2(x)}
+\]
+as $x$ approaches $a$ are identical.
+\end{theorem}
+%-----File: 118.png---Folio 106------
+
+\begin{proof}
+This follows from the identity
+\[
+  \frac{f_1(x) + \varepsilon_1(x)}{f_2(x) + \varepsilon_2(x)}
+  = \frac{f_1(x)}{f_2(x)} \cdot
+  \frac{\left(1 + \dfrac{\varepsilon_1(x)}{f_1(x)}\right)}%
+  {\left(1 + \dfrac{\varepsilon_2(x)}{f_2(x)}\right)},
+\]
+\correction{since}{Since} $\dfrac{\varepsilon_1(x)}{f_1(x)} $ and
+$\dfrac{\varepsilon_2(x)}{f_2(x)}$ are infinitesimal.
+\end{proof}
+
+\begin{corollary}
+If $f_1(x)$ and $f_2(x)$ are infinite at $x=a$, then
+\[
+  \frac{f_1(x) + c} {f_2(x) + d} \quad
+  \text{and} \quad \frac{f_1(x)}{f_2(x)}
+\]
+approach the same values.
+\end{corollary}
+
+\begin{theorem}[66]\hypertarget{thm66}{}
+If $\displaystyle \mathop{L}_{x \doteq a} \dfrac{f_1(x)}{\phi_1(x)} =
+\mathop{L}_{x \doteq a} \dfrac{f_2(x)}{\phi_2(x)} = k$, and if
+$\displaystyle \mathop{L}_{x \doteq a} \frac{\phi_1(x)}{\phi_2(x)} =
+l$\\ is finite, then
+\[
+  k = \mathop{L}_{x \doteq a}
+  \frac{f_1(x) + f_2(x)}{\phi_1(x) + \phi_2(x)}
+  = \mathop{L}_{x \doteq a_1} \frac{f_1(x)}{\phi_1(x)},
+\]
+provided $l \neq-1$ if $k$ is finite, and provided $l>0$ if $k$ is
+infinite.
+\end{theorem}
+
+\begin{proof}
+\begin{align*}
+  &\frac{f_1(x) + f_2(x)}{\phi_1(x) +
+  \phi_2(x)}-\frac{f_2(x)}{\phi_2(x)} =
+  \frac{f_1(x)\phi_2(x)-f_2(x)\phi_1(x)}{\phi_2(x)(\phi_1(x) +
+  \phi_2(x))},\\
+  &\frac{f_1(x) + f_2(x)}{\phi_1(x) + \phi_2(x)} =
+  \frac{f_2(x)}{\phi_2(x)} +
+  \left(\frac{f_1(x)}{\phi_1(x)}-\frac{f_2(x)}{\phi_2(x)}\right) \cdot
+  \left(\frac{1}{1 + \dfrac{\phi_2(x)}{\phi_1(x)}} \right).
+\end{align*}
+In case $k$ is finite, the second term of the right-hand member is
+evidently infinitesimal if $l \neq-1$ and the theorem is proved.  In
+the case where $k$ is infinite we write the above identity in the
+following form:
+\[
+  \frac{f_1(x) + f_2(x)}{\phi_1(x) + \phi_2(x)}
+  = \frac{f_1(x)}{\phi_1(x)} \cdot \frac{1}{1 +
+  \dfrac{\phi_2(x)}{\phi_1(x)}} + \frac{f_2(x)}{\phi_2(x)} \cdot
+  \frac{1}{1 + \dfrac{\phi_1(x)}{\phi_2(x)}}.
+\]
+%-----File: 119.png---Folio 107------
+Both terms of the second member approach $+\infty$ or both $-\infty$
+if $l>0$.
+\end{proof}
+\begin{corollary}
+If $\phi_1(x)$ and $\phi_2(x)$ are both positive for some $V^*(a)$,
+and if $\displaystyle k=\mathop{L}_{x\doteq a}
+\frac{f_1(x)}{\phi_1(x)} = \mathop{L}_{x\doteq a}
+\frac{f_2(x)}{\phi_2(x)}$, then $\displaystyle\mathop{L}_{\text{\correction{$x\doteq a$}{$x=a$}}}
+\frac{f_1(x)+f_2(x)}{\phi_1(x)+\phi_2(x)} = k$
+whenever $k$ is finite. If $k$ is infinite, the condition must be
+added that $\dfrac{\phi_1(x)}{\phi_2(x)}$ has a finite upper and a
+non-zero lower bound.
+\end{corollary}
+
+\begin{theorem}[67]\hypertarget{thm67}{}
+If $f_1(x)$ and $f_2(x)$ are both infinitesimals as $x\doteq a$, then
+a necessary and sufficient condition that
+\[
+  \mathop{L}_{x\doteq a} \frac{f_1(x)}{f_2(x)}
+  =k\qquad \text{($k$ finite and not zero)}
+\]
+is that in the equation $f_1(x)=k\cdot f_2(x) + \varepsilon(x)$,
+$\varepsilon(x)$ is an infinitesimal of higher order than $f_1(x)$ or
+$f_2(x)$.
+\end{theorem}
+
+\begin{proof}
+(1) \emph{The condition is necessary.}---Since
+    $\displaystyle\mathop{L}_{x\doteq a} \frac{f_1(x)}{f_2(x)}=k$,
+\[
+  \frac{f_1(x)}{f_2(x)}=k+\varepsilon'(x),
+\]
+or $f_1(x)=f_2(x)\cdot k+f_2(x)\cdot\varepsilon'(x)$, where
+$\displaystyle\mathop{L}_{x\doteq a} \varepsilon'(x)=0$ (Theorem~\hyperlink{thm31}{31}).
+By Theorems \hyperlink{thm60}{60} and \hyperlink{thm61}{61}, $f_1(x)$ and $f_2(x)\cdot k$ are of the same
+order, since $k\neq0$, while by Theorem~\hyperlink{thm62}{62} $\varepsilon'(x)\cdot
+f_2(x)$ is of higher order than either $f_1 (x)$ or $f_2(x)$. Hence
+the function $\varepsilon(x) = \varepsilon'(x)\cdot f_2(x)$ is
+infinitesimal.
+
+(2) \emph{The condition is sufficient.}---By hypothesis $f_1(x) =
+f_2(x)\cdot k + \varepsilon(x)$, where $f_1(x)$ and $f_2(x)$ are of
+the same order as $x\doteq a$, while $\varepsilon(x)$ is of higher
+order than these. Let $\varepsilon'(x)
+=\dfrac{\varepsilon(x)}{f_2(x)}$, which by hypothesis is an
+infinitesimal. We then have $\dfrac{f_1(x)}{f_2(x)} =k+
+\varepsilon'(x)$. Hence, by Theorem~\hyperlink{thm31}{31}, $\displaystyle
+\mathop{L}_{x\doteq a} \dfrac{f_1(x)}{f_2(x)}=k$.
+\end{proof}
+%-----File: 120.png---Folio 108------
+
+\section[Indeterminate Forms]{Indeterminate Forms.\footnotemark}\hypertarget{chVIsec3}{}%[3]
+\footnotetext{%
+The theorems of this section are to be used in \hyperlink{chVIIsec6}{\S~6} of Chap.~\hyperlink{chapVII}{VII}.}
+
+\begin{lemma}
+If $\dfrac ab$ and $\dfrac cd$ are any two fractions such, that $b$
+and $d$ are both positive or both negative, then the value of
+\[
+  \frac{a + c}{b+d}
+\]
+lies on the interval $\interval{\dfrac ab}{\dfrac cd}$.
+\end{lemma}
+
+\begin{proof}
+Suppose $b$ and $d$ both positive and
+\[
+  \frac ab \geqq \frac{a+c}{b+d},
+\]
+then
+\begin{gather*}
+  ab+ad \geqq ab+bc.\\
+  \therefore ad \geqq bc;\\
+  \therefore cd+ad \geqq cd+bc;\\
+  \therefore \frac{a+c}{b+d} \geqq \frac cd.
+\end{gather*}
+The other cases follow similarly.
+\end{proof}
+
+\begin{theorem}[68]\hypertarget{thm68}{}
+If $f(x)$ and $\phi(x)$, defined on some $V(+\infty)$, are both
+infinitesimal as $x$ approaches $+\infty$, and if for some positive
+number $h$, $\phi(x+h)$ is always less than $\phi(x)$ and
+\[
+  \mathop{L}_{x\doteq\infty} \frac{f(x+h)-f(x)}{\phi(x+h)-\phi(x)}=k,
+\]
+then
+\[
+  \mathop{L}_{x\doteq\infty} \frac{f(x)}{\phi(x)}
+\]
+exists and is equal to $k$.\footnote{%
+  This and the following theorem are due to \textsc{O.~Stolz}, who
+  generalized them from the special cases (stated in our corollaries)
+  due to \textsc{Cauchy}. See \textsc{Stolz} und \textsc{Gmeiner},
+  Functionentheorie, Vol.~1, p.~31. See also the reference to
+  \textsc{Bortolotti} given on page~\pageref{borlottip103}.}
+\end{theorem}
+%-----File: 121.png---Folio 109------
+
+\begin{proof}
+Let $V_1(k)$ and $V_2(k)$ be a pair of vicinities of $k$ such that
+$V_2(k)$ is entirely within $V_1(k)$. By hypothesis there exists an
+$h$ and an $X_2$ such that if $x>X_2$,
+\hypertarget{eq1p109}{\[
+  \frac{f(x+h)-f(x)}{\phi(x+h)-\phi(x)} \tag{1}
+\]}
+is in $V_2(k)$. Since this is true for every $x>X_2$,
+\hypertarget{eq2p109}{\[
+  \frac{f(x+2h)-f(x+h)}{\phi(x+2h)-\phi(x+h)}\tag{2}
+\]}
+is also in $V_2(k)$. From this it follows by means of the lemma that
+\[
+  \frac{f(x+2h)-f(x)}{\phi(x+2h)-\phi(x)},\tag{3}
+\]
+whose value is between the values of \hyperlink{eq1p109}{(1)} and \hyperlink{eq2p109}{(2)}, is also in $V_2(k)$.
+By repeating this argument we have that for every integral value of
+$n$, and for every $x>X_2$,
+\[
+  \frac{f(x+nh)-f(x)}{\phi(x+nh)-\phi(x)}
+\]
+is in $V_2(k)$.
+
+By Theorem~\hyperlink{thm65}{65}, for any $x$
+\[
+  \mathop{L}_{n\doteq\infty} \frac{f(x+nh)-f(x)}{\phi(x+nh)-\phi(x)}
+  = \frac{f(x)}{\phi(x)}.
+\]
+Hence for every $x$ and for every $\varepsilon$ there exists a value
+of $n$, $N_{x\varepsilon}$, such that if $n>N_{x\varepsilon}$,
+\[
+  \left| \frac{f(x+nh)-f(x)}{\phi(x+nh)-\phi(x)}-\frac{f(x)}{\phi(x)} \right| < \varepsilon.
+\]
+Taking $\varepsilon$ less than the distance between the nearest
+end-points of $V_1(k)$ and $V_2(k)$ it is plain that for every
+$x>X_2$, $\dfrac{f(x)}{\phi(x)}$ is
+%-----File: 122.png---Folio 110------
+on $V_1(k)$, which, according to Theorem~\hyperlink{thm26}{26}, proves that
+\[
+  \mathop{L}_{x\doteq\infty} \frac{f(x)}{\phi(x)} = k.\qedhere
+\]
+\end{proof}
+\begin{corollary}
+If $[n]$ is the set of all positive integers and $\phi(n+1)<\phi(n)$
+and $f(n)$ and $\phi(n)$ are both infinitesimal as $n\doteq\infty$,
+then if
+\[
+  \mathop{L}_{n\doteq\infty} \frac{f(n+1)-f(n)}{\phi(n+1)-\phi(n)} =k,
+\]
+it follows that $\displaystyle \mathop{L}_{n\doteq\infty}
+\dfrac{f(n)}{\phi(n)}$ exists and is equal to $k$.
+\end{corollary}
+
+\begin{theorem}[69]\hypertarget{thm69}{}
+If $f(x)$ is bounded on every finite interval of a certain
+$V(+\infty)$, and if $\phi(x)$ is monotonic on the same $V(+\infty)$
+and $\displaystyle \mathop{L}_{x\doteq\infty} \phi(x) = +\infty$, and
+if for some positive number $h$
+\[
+  \mathop{L}_{x\doteq\infty} \frac{f(x+h)-f(x)}{\phi(x+h)-\phi(x)} =k,
+\]
+then
+\[
+  \mathop{L}_{x\doteq\infty} \frac{f(x)}{\phi(x)}
+\]
+exists and is equal to $k$.
+\end{theorem}
+
+\begin{proof}
+By hypothesis, for every pair of vicinities $V_1(k)$ and $V_2(k)$,
+$V_2(k)$ entirely within $V_1(k)$, there exists an $X_2$ such that if
+$x>X_2$, then
+\[
+  \frac{f(x+h)-f(x)}{\phi(x+h)-\phi(x)}
+\]
+is in $V_2(k)$. From this it follows as in the last theorem that
+\[
+  \frac{f(x+nh)-f(x)}{\phi(x+nh)-\phi(x)}
+\]
+is in $V_2(k)$. Now make use of the identity
+%-----File: 123.png---Folio 111------
+\hypertarget{eq1p111}{\begin{align*}
+  \frac{f(x+nh)}{\phi(x+nh)}
+  &= \frac{f(x+nh)-f(x)}{\phi(x+nh)}+\frac{f(x)}{\phi(x+nh)}\\
+  &= \frac{f(x+nh)-f(x)}{\phi(x+nh)-\phi(x)}
+  \left(1-\frac{\phi(x)}{\phi(x+nh)} \right)
+  + \frac{f(x)}{\phi(x+nh)}.
+\tag{1}
+\end{align*}}
+Let $[x']$ be the set of all points on the interval \correction{$\interval{X_2}{X_2+h}$}{$\interval{X_2}{X_2}+h$}, and for this interval let $A_2$ be an upper bound of
+$|f(x')|$ and $B_2$ an upper bound of $\phi(x')$. Then
+\begin{alignat*}{2}
+  \frac{\phi(x')}{\phi(x'+nh)}
+  &= \varepsilon_1(x', n)
+  &&< \frac{B_2}{\phi(X_2+nh)}
+\\
+\intertext{and}
+  \frac{|f(x')|}{\phi(x'+nh)}
+  &= \varepsilon_2(x', n)
+  &&< \frac{A_2}{\phi(X_2+nh)}.
+\end{alignat*}
+Hence for every $\varepsilon$ there exists a value of $n$,
+$N_{\varepsilon_V}$, such that if $n > N_{\varepsilon_V}$
+\hypertarget{eq2p111}{\[
+  \varepsilon_1(x', n) < \varepsilon \qquad \text{ and } \qquad
+  \varepsilon_2(x', n) < \varepsilon
+\tag{2}
+\]}
+independently of $x'$ so long as $x'$ is on \correction{$\interval{X_2}{X_2+h}$}{$\interval{X_2}{X_2}+h$}.
+
+There are then three cases to discuss:
+\begin{align*}
+  (1)&\ k \text{ finite.}
+  & (2)&\ k = +\infty.
+  & (3)&\ k =-\infty.
+\end{align*}
+(1) $k$ {\em finite}. By the preceding argument, for $x > X_2$,
+\[
+  \frac{f(x+nh)-f(x)}{\phi(x+nh)-\phi(x)}
+\]
+is in $V_2(k)$, and hence
+\[
+  \frac{|f(x'+nh)-f(x')|}{\phi(x'+nh)-\phi(x')}
+  < K + \varepsilon_{V_2},
+\]
+where $\varepsilon_{V_2}$, is the length of the interval $V_2(k)$ and
+$K$ the absolute value of $k$.
+
+Then, in view of \hyperlink{eq1p111}{(1)},
+\[
+  \left|\frac{f(x'+nh)}{\phi(x'+nh)}
+  -\frac{f(x'+nh)-f(x')}{\phi(x'+nh)-\phi(x')}
+  \right|
+  < (K+\varepsilon_{V_2}) \varepsilon_1(x',n) + \varepsilon_2(x',n).
+\]
+%-----File: 124.png---Folio 112------
+Now take $\varepsilon_V$ smaller in absolute value than the length of
+the interval between the closer end-points of $V_1(k)$ and
+$V_2(k)$. By \hyperlink{eq2p111}{(2)} there exists a value of $n$, $N_{\varepsilon_V}$,
+such that if $n>N_{\varepsilon_V}$,
+\begin{align*}
+  \varepsilon_1(x', n) &< \frac{\varepsilon_V}{2(K+\varepsilon_{V_2})}\\
+\intertext{and}
+  \varepsilon_2(x', n) &< \frac{\varepsilon_V}{2}
+\end{align*}
+for all values of $x'$ on \correction{$\interval{X_2}{X_2+h}$}{$\interval{X_2}{X_2}+h$}.
+
+Hence for $n > N_{\varepsilon_V}$
+\[
+  \left|\frac{f(x'+nh)}{\phi(x'+nh)}
+  -\frac{f(x'+nh)-f(x')}{\phi(x'+nh)-\phi(x')}
+  \right|
+  < (K + \varepsilon_{V_2}) \frac{\varepsilon_V}{2(K+\varepsilon_{V_2})}
+  + \frac{\varepsilon_V}{2}
+  = \varepsilon_V,
+\]
+and since for $x > X_2 + N_{\varepsilon_V}h$ there is an
+$n>N_{\varepsilon_V}$ and an $x'$ between $X_2$ and $X_2 + h$ such
+that
+\[
+  x' + nh = x,
+\]
+it follows that if $x > X_2 + N_{\varepsilon_V}$,
+\[
+  \left|
+    \frac{f(x)}{\phi(x)}
+  -\frac{f(x'+nh)-f(x')}{\phi(x'+nh)-\phi(x)}
+  \right|
+  < \varepsilon_V,
+\]
+and therefore, $\dfrac{f(x)}{\phi(x)}$ is on $V_1(k)$.
+
+This means, according to Theorem~\hyperlink{thm26}{26}, that
+\[
+  \mathop{L}_{x\doteq\infty} \frac{f(x)}{\phi(x)} = k.
+\]
+
+(2) $k = +\infty$.
+
+If the numbers $m_1$ and $m_2$ are the lower end points of $V_1(k)$
+and $V_2(k)$, then
+\[
+  \frac{f(x'+nh)-f(x')}{\phi(x'+nh)-\phi(x')} > m_2 \quad \text{for} \quad
+  x' > X_2.
+\]
+%-----File: 125.png---Folio 113------
+If $\varepsilon_V$ is then chosen less than $m_2-m_1$, there will
+exist a value of $N_{\varepsilon_V}$ such that
+\[
+  \varepsilon_1(x', n) < \frac{\varepsilon_V}{2m_2}
+  \qquad \text{and} \qquad
+  \varepsilon_2(x', n) < \frac{\varepsilon_V}{2m_1}
+\]
+for all values of $n > N_{\varepsilon_V}$ independently of $x'$ so
+long as $x'$ is in \correction{$\linterval{X_2}{X_2+h}$}{$\linterval{X_2}{X_2}+h$}. Then, in view of \hyperlink{eq1p111}{(1)},
+\[
+  \frac{f(x'+nh)}{\phi(x'+nh)}
+  > m_2 \left(1-\frac{\varepsilon_V}{2m_2} \right)
+          -\frac{\varepsilon_V}{2m_2}
+  > m_2-\frac{\varepsilon_V}{2} \left(1+\frac{1}{m_2} \right).
+\]
+Since there is no loss of generality if $m_2 > +1$, this proves that
+for $x > X_2 + N_{\varepsilon_V} n$,
+\[
+  \frac{f(x)}{\phi(x)} > m_2-\varepsilon_V > m_1,
+\]
+and hence $\dfrac{f(x)}{\phi(x)}$ is on $V_1(k)$.
+
+(3) $k =-\infty$ is treated in an analogous manner.
+\end{proof}
+\begin{ncorollary}[1]
+If $[n]$ is the set of all positive integers and if
+\[
+  \phi(n+1) > \phi(n) \qquad \text{and} \qquad
+  \mathop{L}_{n=\infty} \phi(n) = \infty,
+\]
+then if
+\[
+  \mathop{L}_{n=\infty}{L} \frac{f(n+1)-f(n)}{\phi(n+1)-\phi(n)} = k,
+\]
+it follows that $\displaystyle{\mathop{L}_{n = \infty}}
+\dfrac{f(n)}{\phi(n)}$ exists and is equal to $k$\correction{.}{}
+\end{ncorollary}
+
+\begin{ncorollary}[2]
+If $f(x)$ is bounded on every interval, \correction{$\interval{x}{(x+1)}$}{$\interval{x}{(x}+1)$}, and if
+\[
+  \mathop{L}_{x=\infty} f(x+1)-f(x) = k,
+\]
+then
+\[
+  \mathop{L}_{x=\infty} \frac{f(x)}{x}
+\]
+exists and is equal to $k$.
+\end{ncorollary}
+%-----File: 126.png---Folio 114------
+
+\section{Rank of Infinitesimals and Infinites.}\hypertarget{chVIsec4}{}%[4]
+\index{Rank of infinitesimals and infinites}
+\begin{definition}
+If on some $V^*(a)$ neither $f_1(x)$ nor $f_2(x)$ vanishes, and
+$\displaystyle\left|\frac{f_1(x)}{f_2(x)} \right|$ and
+$\displaystyle\left|\frac{f_2(x)}{f_1(x)} \right|$ are both bounded as
+$x$ approaches $a$, then $f_1(x)$ and $f_2(x)$ are of the same
+\textit{rank} whether $\displaystyle{\mathop{L}_{x \doteq a}}
+\frac{f_1(x)}{f_2(x)}$ exists or not.\footnote{%
+  $x$ and $x \cdot (\sin\dfrac1{x}+2)$ are of the same rank but not of
+  the same order as $x$ approaches zero.}
+\end{definition}
+
+The following theorem is obvious.
+\begin{theorem}[70]\hypertarget{thm70}{}
+If $f_1(x)$ and $f_2(x)$ are of the same order, they are of the same
+rank, and if $f_1(x)$ and $f_2(x)$ are of different orders, they are
+not of the same rank. If $f_1(x)$ and $f_2(x)$ are of the same rank,
+they may or may not be of the same order.
+\end{theorem}
+
+\begin{theorem}[71]\hypertarget{thm71}{}
+If $f_1(x)$ and $f_2(x)$ are of the same rank as $x$ approaches $a$,
+then $c\cdot f_1(x)$ and $f_2(x)$ are of the same rank, $c$ being any
+constant not zero.
+\end{theorem}
+
+\begin{proof}
+By hypothesis for some positive number $M$,
+\begin{gather*}
+  \left|\frac{f_1(x)}{f_2(x)} \right|< M \text{ and }
+  \left|\frac{f_2(x)}{f_1(x)} \right|< M,\\
+\intertext{hence}
+  \left|\frac{c \cdot f_1(x)}{f_2(x)} \right|< M \cdot|c| \text{ and }
+  \left|\frac{f_2(x)}{c \cdot f_1(x)} \right|< \frac{M}{|c|}.\qedhere
+\end{gather*}
+\end{proof}
+
+\begin{theorem}[72]\hypertarget{thm72}{}
+If $f_1(x)$ and $f_2(x)$ are of the same rank and $f_2(x)$ and
+$f_3(x)$ are of the same rank as $x$ approaches $a$, then $f_1(x)$ and
+$f_3(x)$ are of the same rank as $x$ approaches $a$.
+\end{theorem}
+
+\begin{proof}
+By hypothesis,
+\[
+  \left|\frac{f_1(x)}{f_2(x)} \right|< M_1 \text{ and }
+  \left|\frac{f_2(x)}{f_3(x)} \right|< M_2
+\]
+in some neighborhood of $x=a$. Therefore
+\[
+  \left|\frac{f_1(x)}{f_2(x)} \right|\cdot
+  \left|\frac{f_2(x)}{f_3(x)} \right|< M_1 \cdot M_2 \text{ or }
+  \left|\frac{f_1(x)}{f_3(x)} \right|< M_1 \cdot M_2.\qedhere
+\]
+%-----File: 127.png---Folio 115------
+In the same manner
+\[
+  \left|\frac{f_2(x)}{f_1(x)} \right|< M_1 \text{ and }
+  \left|\frac{f_3(x)}{f_2(x)} \right|< M_2, \text{ whence }
+  \left|\frac{f_3(x)}{f_1(x)} \right|< M_1 \cdot M_2.
+\]
+\end{proof}
+\begin{theorem}[73]\hypertarget{thm73}{}
+If $f_1(x)$ is infinitesimal (infinite) and does not vanish on some
+$V^*(a)$, and if $f_2(x)$ and $f_3(x)$ are infinitesimal (infinite) of
+the same rank as $x$ approaches $a$, then $f_1(x) \cdot f_2(x)$ is of
+higher order than $f_3(x)$, and $f_1(x) \cdot f_3(x)$ is of higher
+order than $f_2(x)$. Conversely, if for every function, $f_1(x)$,
+infinitesimal (infinite) at $a$, $f_1(x) \cdot f_2(x)$ is of higher
+order than $f_3(x)$, and $f_1(x) \cdot f_3(x)$ is of higher order than
+$f_2(x)$, then $f_2(x)$ and $f_3(x)$ are of the same rank.
+\end{theorem}
+
+\begin{proof}
+Since $\displaystyle\left|\frac{f_1(x)}{f_3(x)} \right|$ is bounded as
+$x$ approaches $a$, it follows by Theorem~\hyperlink{thm33}{33} that
+\[
+  \mathop{L}_{x\doteq a} \frac{f_1(x)\cdot f_2(x)}{f_3(x)} = 0,
+\]
+which proves the first part of the theorem.
+
+Since likewise $\displaystyle\left|\frac{f_3(x)}{f_2(x)} \right|$ is
+bounded, we have that
+\[
+  \mathop{L}_{x\doteq a} \frac{f_1(x)\cdot f_3(x)}{f_2(x)} = 0.
+\]
+
+Suppose that for every $f_1(x)$
+\[
+  \mathop{L}_{x\doteq a} \frac{f_1(x)\cdot f_2(x)}{f_3(x)} = 0
+\text{ and }
+  \mathop{L}_{x\doteq a} \frac{f_1(x)\cdot f_3(x)}{f_2(x)} = 0,
+\]
+and that $f_2(x)$ and $f_3(x)$ are not of the same rank. Then, on a
+certain subset $[x']$, $\displaystyle \mathop{L}_{x\doteq a}
+\frac{f_2(x')}{f_3(x')} = 0$, or on some other subset $[x'']$,
+$\displaystyle \mathop{L}_{x\doteq a} \frac{f_3(x'')}{f_2(x'')} =
+0$. Let $f_1(x) = \dfrac{f_2(x)}{f_3(x)}$ on the set $[x']$ for which
+$\displaystyle \mathop{L}_{x\doteq a} \frac{f_2(x)}{f_3(x)} = 0$, and
+$x-a$ on the other points of the continuum;
+%-----File: 128.png---Folio 116------
+then $f_1(x)$ is an infinitesimal as $x$ approaches $a$, while for the
+set $[x']$
+\[
+  \mathop{L}_{x \doteq a} \frac{f_1{(x')} \cdot f_3(x')}{f_3(x')} =
+  \mathop{L}_{x \doteq a} \frac{f_2(x')}{f_3(x')} \cdot
+  \frac{f_3(x')}{f_2(x')} = 1,
+\]
+which contradicts the hypothesis that
+\[
+  \mathop{L}_{x \doteq a} \frac{f_1(x) \cdot f_3(x)}{f_2(x)} = 0.
+\]
+Similarly if on a certain subset $\displaystyle{\mathop{L}_{x \doteq
+a}} \dfrac{f_3(x)}{f_2(x)} = 0$, we obtain a contradiction by putting
+$f_1(x) = \dfrac{f_3(x)}{f_2(x)}$.
+\end{proof}
+%-----File: 129.png---Folio 117------
+
+
+\chapter{DERIVATIVES AND DIFFERENTIALS.}\hypertarget{chapVII}{}%[VII]
+
+\section{Definition and Illustration of Derivatives.}\hypertarget{chVIIsec1}{}%[1]
+
+\begin{definition}\index{Derivative}
+If the ratio $\frac{f(x)-f(x_1)}{x-x_1}$ approaches a definite limit,
+finite or infinite, as $x$ approaches $x_1$, the \textit{derivative}
+of $f(x)$ at the point $x_1$ is the limit
+\[
+  \mathop{L}_{x \doteq x_1} \frac{f(x)-f(x_1)}{x-x_1}.
+\]
+\end{definition}
+\begin{figure}[!hbtp]\label{fig14}\hypertarget{fig14}{}
+\centering
+\setlength{\unitlength}{0.15\textwidth}
+\begin{picture}(6,4)(0,0)
+\put(0,0.25){\line(1,0){6}}
+\put(0,0.25){\line(0,1){3.5}}
+\qbezier(1,1.25)(1.5,2)(2,2.5)
+\qbezier(2,2.5)(2.5,3)(3,3.25)
+\qbezier(3,3.25)(4,3.75)(5,2.5)
+\path(2,0.25)(2,2.5)(3,3.25)(3,0.25)
+\path(2,2.5)(3,2.5)
+\put(2,0.23){\makebox(0,0)[tc]{$x_1$}}
+\put(3,0.23){\makebox(0,0)[tc]{$x$}}
+\put(1.9,2.5){\makebox(0,0)[br]{$A$}}
+\put(2.9,3.25){\makebox(0,0)[br]{$B$}}
+\put(3.1,3.25){\makebox(0,0)[tl]{$f(x)$}}
+\put(3.1,2.5){\makebox(0,0)[lc]{$f(x_1)$}}
+\put(3,0){\makebox(0,0)[tc]{\sc Fig.~14.}}
+\end{picture}
+\end{figure}
+It is implied that the function $f(x)$ is a single-valued function of
+$x$. $x-x_1$ is sometimes denoted by $\Delta x_1$, and $f(x)-f(x_1)$
+by $\Delta f(x_1)$, or, if $y=f(x)$, by $\Delta y_1$.
+
+An obvious illustration of a derivative occurs in Cartesian geometry
+when the function is represented by a graph (Fig.~\hyperlink{fig14}{14}).
+%-----File: 130.png---Folio 118------
+$\dfrac{f(x)-f(x_1)}{x-x_1}$ is the slope of the line $AB$. If we
+suppose that the line $AB$ approaches a fixed direction (which in this
+figure would obviously be the case) as $x$ approaches $x_1$, then
+$\displaystyle\mathop{L}_{x \doteq x_1} \frac{f(x)-f(x_1)}{x-x_1}$
+will exist and will be equal to the slope of the limiting position of
+$AB$.
+
+If the point $x$ were taken only on one side of $x_1$, we should have
+two similar limiting processes. It is quite conceivable, however, that
+limits should exist on each side, but that they should differ. That
+case occurs if the graph has a cusp as in Fig.~\hyperlink{fig15}{15}.
+
+\begin{figure}[!hbtp]\label{fig15}\hypertarget{fig15}{}
+\centering
+\setlength{\unitlength}{0.15\textwidth}
+\begin{picture}(5,3)(0,-0.5)
+\put(0,0){\line(1,0){5}}
+\put(2,0){\line(0,1){2.5}}
+\qbezier(1,1)(1.85,1.55)(2,2.5)
+\qbezier(2,2.5)(2.5,1.5)(3,1.25)
+\qbezier(3,1.25)(4,0.75)(4.5,0.6)
+\path(1,0)(1,1)(2,1)(2,1.25)(3,1.25)(3,0)
+\put(1,-0.1){\makebox(0,0)[tc]{$x$}}
+\put(2,-0.1){\makebox(0,0)[tc]{$x_1$}}
+\put(3,-0.1){\makebox(0,0)[tc]{$x$}}
+\put(1,1){\makebox(0,0)[br]{$B$}}
+\put(2,2.5){\makebox(0,0)[cb]{$A$}}
+\put(3,1.25){\makebox(0,0)[lb]{$B$}}
+\put(2.1,1){\makebox(0,0)[lc]{$f(x)$}}
+\put(1.9,1.25){\makebox(0,0)[rc]{$f(x)$}}
+\put(2.1,2.5){\makebox(0,0)[cl]{$f(x_1)$}}
+\put(2.5,-0.5){\makebox(0,0)[bc]{\sc Fig.~15.}}
+\end{picture}
+\end{figure}
+
+
+These\index{Derivative!progressive and regressive}\index{Progressive derivative}\index{Regressive derivative} two cases are distinguished by the terms progressive
+and regressive derivatives. When the independent variable
+approaches its limit from below we speak of the progressive
+derivative, and when from above we speak of the regressive
+derivative. It follows from the definition of derivative that,
+except in one singular case, it exists only when both these
+limits exist and are equal. The exception is the case of a
+derivative of a function at an end-point of an interval upon
+which the function is defined. Obviously both the progressive
+and the regressive derivative cannot exist at such a point. In
+%-----File: 131.png---Folio 119------
+this case we say the derivative exists if either the progressive or
+the regressive derivative exists.
+
+Whether the progressive and regressive derivatives exist or not, there
+exist always four so-called derived numbers (which may be
+$\pm\infty$), namely, the upper and lower bounds of indetermination of
+\[
+  \frac{f(x)-f(x_1)}{x-x_1},
+\]
+as $x \doteq x_1$ from the right or from the left. (Compare
+page~\pageref{limp84}, Chapter~\hyperlink{chapIV}{IV}.) The derived numbers are denoted by
+the symbols.
+\[
+  \overrightarrow{D},\ \underrightarrow{D},\
+  \overleftarrow{D},\ \underleftarrow{D},
+\]
+analogous to the symbols on page~\pageref{limp84}. Of course, in every
+case,
+\[
+  \overrightarrow{D}\geqq \underrightarrow{D} \text{ and }
+  \overleftarrow{D} \geqq \underleftarrow{D}.
+\]
+
+If we consider the curve representing the function
+\[
+  y=x \cdot \sin \frac1x
+\]
+at the point $x=0$, it is apparent that the limiting position of $AB$
+does not exist, although the function is continuous at the point $x=0$
+if defined as zero for $x=0$. For at every maximum and minimum of the
+curve $\sin\dfrac{1}{x}$, $x \cdot \sin\dfrac{1}{x} = \pm x$, and the
+curve touches the lines $x=y$ and $x=-y$. That is,
+$\dfrac{f(x)-f(x_1)}{x-x_1}$ approaches every value between $1$ and
+$-1$ inclusive, as $x$ approaches zero.
+
+The notion \textit{derivative} is fundamental in physics as well as in
+geometry. If, for instance, we consider the motion of a body, we may
+represent its distance from a fixed point as a function of time,
+$f(t)$. At a certain instant of time $t_1$ its distance from the fixed
+point is $f(t_1)$, and at another instant $t_2$ it is $f(t_2)$; then
+\[
+  \frac{f(t_1)-f(t_2)}{t_1-t_2}
+\]
+%-----File: 132.png---Folio 120------
+is the average velocity of the body during the interval of time
+$t_1-t_2$ in a direction from or toward the assumed fixed point.
+Whether the motion be from or toward the fixed point is of course
+indicated by the sign of the expression
+$\dfrac{f(t_1)-f(t_2)}{t_1-t_2}$.  If we consider this ratio as the
+time interval is taken shorter and shorter, that is, as $t_2$
+approaches $t_1$, it will in ordinary physical motion approach a
+perfectly definite limit. This limit is spoken of as the velocity of
+the body at the instant $t_1$.
+
+\begin{definition}\index{Derived function}
+The derivative of a function $y=f(x)$ is denoted by $f'(x)$ or by
+$D_xf(x)$ or $\dfrac{df(x)}{dx}$ or $\dfrac{dy}{dx}$. $f'(x)$ is also
+referred to as the \emph{derived function} of $f(x)$.
+\end{definition}
+
+\section{Formulas of Differentiation.}\hypertarget{chVIIsec2}{}%[2]
+
+\begin{theorem}[74]\hypertarget{thm74}{}
+The derivative of a constant is zero. More precisely: If there exists
+a neighborhood of $x_1$ such that for every value of $x$ on this
+neighborhood $f(x) =f(x_1)$, then $f'(x_1) =0$.
+\end{theorem}
+
+\begin{proof}
+In the neighborhood specified $\dfrac{f(x)-f(x_1)}{x-x_1}=0$ for every
+value of $x$.
+\end{proof}
+
+\begin{corollary}
+If $f'(x_1)$ exists and if in every $V^*(x_1)$ there is a value of $x$
+such that $f(x) =f(x_1)$, then $f'(x_1) = 0$.
+\end{corollary}
+
+\begin{theorem}[75]\hypertarget{thm75}{}
+When for two functions $f_1(x)$ and $f_2(x)$ the derived functions
+$f_1'(x)$ and $f_2'(x)$ exist at $x_1$ it follows that, except in the
+indeterminate case $\infty-\infty$,
+\begin{enumerate}
+\item[\textnormal{(\textit{a})}] If $f_3(x) = f_1(x) + f_2(x)$, then $f_3(x)$ has a
+derivative at $x_1$ and
+\[
+  f_3'(x_1) =f_1'(x_1) + f_2'(x_1).
+\]
+
+\item[\textnormal{(\textit{b})}] If $f_3(x) = f_1(x) \cdot f_2(x)$, then $f_3(x)$
+has a derivative at $x_1$ and
+\[
+  f_3'(x_1) = f_1'(x_1) \cdot f_2(x_1) + f_1(x_1) \cdot f_2'(x_1).
+\]
+
+\item[\textnormal{(\textit{c})}]If $f_3(x) = \dfrac{f_1(x)}{f_2(x)}$, then,
+provided there is a $V(x_1)$ upon which $f_2(x) \neq 0$, $f_3(x)$ has
+a derivative and
+\[
+  f_3'(x_1)
+  = \frac{f_1'(x_1) \cdot f_2(x_1)-f_1(x_1) \cdot f_2'(x_1)}
+       {\{f_2(x_1)\}^2}.
+\]
+\end{enumerate}
+\end{theorem}
+%-----File: 133.png---Folio 121------
+
+\begin{proof}
+By definition and the theorems of Chapter~\hyperlink{chapIV}{IV} (which exclude the case
+$\infty-\infty$),
+\begin{enumerate}
+\item[(\textit a)]
+\begin{align*}
+  f_1'(x_1) + f_2'(x_1)
+  &= \mathop{L}_{x\doteq x_1} \frac{f_1(x)-f_1(x_1)}{x-x_1}
+  + \mathop{L}_{x\doteq x_1} \frac{f_2(x)-f_2(x_1)}{x-x_1}
+\tag{1}
+\\
+&= \mathop{L}_{x\doteq x_1}
+   \left\{ \frac{f_1(x)-f_1(x_1)}{x-x_1}
+         + \frac{f_2(x)-f_2(x_1)}{x-x_1} \right\}
+\tag{2}
+\\
+&= \mathop{L}_{x\doteq x_1}
+   \frac{f_1(x)+f_2(x)-f_1(x_1)-f_2(x_1)}{x-x_1}
+\tag{3}
+\\
+&= \mathop{L}_{x\doteq x_1} \frac{f_3(x)-f_3(x_1)}{x-x_1}.
+\end{align*}
+But by definition,
+\[
+  f_3'(x_1) = \mathop{L}_{x\doteq x_1} \frac{f_3(x)-f_3(x_1)}{x-x_1}.
+\tag{4}
+\]
+Hence $f_3'(x_1)$ exists, and $f_3'(x_1) =f_1'(x_1) +f_2'(x_1)$.
+\item[(\textit b)]
+$f_3(x)=f_1(x)\cdot f_2(x)$.\\
+Whenever $x\neq x_1$ we have the identity
+\begin{align*}
+&\frac{f_3(x)-f_3(x_1)}{x-x_1}
+= \frac{f_1(x)\cdot f_2(x)-f_1(x_1)\cdot f_2(x_1)}{x-x_1}
+\\
+=\,&\frac{f_1(x )\cdot f_2(x)-f_1(x_1)\cdot f_2(x )
+         + f_1(x_1)\cdot f_2(x)-f_1(x_1)\cdot f_2(x_1) }{x-x_1}
+\\
+=\, &f_2(x) \left\{ \frac{f_1(x)-f_1(x_1)}{x-x_1} \right\}
+   + f_1(x) \left\{ \frac{f_2(x)-f_2(x_1)}{x-x_1} \right\}.
+\end{align*}
+But the limit of the last expression exists as $x\doteq x_1$ (except
+perhaps in the case $\infty-\infty$) and is equal to
+\[
+  f_2(x_1)\cdot f_1'(x_1) + f_1(x_1)\cdot f_2'(x_1).
+\]
+%-----File: 134.png---Folio 122------
+Hence
+\[
+  \mathop{L}_{x \doteq x_1} \frac{f_3(x)-f_3(x_1)}{x-x_1}
+\]
+exists and
+\[
+  f_3'(x_1) = f_2(x_1)\cdot f_1'(x_1) + f_2'(x_1) \cdot
+  f_1(x_1).\]
+\item[(\textit c)]
+\[
+  f_3(x)= \frac {f_1(x)}{f_2(x)}.
+\]
+The argument is based on the identity
+\[
+  \frac{\frac{f_1(x)}{f_2(x)}-\frac{f_1(x_1)}{f_2(x_1)} }{x-x_1}
+  = \frac{f_1(x) \cdot f_2(x_1)-f_2(x) \cdot f_1(x_1) }
+       { f_2(x) \cdot f_2(x_1) \cdot (x-x_1) },
+\]
+which holds when $x \neq x_1$ and when $f_2(x) \neq 0$.  But
+\begin{align*}
+  &\frac{f_1(x)\cdot f_2(x_1)-f_2(x) \cdot f_1(x_1)}
+       {f_2(x)\cdot f_2(x_1) (x-x_1)}
+\\
+  &= \frac{f_1(x )\cdot f_2(x_1)-f_1(x_1)\cdot f_2(x_1)
+       + f_1(x_1)\cdot f_2(x_1)-f_2(x )\cdot f_1(x_1)}
+       {f_2(x)\cdot f_2(x_1) (x-x_1)}
+\\
+  &= \frac{f_2(x_1) \left\{f_1(x)-f_1(x_1) \right\}
+      -f_1(x_1) \left\{ f_2(x)-f_2(x_1) \right\}}
+       {f_2(x)\cdot f_2(x_1) (x-x_1)}.
+\end{align*}
+As before (excluding the case $\infty-\infty$) we have
+\[
+  f_3'(x_1)
+= \frac{f_2(x_1)\cdot f_1'(x_1)-f_2'(x_1) \cdot f_1(x_1)}
+       {\left\{ f_2(x_1) \right\}^2 }\text{\correction{.}{,}}
+\]
+\end{enumerate}
+\end{proof}
+
+\begin{corollary}
+It follows from Theorems \hyperlink{thm74}{74} and \hyperlink{thm75}{75} of this chapter that if
+$f_2(x)=a\cdot f_1(x)$ where $f_1'(x)$ exists, then
+\[
+  f_2'(x)=a\cdot f_1'(x).
+\]
+\end{corollary}
+
+\begin{theorem}[76]\hypertarget{thm76}{}\label{p122t76}
+If $x>0$, then $\dfrac{d}{dx}x^k =k\cdot x^{k-1}$.
+\end{theorem}
+%-----File: 135.png---Folio 123------
+\label{p123}
+\begin{enumerate}
+\item[(\textit{a})] If $k$ is a positive integer, we have
+\begin{align*}
+   \mathop{L}_{x\doteq x_1} \frac{x^k-x_1^k}{x-x_1}
+&= \mathop{L}_{x\doteq x_1}
+   \bigl\{ x^{k-1} + x^{k-2}\cdot x_1 + \ldots
+         + x^k\cdot x_1^{k-2}+x_1^{k-1} \bigr\}
+\\
+&= k\cdot x_1^{k-1}.
+\end{align*}
+
+\item[(\textit{b})] If $k$ is a positive rational fraction
+$\dfrac{m}{n}$, we have
+\begin{gather*}
+  \mathop{L}_{x\doteq x_1}
+  \frac{x^{\frac mn}-{x_1}^{\frac mn}}{x-x_1}
+= \mathop{L}_{x\doteq x_1}
+  \frac{\bigl(x^{\frac1n}\bigr)^m
+      -\bigl({x_1}^{\frac1n}\bigr)^m}
+       {\bigl(x^{\frac1n}\bigr)^n
+      -\bigl({x_1}^{\frac1n}\bigr)^n}
+\\
+= \mathop{L}_{x\doteq x_1}
+  \frac{1}{\bigl(x^{\frac1n}\bigr)^{n-1}
+          + \bigl(x^{\frac1n}\bigr)^{n-2}\cdot
+            \bigl({x_1}^{\frac1n}\bigr) + \ldots
+          + \bigl({x_1}^{\frac1n}\bigr)^{n-1}}
+  \cdot
+  \frac{\bigl(x^{\frac1n}\bigr)^m
+      -\bigl({x_1}^{\frac1n}\bigr)^m}
+       {x^{\frac1n}-{x_1}^{\frac1n}}
+\\
+= \frac{1}{n\cdot \bigl({x_1}^{\frac1n}\bigr)^{n-1}} \cdot
+  m \bigl({x_1}^{\frac1n}\bigr)^{m-1},
+\end{gather*}
+by the preceding case.\\
+But
+\[
+  \frac{1}{n\cdot \bigl({x_1}^{\frac1n}\bigr)^{n-1}} \cdot
+  m \bigl({x_1}^{\frac1n}\bigr)^{m-1}
+= \frac mn{x_1}^{\frac mn-1}
+= k\cdot {x_1}^{k-1}.
+\]
+
+\item[(\textit{c})] If $k$ is a negative rational number and equal to
+$-m$, then, by the two preceding cases,
+\begin{align*}
+  \mathop{L}_{x\doteq x_1} \frac{x^{-m}-{x_1}^{-m}}{x-x_1}
+=-\mathop{L}_{x\doteq x_1} \cdot
+  \frac{1}{x^m \cdot x_1^m} \cdot
+  \frac{x^m-x_1^m}{x-x_1}
+&=-\frac{1}{x_1^{2m}} \cdot mx_1^{m-1}
+\\
+&=-m{x_1}^{-m-1}\text{\correction{.}{}}
+\end{align*}
+But
+\[
+  -m{x_1}^{-m-1} = k\cdot x^{k-1}.
+\]
+
+\item[(\textit{d})] If $k$ is a positive irrational number, we proceed
+as follows:
+%-----File: 136.png---Folio 124------
+
+Consider values of $x$ greater than or equal to unity. Let $x$
+approach $x_1$ so that $x>x_1$. Since, by Theorem~\hyperlink{thm23}{23}, $x^k$ is a
+monotonic increasing function of $k$ for $x > 1$, it follows that
+\[
+  \frac{x^k-x_1^k}{x-x_1}
+= x_1^k \cdot \frac{\left(\dfrac{x}{x_1}\right)^k-1}{x-x_1}
+> x_1^{k'} \cdot \frac{\left(\dfrac{x}{x_1}\right)^{k'}-1}{x-x_1}
+\]
+for all values of $k'$ less than $k$, and all values of $x$ greater
+than $x_1$.  If $k'$ is a rational number, we have by the preceding
+cases that
+\[
+  \mathop{L}_{x\doteq x_1}
+  x_1^{k'} \cdot \frac{\left(\dfrac{x}{x_1}\right)^{k'}-1}{x-x_1}
+= k'x_1^{k'-1}.
+\]
+Since $x_1^{k-1}$ is a continuous function of $k$, it follows that for
+every number $N$ less than $kx_1^{k-1}$ there exists a rational number
+$k_1'$ less than $k$ such that
+\[
+  N < k_1'\cdot x_1^{k'-1} < k\cdot x_1^{k-1}.
+\]
+Hence, by Theorem~\hyperlink{thm40}{40},
+\[
+  x_1^k \cdot \frac{\left(\dfrac{x}{x_1}\right)^k-1}{x-x_1}
+\]
+cannot approach a value $N$ less than $kx_1^{k-1}$ as $x$ approaches $x_1$.
+
+By a precisely similar argument we show that a number greater than
+$kx_1^{k-1}$ cannot be a value approached. Since there is always at
+least one value approached, we have that
+\[
+  \mathop{L}_{x\doteq x_1} \frac{x^k-x_1^k}{x-x_1} = k\cdot x_1^{k-1}.
+\]
+
+If $x<x_1$ as $x$ approaches $x_1$, we write
+\[
+  \frac{x^k-x_1^k}{x-x_1}
+= x^k\cdot\frac{\left(\dfrac{x_1}{x}\right)^k-1}{x_1-x}
+\]
+and proceed as before. If $k$ is a negative number we proceed
+%-----File: 137.png---Folio 125------
+as under (\textit{c}). The case in which $x_1 < 1$ is treated
+similarly. For another proof see page~\pageref{pt76p127}.
+\end{enumerate}
+
+\begin{theorem}[77]\hypertarget{thm77}{}
+$\dfrac{d}{dx}\log_a x = \dfrac{1}{x} \cdot \log_a e$.
+\end{theorem}
+
+\begin{proof}
+\begin{align*}
+\frac{\log_a(x + \Delta x)-\log_a x}{\Delta x}
+  & = \frac{1}{\Delta x} \log_a\frac{x+\Delta x}{x} \\
+  & = \frac1x \cdot \log_a \left(1+\frac{\Delta x}{x}\right) ^
+    {\frac{x}{\Delta x}}.
+\end{align*}
+But, by Theorem~\hyperlink{thm57}{57},
+\[
+\mathop{L}_{\Delta x \doteq 0}
+  \left(1 + \frac{\Delta x}{x} \right)^{\frac{x}{\Delta x}} = e.
+\]
+Therefore
+\[
+\mathop{L}_{\text{\correction{$\Delta x\doteq 0$}{$\Delta x\doteq a$}}}
+  \frac{\log_a(x + \Delta x)-\log_a x}{\Delta x} =
+  \frac1x\cdot\log_a e.\qedhere
+\]
+\end{proof}
+\begin{corollary}
+\[
+  \frac{d}{dx}\log_a x = \frac1x.
+\]
+\end{corollary}
+\begin{theorem}[78]\hypertarget{thm78}{}
+If $f'_1(x)$ exists and if there is a $V(x_1)$ upon which $f_1(x)$ is
+continuous and possesses a single-valued inverse $x=f_2(y)$, then
+$f_2(y)$ is differentiable and
+\[
+f'_1(x_1) = \frac{1}{f'_2(y_1)},\ \text{where}\ y_1=f_1(x_1).\footnote{%
+  Theorem~\hyperlink{thm78}{78} gives a sufficient condition for the equality
+  \[
+    \frac{dy}{dx} = \frac{1}{\frac{dx}{dy}}.
+  \]}
+\]
+
+If $f'(x)$ is $0$ or $+\infty$ or $-\infty$ the convention
+$\dfrac{1}{+\infty} = \dfrac{1}{-\infty} = 0$ is understood. Cf.\
+Theorem~\hyperlink{thm37}{37}.
+\end{theorem}
+
+\begin{proof}
+To prove this theorem we observe that
+\[
+f'_1(x_1) =
+  \mathop{L}_{x \doteq x_1}\frac{f_1(x)-f_1(x_1)}{x-x_1} =
+  \mathop{L}_{x \doteq x_1}\frac{1}{\dfrac{x-x_1}{f_1(x)-f_1(x_1)}}.
+\]
+By the definition of single-valued inverse (p.~\pageref{dp45}),
+\[
+  \frac{x-x_1}{f_1(x)-f(x_1)} = \frac{f_2(y)-f_2(y_1)}{y-y_1}.
+\]
+%-----File: 138.png---Folio 126------
+Hence, by Theorems \hyperlink{thm38}{38} and \hyperlink{thm34}{34} and \hyperlink{thm37}{37},
+\[
+  \mathop{L}_{x\doteq x_1} \dfrac{1}{ \dfrac{x-x_1}{f(x)-f(x_1)} }
+= \mathop{L}_{y\doteq y_1} \dfrac{1}{ \dfrac{f_2(y)-f_2(y_1)}{y-y_1} }
+= \frac{1}{f_2'(y)}.\qedhere
+\]
+\end{proof}
+\begin{theorem}[79]\hypertarget{thm79}{}
+If \index{Change of variable}
+\begin{enumerate}
+\item[\textnormal{(1)}]\hypertarget{item1p126}{} $f_1'(x)$ exists and is finite for $x = x_1$, and $f_1(x)$
+is continuous at $x = x_1$,
+\item[\textnormal{(2)}]\hypertarget{item2p126}{} $f_2'(y)$ exists and is finite for $y_1 =f_1(x_1)$,
+\end{enumerate}
+then
+\[
+  \frac{d}{dx_1} f_2\{ f_1(x_1) \}
+= f_2'(y_1) \cdot f_1'(x_1).\footnote{%
+Theorem~\hyperlink{thm79}{79} gives a sufficient condition for the equality
+\[
+  \frac{dz}{dx} = \frac{dz}{dy} \frac{dy}{dz}.
+\]}
+\]
+\end{theorem}
+
+\begin{proof}
+We prove this theorem first for the case when there is a $V^*(x_1)$
+upon which $f_1(x) \neq f_1(x_1)$. In this case the following is an
+identity in $x$:
+\hypertarget{eq1p126}{\[
+  \frac{f_2\{f_1(x)\}-f_2\{f_1(x_1)\} }{ x-x_1 }
+= \frac{f_2\{f_1(x)\}-f_2\{f_1(x_1)\} }{ f_1(x)-f_1(x_1) }
+\cdot \frac{f_1(x)-f_1(x_1) }{ x-x_1 }.
+\tag{1}
+\]}
+By hypothesis~\hyperlink{item2p126}{(2)} and Theorem~\hyperlink{thm38}{38},
+\[
+  f_2'(y_1)
+= \mathop{L}_{y\doteq y_1} \frac{f_2(y)-f_2(y_1) }{ y-y_1 }
+= \mathop{L}_{x\doteq x_1}
+    \frac{f_2\{f_1(x)\}-f_2\{f_1(x_1)\} }{ f_1(x)-f_1(x_1) }.
+\]
+By hypothesis~\hyperlink{item1p126}{(1)},
+\[
+  f_1'(x)
+= \mathop{L}_{x\doteq x_1} \frac{f_1(x)-f_1(x_1) }{ x-x_1 }.
+\]
+Hence, by equation~\hyperlink{eq1p126}{(1)} and Theorem~\hyperlink{thm34}{34}, we have the existence of
+\[
+  \frac{d}{dx} f_2\{f_1(x)\}
+= \mathop{L}_{x\doteq x_1}
+    \frac{f_2\{f_1(x)\}-f_2\{f_1(x_1)\} }{ x-x_1 }
+= f_2'(y_1) \cdot f_1'(x_1).
+\]
+
+If $f_1(x) = f_1(x_1)$ for values of $x$ on every neighborhood of $x =
+x_1$, then, by hypothesis~(1) and the corollary of Theorem~\hyperlink{thm74}{74},
+\[
+  f'(x_1) = 0.
+\]
+%-----File: 139.png---Folio 127------
+Let $[x']$ be the set of points upon which $f_1(x)\neq
+f_1(x_1)$. (There is such a set unless $f(x)$ is constant in the
+neighborhood of $x=x_1$.) Then, by the same argument as in the first
+case, we have
+\[
+  \frac{d}{dx'}f_2\{f_1(x_1)\}=f_2'(y_1)\cdot f_1'(x_1) =0\text{ for $x$
+  on the set $[x']$.}
+\]
+Let $[x'']$ be the set of values of $x$ not included in $[x']$. Then
+\[
+  \frac{d}{dx''}f_2\{f_1(x_1)\}
+= \mathop{L}_{x''\doteq x'} \frac{f_2\{f_1(x'')\}-f_2\{f_1(x_1)\}}{x''-x_1} = 0,
+\]
+since the limitand function is zero. Hence both for the set $[x']$ and
+for the set $[x'']$ the conclusion of our theorem is that the
+derivative required is zero.
+\end{proof}
+\begin{theorem}[80]\hypertarget{thm80}{}
+\[
+  \frac{d}{dx}a^x=a^x\log a.
+\]
+\end{theorem}
+
+\begin{proof}
+Let
+\begin{align*}
+  y&=a^x,\\
+\intertext{therefore}
+  \log y &= x\cdot\log a\\
+\intertext{and, by Theorem~\hyperlink{thm77}{77},}
+  \frac{\dfrac{dy}{dx}}y&= \log a,\\
+\intertext{whence}
+  \frac{dy}{dx}&=y\cdot \log a = a^x\log a.\qedhere
+\end{align*}
+\end{proof}
+
+This method also affords an elegant proof of
+Theorem~\hyperlink{thm76}{76}\label{pt76p127}, viz.,
+\[
+  \frac{d}{dx}x^n=nx^{n-1}.
+\]
+Let
+\begin{align*}
+  y&=x^n,\\[2ex]
+  \log y &=n \log x,\\
+  \frac{\dfrac{dy}{dx}}{y}&=\frac nx,\\
+  \frac{dy}{dx}&=n\cdot\frac yx=n\cdot x^{n-1}.
+\end{align*}
+%-----File: 140.png---Folio 128------
+\section{Differential Notations.}\hypertarget{chVIIsec3}{}%[3]
+\index{Differential}
+If
+\[
+  y=f(x) \quad\text{and}\quad
+  \mathop{L}_{x \doteq a} \frac{f(x)-f(x_1)}{x-x_1}=K,
+\]
+we denote $f(x)-f(x_1)$ by $\Delta y$, and $x-x_1$ by $\Delta
+x$. Then, by Theorem~\hyperlink{thm31}{31},
+\[
+  \Delta y = \Delta x \cdot K + \Delta x \cdot \varepsilon(x),
+\]
+where $\Delta x \cdot \varepsilon(x)$ is an infinitesimal with respect
+to $\Delta y$ and $\Delta x$ for $x \doteq a$. This fact is expressed
+by the equation
+\[
+  dy = K \cdot dx,\ \text{where}\ K=f'(x).
+\]
+Here $dy$ and $dx$ are any numbers that satisfy this equation.  There
+is no condition as to their being small, either expressed or implied,
+and $dx$ and $dy$ may be regarded as variable or
+\begin{figure}[!hbtp]\label{fig16}\hypertarget{fig16}{}
+\setlength{\unitlength}{0.1\textwidth}
+\centering
+\begin{picture}(8,6.5)(-0.5,-1)
+\put(-0.5,0){\line(1,0){8}}
+\put(0,-0.5){\line(0,1){6}}
+\put(2.5,0){\line(0,1){2}}
+\put(4.5,0){\line(0,1){3}}
+\dashline{0.05}(4.5,3)(4.5,4)
+\put(5.5,2){\line(0,1){3}}
+\put(2.5,2){\line(1,0){3}}
+\put(2,1.5){\line(1,1){3.5}}
+\qbezier(2.5,2)(3.5,3)(4.5,3)
+\qbezier(4.5,3)(5.5,3)(6,2.5)
+\put(2.5,1.9){\makebox(0,0)[tl]{$A$}}
+\put(2.5,-0.1){\makebox(0,0)[tc]{$x_1$}}
+\put(4.5,1.9){\makebox(0,0)[tl]{$B'$}}
+\put(4.5,-0.1){\makebox(0,0)[tc]{$x$}}
+\put(4.5,2.9){\makebox(0,0)[tl]{$D'$}}
+\put(4.5,4){\makebox(0,0)[tl]{$C'$}}
+\put(5.5,1.9){\makebox(0,0)[tl]{$B$}}
+\put(5.6,5){\makebox(0,0)[cl]{$C$}}
+\put(3.5,2.1){\makebox(0,0)[bc]{$dx$}}
+\put(5.6,3.5){\makebox(0,0)[lm]{$dy$}}
+\put(3.5,-0.6){\makebox(0,0)[tc]{\sc Fig.~16}}
+\end{picture}
+\end{figure}
+constant, large or small, as may be found convenient. When either $dx$
+or $dy$ is once chosen, the other is, of course, determined.  The
+numbers $dx$ and $dy$ are called the differentials of $x$ and $y$
+respectively.
+%-----File: 141.png---Folio 129------
+
+In Fig.~\hyperlink{fig16}{16}, $f'(x_1)$ is the tangent of the angle $CAB$, $dx$ is the
+length of any segment $\overline{AB}$ with one extremity at $A$ and
+parallel to the $x$-axis, and $dy$ is the length of the segment
+$\overline{BC}$. If $x$ is regarded as approaching $x_1$, then
+$\overline{AB'}$ is the infinitesimal $\Delta x$, $\overline{B'D'}$ is
+$\Delta y$, while $\overline{D'C'}$ is $\varepsilon(x) \cdot \Delta
+x$. Hence, by Theorem~\hyperlink{thm73}{73}, $\overline{D'C'}$ is an infinitesimal of
+higher order than $\Delta x$ or $\Delta y$.
+
+We thus obtain a complete correspondence between derivatives and the
+ratios of differentials. Accordingly, for any formula in derivatives
+there is a corresponding formula in differentials.  Thus corresponding
+to Theorem~\hyperlink{thm75}{75} we have:
+
+\begin{theorem}[81]\hypertarget{thm81}{}
+When for two functions $f_1(x)$ and $f_2(x)$
+\[
+  df_1(x)=f'_1(x)\cdot dx\text{ and }df_2(x)=f_2(x)\cdot dx\text{ at }x_1,
+\]
+it follows that
+\begin{enumerate}
+\item[\textnormal{(\textit{a})}] If $f_3(x)=f_1(x)+f_2(x)$, then
+\begin{align*}
+  df_3(x_1)&=\{f'_1(x_1)+f'_2(x_1)\}dx \\
+  &= df_1(x_1)+df_2(x_1).
+\end{align*}
+\item[\textnormal{(\textit{b})}] If $f_3(x)=f_1(x)-f_2(x)$, then
+\begin{align*}
+  df_3(x_1)&=\{f'_1(x_1)-f'_2(x_1)\}dx \\
+  &= df_1(x_1)-df_2(x_1).
+\end{align*}
+\item[\textnormal{(\textit{c})}] If $f_3(x)=f_1(x) \cdot f_2(x)$, then
+\begin{align*}
+  df_3(x_1)&=\{f_1(x_1)\cdot f'_2(x_1) + f_2(x_1)+f'_1(x_1)\}\cdot dx
+\\ &= f_1(x_1)\cdot df_2(x_1)+f_2(x_1)\cdot df_1(x_1).
+\end{align*}
+\item[\textnormal{(\textit{d})}] If $f_3(x)=\dfrac{f_1(x)}{f_2(x)}$, then
+\begin{align*}
+  df_3(x_1)&=\dfrac{\{f_2(x_1)\cdot f'_1(x_1)-f_1(x_1)\cdot f'_2(x_1)\}
+\cdot dx}{\{f_2(x_1)\}^2} \\
+  &= \dfrac{f_2(x_1)\cdot df_1(x_1)-f_1(x_1)df_2(x_1)}{\{f_2(x_1)\}^2}.
+\end{align*}
+\end{enumerate}
+\end{theorem}
+The rule obtained on page~\pageref{p122t76} et seq.\ that the derivative of $x^k$ is
+$k \cdot x^{k-1}$ corresponds to the equation $dx^k=k\cdot
+x^{k-1}\cdot dx$. If, in the
+%-----File: 142.png---Folio 130------
+equation $dy=f'(x)dx$, $dx$ is regarded as a constant while $x$
+varies, then $dy$ is a function of $x$. We then obtain a differential
+$d_2(dy) = \{f''(x)\cdot dx\}d_2x$ in precisely the same manner that
+we obtain $dy=f'(x)\cdot dx$. Since $d_2x$ may be chosen arbitrarily,
+we choose it equal to $dx$. Hence $d(dy) =f''(x)dx^2$. We write this
+\[
+  d^2y=f''(x)\cdot dx^2.
+\]
+The \index{Differential coefficient}\textit{differential coefficient} $f''(x)$ is clearly identical
+with the \textit{derivative of} $f'(x)$. In this manner we obtain
+successively
+\[
+  d^3y=f^{(3)}(x)\cdot dx^3,\ \text{etc.}
+\]
+
+We may write these results,
+\[
+  \frac{dy}{dx}=f'(x),\ \frac{d^2y}{dx^2}=f''(x),\ldots,\
+  \frac{d^ny}{dx^n}f^{(n)}(x).
+\]
+Evidently the existence of the differential coefficient is coextensive
+with the existence of the derivative.
+
+\section{Mean-value Theorems.}\hypertarget{chVIIsec4}{}%[4]
+
+\begin{theorem}[82]\hypertarget{thm82}{}
+If $f(x)$ has a unique and finite derivative at $x=x_1$, then $f(x)$
+is continuous at $x_1$.
+\end{theorem}
+
+\begin{proof}
+The proof depends upon the evident fact that if $f(x)-f(x_1)$ approach
+anything but zero as $x$ approaches $x_1$, then one of the values
+approached by
+\[
+  \frac{f(x)-f(x_1)}{x-x_1}
+\]
+is $+\infty$ or $-\infty$.
+\end{proof}
+
+\begin{definition}\index{Maximum of a function}
+The function $f(x)$ is said to have a \textit{maximum} at $x=x_1$ if
+there exists a neighborhood $V(x_1)$ such that
+\begin{enumerate}
+\item[(1)] No value of $f(x)$ in $V(x_1)$ is greater than $f(x_1)$.
+
+\item[(2)] There is a value of $x$, $x_2$, in $V(x_1)$ such that $x_2<x_1$
+and $f(x_2)<f(x_1)$.
+%-----File: 143.png---Folio 131------
+
+\item[(3)] There is a value of $x$, $x_3$, in $V(x_1)$ such that $x_3 > x_1$ \correction{and}{\textit{and}}
+$f(x_3) < f(x_1)$.
+\end{enumerate}
+
+Similarly we define a \index{Minimum of a function}\textit{minimum} of a function.
+\end{definition}
+This definition allows any point of a constant stretch like $a$,
+Fig.~\hyperlink{fig17}{17}, to be a maximum, but does not allow any point of $b$ to be
+either a maximum or a minimum.
+
+\begin{figure}[!hbtp]\label{fig17}\hypertarget{fig17}{}
+\centering
+\setlength{\unitlength}{0.08\textwidth}
+\begin{picture}(10,7)(0,-0.5)
+\path(0,6.5)(0,0)(10,0)
+\path(0.25,2.5)(1.5,3.5)(3.5,3.5)(5,2)
+\path(5.25,2.25)(6,3.5)(8.5,3.5)(9.5,5.75)
+\put(2.5,3.55){\makebox(0,0)[bc]{$a$}}
+\put(7.25,3.55){\makebox(0,0)[bc]{$b$}}
+\put(5,-0.5){\makebox(0,0)[bc]{\sc Fig.~17.}}
+\end{picture}
+\end{figure}
+
+\begin{theorem}[83]\hypertarget{thm83}{}\label{t83p131}
+If $f'(x_1)$ exists and if $f(x)$ has a maximum or a minimum at $x =
+x_1$, then $f'(x_1) = 0$.
+\end{theorem}
+
+\begin{proof}
+In case of a maximum at $x_1$, it follows directly from the hypothesis
+that
+\[
+  \mathop{L}_{\stackrel{x\doteq x_1}{x > x_1}}
+  \frac{f(x)-f(x_1)}{x-x_1} \qqle 0, \text{ and also }
+  \mathop{L}_{\stackrel{x\doteq x_1}{x < x_1}}
+  \frac{f(x)-f(x_1)}{x-x_1} \qqge  0,
+\]
+Since $f'(x_1)$ exists these limits are equal, that is, the derivative
+is equal to zero. Similarly in case of a minimum.
+\end{proof}
+
+\begin{theorem}[84]\index{Rolle's theorem}\hypertarget{thm84}{}
+If $f(x_1) = f(x_2)$, $f(x)$ being continuous on the
+%-----File: 144.png---Folio 132------
+interval $\interval{x_1}{x_2}$, and if the derivative
+exists\footnote{%
+    Not necessarily finite.}
+at every point between $x_1$ and $x_2$, then there is a value $\xi$
+between $x_1$ and $x_2$ such that $f'(\xi) =0$.  The derivative need
+not exist at $x_1$ and $x_2$.
+\end{theorem}
+
+\begin{proof}
+\begin{enumerate}
+\item[(\textit{a})] The function may be a constant between $x_1$ and
+$x_2$, in which case $f'(x)=0$ for all values of $x$ between $x_1$ and
+$x_2$ by Theorem~\hyperlink{thm74}{74}.
+
+\item[(\textit b)] There may be values of the function between $x_1$
+and $x_2$ which are greater than $f(x_1)$ and $f(x_2)$. Since the
+function is continuous on the interval $\interval{x_1}{x_2}$, it
+reaches a least upper bound on this interval at some point $x_3$
+(different from $x_1$ and $x_2$).  By Theorem~\hyperlink{thm83}{83},
+\[
+  f'(x_3)=0.
+\]
+\item[(\textit{c})] In case there are values of the function on the
+interval $\interval{x_1}{x_2}$ less than $f(x_1)$, the derivative is
+zero at the minimum point in precisely the same manner as under case
+(\textit{b}).
+\end{enumerate}
+\end{proof}
+
+\begin{figure}[!hbtp]\label{fig18}\hypertarget{fig18}{}
+\centering
+\setlength{\unitlength}{0.06\textwidth}
+\begin{picture}(7,6.5)(-1,-1.5)
+\put(-1,0){\line(1,0){7}}
+\put(0,0){\line(0,1){5}}
+\put(5,0){\line(0,1){4}}
+\put(0,5){\line(5,-1){5}}
+\put(1.5,3){\line(5,-1){3}}
+\dashline{0.1}(3,0)(3,2.7)
+\qbezier(0,5)(1.5,3)(3,2.7)
+\qbezier(3,2.7)(4,2.5)(5,4)
+\put(0,-0.25){\makebox(0,0)[tc]{$x_1$}}
+\put(0,5.25){\makebox(0,0)[bc]{$A$}}
+\put(3,-0.25){\makebox(0,0)[tc]{$\xi$}}
+\put(5,-0.25){\makebox(0,0)[tc]{$x_2$}}
+\put(5,4.25){\makebox(0,0)[bc]{$B$}}
+\put(2.5,-1){\makebox(0,0)[tc]{\textsc{Fig.~18.}}}
+\end{picture}
+\end{figure}
+
+This theorem is called \textsc{Rolle's} Theorem. The restriction
+that $f(x)$ shall be continuous is unnecessary if the derivative
+%-----File: 145.png---Folio 133------
+exists, but simplifies the argument. The proof without this
+restriction is suggested as an exercise for the reader.
+
+The geometric interpretation is that any curve representing a
+continuous function, $f(x)$, such that $f(x_1) = f(x_2)$, and having a
+tangent at every point \correction{between}{betweeen} $x_1$ and $x_2$
+has a horizontal tangent at some point between them. An immediate
+generalization of this is that between any two points $A$ and $B$ on a
+curve which satisfies the hypothesis of this theorem there is a
+tangent to the curve which is parallel to the line $AB$.  The
+following theorem is a corresponding analytical generalization:
+
+\begin{theorem}[85]\index{Mean-value theorem!of the differential calculus}
+\hypertarget{thm85}{}If $f(x)$ is continuous on the interval $\interval{x_1}{x_2}$, and if
+the derivative exists at every point between $x_1$ and $x_2$, then
+there is a value of $x$, $x = \xi$, between $x_1$ and $x_2$ such that
+\[
+  f'(\xi) = \frac{f(x_1)-f(x_2)}{x_1-x_2}.
+\]
+\end{theorem}
+
+\begin{proof}
+Consider a function $f_1(x)$ such that
+\[
+  f_1(x) = f(x)-(x-x_2)\cdot \frac{f(x_1)-f(x_2)}{x_1-x_2};
+\]
+then $f_1(x_1)= f(x_2)$ and $f_1(x_2) = f(x_2)$. Therefore $f_1(x_1) = f_1(x_2)$.
+Hence, by Theorem~\hyperlink{thm84}{84}, there is an $x$, $x = \xi$ on the segment $\overline{x_1\ x_2}$
+such that $f_1'(\xi) = 0$.
+That is,
+\[
+  f_1'(\xi) = f'(\xi)-\frac{f(x_1)-f(x_2)}{x_1-x_2} = 0.
+\]
+Therefore
+\[
+  f'(\xi) = \frac{f(x_1)-f(x_2)}{x_1-x_2}.\qedhere
+\]
+\end{proof}
+This is the ``mean-value theorem.'' Its content may also be expressed
+by the equation
+\[
+  f(x_2) = f(x_1) + (x_2-x_1) f'(\xi).
+\]
+%-----File: 146.png---Folio 134------
+Denoting $x_1-x$ by $dx$ and $\xi$ by $x+\theta dx$, where
+$0<\theta<1$, it takes the form
+\[
+  f(x_1 +dx) = f(x_1) + f'(x_1+\theta dx)dx.
+\]
+
+\begin{theorem}[86]\hypertarget{thm86}{}
+If $f_1(x)$ and $f_2(x)$ are continuous on an interval $\interval{a}{b}$, and if $f_1'(x)$ and $f_2'(x)$ exist between $a$ and $b$,
+$f_2'(x)\neq\pm\infty$, and $f_2'(x)\neq 0$, $f_2(a)\neq f_2(b)$, then
+there is a value of $x$, $x=\xi$ between $a$ and $b$ such that
+\[
+  \frac{f_1(a)-f_1(b)}{f_2(a)-f_2(b)} = \frac{f_1'(\xi)}{f_2'(\xi)}.
+\]
+\end{theorem}
+
+\begin{proof}
+Consider a function
+\[
+  f_3(x)= \frac{f_1(a)-f_1(b)}{f_2(a)-f_2(b)} \{ f_2(x)-f_2(b) \}-\{
+  f_1(x)-f_1(b) \}.
+\]
+Since $f_3(a)=0$ and $f_3(b)=0$, we have as before $f_3'(\xi)=0$.\\
+But
+\[
+  f_3'(\xi)= \frac{f_1(a)-f_1(b)}{f_2(a)-f_2(b)} \cdot
+  f'_2(\xi)-f'_1(\xi).
+\]
+Therefore
+\[
+  \frac{f_1(a)-f_1(b)}{f_2(a)-f_2(b)} = \frac{f_1'(\xi)}{f_2'(\xi)}.\qedhere
+\]
+\end{proof}
+
+This is called the second mean-value theorem. The first mean-value
+theorem has a very important extension to ``Taylor's series with a
+remainder,'' which follows as Theorem~\hyperlink{thm87}{87}.
+
+
+\section{Taylor's Series.}\hypertarget{chVIIsec5}{}%[5]
+\index{Taylor's series}\index{Series!Taylor's}
+The derivative of $f'(x)$ is denoted by $f''(x)$ and is called the
+second \correction{derivative}{derviative} of $f(x)$. In general the
+$n$th derivative is the derivative of the $n-1$st derivative and is
+denoted by $f^{(n)}(x)$.
+
+\begin{theorem}[87]\hypertarget{thm87}{}
+If the first $n$ derivatives of the function $f(x)$ exist and are
+finite upon the interval $\interval{a}{b}$, there is a value of $x$,
+$x_n$ on the interval $\interval{a}{b}$ such that
+%-----File: 147.png---Folio 135------
+\begin{multline*}
+  f(b) = f(a)
+  + \frac{(b-a)}{1!} f'(a)
+  + \frac{(b-a)^2}{2!} f''(a) + \ldots
+\\
+  + \frac{(b-a)^{n-1}}{(n-1)!}\cdot f^{(n-1)}(a)
+  + \frac{(b-a)^n}{n!} f^{(n)}(x_n).
+\end{multline*}
+\end{theorem}
+\begin{proof}
+Let $R_n$ be a constant such that
+\begin{multline*}
+  F(x) = f(x)-f(a)-(x-a)f'(a)-\frac{(x-a)^2}{2!}f''(a)-\ldots
+\\
+  -\frac{(x-a)^{n-1}}{(n-1)!}f^{(n-1)}(a)-\frac{(x-a)^n}{n!}R_n
+\end{multline*}
+is equal to zero for $x=b$. Since $F(x)=0$ for $x=a$, there is, by
+Theorem~\hyperlink{thm84}{84}, some value of $x$, $x_1$, $a<x_1<b$ such that $F'(x_1)=0$.
+That is,
+\begin{multline*}
+  F'(x) = f'(x)-f'(a)-(x-a)f''(a)-\ldots
+\\
+  -\frac{(x-a)^{n-2}}{(n-2)!}f^{(n-1)}(a)-
+  \frac{(x-a)^{n-1}}{(n-1)!}R_n
+\end{multline*}
+is equal to zero for $x=x_1$. Since also $F'(a) =0$, there is a value
+of $x$, $x_2$, $a<x_2<x_1$ such that $F''(x_2)=0$. Proceeding in this
+manner we obtain a value of $x$, $x_n$, $a<x_n<x_{n-1}$ such that
+\[
+  F^{(n)}(x_n) = 0.
+\]
+But
+\[
+  F^{(n)}(x_n) = f^{(n)}(x_n)-R_n = 0.
+\]
+Therefore
+\[
+  R_n = f^{(n)}(x_n),
+\]
+whence the theorem.
+\end{proof}
+
+\begin{corollary}
+In Theorem~\hyperlink{thm87}{87}, $f^{(n)}(x)$ need be supposed to exist only on
+\correction{$\overline{a\ b}$}{$\overline{ab}$}.
+\end{corollary}
+
+\begin{definition}
+The expression
+\[
+  \frac{(b-a)^n}{n!}R_n =
+  \frac{(b-a)^n}{n!}f^n(x_n) =
+  f(b)-\sum_{k=0}^{n-1} \frac{(b-a)}{k!} f^{(k)}(a)
+\]
+is called the \textit{remainder}, and the infinite series
+\[
+  \sum_{k=0}^\infty \frac{\text{\correction{$(b-a)^n$}{$(b-a^n)$}}}{n!} f^{(n)}(a)
+\]
+is called \index{Series!Taylor's}\textit{Taylor's Series}.
+\end{definition}
+%-----File: 148.png---Folio 136------
+If
+\[
+  \mathop{L}_{n=\infty} \frac{f^{(n)}(x_n)(b-a)^n}{n!} = c,
+\]
+a constant different from zero,\\
+then
+\[
+  \sum_{n=0}^\infty \frac{f^{(n)}(a)(b-a)^n}{n!}
+\]
+is convergent but not equal to $f(b)$, i.e.,
+\[
+  \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} \cdot (b-a)^n = f(b)-c.
+\]
+If
+\[
+  \mathop{L}_{n=\infty} \frac{f^{(n)}(x_n)}{n!} \cdot (b-a)^n
+\]
+fails to exist and be finite, then
+\[
+  \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} \cdot (b-a)^n
+\]
+is a divergent series.
+
+Hence an obvious necessary and sufficient condition that for a
+function $f(x)$ all of whose derivatives exist for the values of $x$,
+$a\qqle x\qqle b$,
+\[
+  f(b) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} \cdot (b-a)^n,
+\]
+is that
+\[
+\mathop{L}_{n\doteq\infty} \frac{f^{(n)}(x_n)}{n!} (b-a)^n =
+0.\footnotemark
+\]
+\footnotetext{%
+  \[
+    \mathop{L}_{n\text{\correction{$\doteq$}{$=$}}\infty} \frac{f^{(n)}(x_n)}{n!} (b-a)^n = 0.
+  \]
+  for every value of $x$ on $\interval{a}{b}$ is not sufficient, since
+  $x_n$ depends upon $n$.
+  }
+
+This leads at once, by Theorem~\hyperlink{thm33}{33}, to the following sufficient
+condition:
+%-----File: 149.png---Folio 137------
+
+\begin{theorem}[88]\hypertarget{thm88}{}
+If $f^{(n)}(x)$ exists and $\left|f^{(n)}(x) \right|$ is less than a
+fixed quantity $M$ for every $x$ on the interval $\interval{a}{b}$ and
+for every $n$ $(n = 1, 2, \ldots)$, then
+\[
+  f(b) = f(a) + \frac{(b-1)}{1!}f'(a) + \ldots +
+  \frac{(b-a)^n}{n!}f^{(n)}(a) + \ldots.
+\]
+\end{theorem}
+
+Functions are well known all of whose derivatives exist at every point
+on an interval $\interval{a}{b}$, but such that for some point on this
+interval
+\[
+  \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x-a)^n = f(x) + R(x),
+\]
+where $R$ is a function of $x$ not identically zero. Other functions
+are known for which the series is divergent. The classical example of
+the former is that given by Cauchy,\footnote{%
+  \textsc{Cauchy}, \textit{Collected Works}, 2d series, Vol.~4,
+  p.~250.}
+$e^{-\frac{1}{x^2}}$ at the point $x=0$. If this function is defined
+to be zero for $x=0$, all its derivatives are zero for $x=0$, whence
+Taylor's development gives a function which is zero for all values of
+$x$.
+
+\textsc{Pringsheim}\footnote{%
+  \textsc{A.~Pringsheim}, Mathematische Annalen, Vol.~44 (1893), p.\
+  52, 53.  See also \textsc{K\"onig}, Mathematische Annalen, Vol.~23,
+  p.~450.}
+has given a set of necessary and sufficient conditions that a function
+shall be representable for the values of $h$, $0<h<R$, by means of the
+series
+\[
+  \sum_{n=0}^\infty \frac{1}{n!} \cdot f^{(n)}(0) \cdot h^n.
+\]
+
+It was remarked above, p.~\pageref{t83p131}, that a necessary condition for $f(x)$
+to be a maximum at $x = a$ is $f'(a) = 0$ if the derivative exists.
+Taylor's series permits us to extend this as follows:
+
+\begin{theorem}[89]\hypertarget{thm89}{}
+If on some $V(a)$ the first $n$ derivatives of $f(x)$ exist and are
+finite and on $V^*(a) f^{(n+1)}(x)$ exists and is bounded,\footnote{%
+    Instead of assuming the existence of $f^{(n+1)}(x)$ we might have
+    assumed $f^{(n)}(x)$ continuous without essentially changing the
+    proof.}
+and if
+%-----File: 150.png---Folio 138------
+\begin{gather*}
+  0=f'(a)=f''(a)=\ldots=f^{(n-1)}(a),\\
+  f^{(n)}(a)\neq0,
+\end{gather*}
+\emph{then}:
+\begin{enumerate}
+\item[\textnormal{(1)}] If $n$ is odd, $f(x)$ has neither a maximum nor a minimum
+at $a$;
+
+\item[\textnormal{(2)}] If $n$ is even, $f(x)$ has a maximum or a minimum according
+as $f^{(n)}(a)<0$ or $f^{(n)}(a)>0$.
+\end{enumerate}
+\end{theorem}
+\begin{proof}
+By Taylor's theorem, for every $x$ in the vicinity of \correction{$a$}{a}
+\[
+  f(x)=f(a) + (x-a)^nf^{(n)}(a)+(x-a)^{n+1}\cdot f^{(n+1)}(\xi_x),
+\]
+where $\xi_x$ is between $x$ and $a$.  Hence
+\[
+  f(x)-f(a) = (x-a)^n\{f^{(n)}(a)+(x-a)f^{(n+1)}(\xi_x)\}.
+\]
+But since $f^{(n+1)}(\xi_x)$ is bounded and $x-a$ is infinitesimal,
+there exists a $\overline{V^*}(a)$ such that if $x$ is in
+$\overline{V^*}(a)$,
+\[
+  f(x)-f(a)
+\]
+is positive or negative according as
+\[
+  (x-a)^n\cdot f^{(n)}(a)
+\]
+is positive or negative.
+\begin{enumerate}
+\item[(1)] If $n$ is odd, $(x-a)^n$ is of the same sign as $x-a$, and
+hence for $f^{(n)}(a)>0$
+\begin{gather*}
+f(x)-f(a)>0 \quad \text{if }  x>a,\\
+f(x)-f(a)<0 \quad \text{if }  x<a;
+\end{gather*}
+while for $f^{(n)}(a)<0$
+\begin{gather*}
+f(x)-f(a)>0 \quad \text{if }  x<a,\\
+f(x)-f(a)<0 \quad \text{if }  x>a.
+\end{gather*}
+
+\item[(2)] If $n$ is even, $(x-a)^n$ is always positive, and hence if
+$f^{(n)}(a) >0$,
+\[
+  \left.
+  \begin{aligned}
+  f(x)-f(a)>0 \quad &\text{if }  x>a,\\
+  f(x)-f(a)>0 \quad &\text{if }  x<a;
+  \end{aligned}\right\}
+  \text{ then $f(a)$ is a maximum.}
+\]
+%-----File: 151.png---Folio 139------
+If $f^{(n)}(a)<0$,
+\[
+\left.
+\begin{aligned}
+  f(x)-f(a)<0 &\quad \text{if } x>a, \\
+  f(x)-f(a)<0 &\quad \text{if } x<a;
+\end{aligned}
+\right\} \text{ then $f(a)$ is a minimum.}
+\]
+\end{enumerate}
+\end{proof}
+
+
+\section{Indeterminate Forms.}\hypertarget{chVIIsec6}{}%[6]
+
+The mean-value theorems have an important application in the
+derivation of \index{LHosp@L'Hospital's rule}\textsc{l'Hospital's} rule for calculating
+``indeterminate forms.'' There are seven cases.
+
+\begin{enumerate}
+\item[(1)]\hypertarget{case1}{} $\dfrac{0}{0}$, i.e., to compute
+$\displaystyle\mathop{L}_{x \doteq a}\frac{f(x)}{\phi(x)}$ if
+$\displaystyle\mathop{L}_{x \doteq a}f(x)=0$ and
+$\displaystyle\mathop{L}_{x \doteq a}\phi(x)=0$.
+
+\item[(2)]\hypertarget{case2}{} $\dfrac{\infty}{\infty}$, i.e., to compute
+$\displaystyle\mathop{L}_{x \doteq a}\frac{f(x)}{\phi(x)}$ if
+$\displaystyle\mathop{L}_{x \doteq a}f(x)=\pm\infty$ and
+$\displaystyle\mathop{L}_{x \doteq a}\phi(x)=\pm\infty$.
+
+\item[(3)]\hypertarget{case3}{} $\infty-\infty$, i.e., to compute
+$\displaystyle\mathop{L}_{x \doteq a}\{f(x)-\phi(x)\}$ if
+$\displaystyle\mathop{L}_{x \doteq a}f(x)=\pm\infty$ and
+$\displaystyle\mathop{L}_{x \doteq a}\phi(x)=\pm\infty$.
+
+\item[(4)]\hypertarget{case4}{} $0 \cdot \infty$, i.e., to compute
+$\displaystyle\mathop{L}_{x \doteq a}f(x)\cdot\phi(x)$ if
+$\displaystyle\mathop{L}_{x \doteq a}f(x)=0$ and
+$\displaystyle\mathop{L}_{x \doteq a}\phi(x)=\pm\infty$.
+
+\item[(5)]\hypertarget{case5}{} $1^\infty$, i.e., to compute
+$\displaystyle\mathop{L}_{x \doteq a}f(x)^{\phi(x)}$ if
+$\displaystyle\mathop{L}_{x \doteq a}f(x)=1$ and
+$\displaystyle\mathop{L}_{x \doteq a}\phi(x)=\pm\infty$.
+
+\item[(6)]\hypertarget{case6}{} $0^0$, i.e., to compute
+$\displaystyle\mathop{L}_{x \doteq a}f(x)^{\phi(x)}$ if
+$\displaystyle\mathop{L}_{x \doteq a}f(x)=0$ and
+$\displaystyle\mathop{L}_{x \doteq a}\phi(x)=0$.
+
+\item[(7)]\hypertarget{case7}{} $\infty^0$, i.e., to compute
+$\displaystyle\mathop{L}_{x \doteq a}f(x)^{\phi(x)}$ if
+$\displaystyle\mathop{L}_{x \doteq a}f(x)=\pm\infty$ and
+$\displaystyle\mathop{L}_{x \doteq a}\phi(x)=0$.
+\end{enumerate}
+
+These problems may all be reduced to one or the other of the first
+two. The third may be written (since $f(x)\neq 0$ on some $V^*(a)$)
+\[
+  f(x)-\phi(x) = \frac{1}{\dfrac{1}{f(x)}}-\phi(x) =
+  \frac{1-\dfrac{\phi(x)}{f(x)}}{\dfrac{1}{f(x)}},
+\]
+which is either determinate or of type~\hyperlink{case1}{(1)}.
+%-----File: 152.png---Folio 140------
+
+To the cases \hyperlink{case5}{(5)}, \hyperlink{case6}{(6)}, and \hyperlink{case7}{(7)} we may apply the corollaries of
+Theorem~\hyperlink{thm39}{39} of Chapter~\hyperlink{chapIV}{IV}, from which it follows (provided $f(x) \neq
+0$ on some $V^*(a)$), that
+\[
+  \mathop{L}_{x\doteq a} f(x)^{\phi(x)}
+\]
+exists if and only if
+\[
+  \log \mathop{L}_{x\doteq a} f(x)^{\phi(x)}
+= \mathop{L}_{x \doteq a} \log f(x)^{\phi(x)}
+= \mathop{L}_{x\doteq a} \phi(x) \log f(x) \text{ exists.}
+\]
+The evaluation of
+\[
+  \mathop{L}_{x \doteq a} \frac{\log f(x)}{\dfrac{1}{\phi(x)}}
+\]
+comes under case~\hyperlink{case1}{(1)} or case~\hyperlink{case2}{(2)}.
+
+The evaluation of cases \hyperlink{case1}{(1)} and \hyperlink{case2}{(2)} is effected by the following
+theorems:
+
+\begin{theorem}[90]\hypertarget{thm90}{}
+If $f(x)$ and $\phi(x)$ are continuous and differentiable and
+$\phi(x)$ is monotonic and $\phi'(x) \neq 0$ and $\phi'(x)\neq \infty$
+and
+\begin{enumerate}
+\item[\textnormal{(1)}] if $\displaystyle \mathop{L}_{x \doteq \infty} f(x)=0$ and
+ $\displaystyle \mathop{L}_{x \doteq \infty} \phi(x)=0$ or
+\item[\textnormal{(2)}] if $\displaystyle\mathop{L}_{x \doteq \infty} \phi(x) =
+  \pm\infty$,\footnote{%
+  It is not necessary that $Lf(x)=\infty$; cf.\ Theorem~\hyperlink{thm69}{69}.}
+\end{enumerate}
+then if
+\begin{align*}
+  &\mathop{L}_{x \doteq \infty} \frac{f'(x)}{\phi'(x)} = K,\\
+  &\mathop{L}_{x \doteq \infty} \frac{f(x)}{\phi(x)}
+\end{align*}
+exists and is equal to K.
+\end{theorem}
+
+\begin{proof}
+For every positive $h$ we have, by the second mean-value theorem,
+\[
+  \frac{f(x+h)-f(x)}{\phi(x+h)-\phi(x)} =
+  \frac{f'(\xi_x)}{\phi'(\xi_x)},
+\]
+where $\xi_x$ lies between $x$ and $x+h$. But since $\xi_x$ takes on
+values which are a subset of the values of $x$, and since
+$\displaystyle\mathop{L}_{x\doteq\infty} \xi_x = \infty,$
+%-----File: 153.png---Folio 141------
+\[
+  \mathop{L}_{x \doteq \infty} \frac{f'(x)}{\phi'(x)} = K \quad
+  \text{implies} \quad \mathop{L}_{x \doteq \infty}
+  \frac{f'(\xi_x)}{\phi'(\xi_x)} = K,
+\]
+which in turn implies
+\[
+  \mathop{L}_{x \doteq \infty}
+  \dfrac{f(x+h)-f(x)}{\phi(x+h)-\phi(x)}=K,
+\]
+and this, according to Theorems \hyperlink{thm68}{68} and \hyperlink{thm69}{69}, gives
+\[
+  \mathop{L}_{x \doteq \infty} \frac{f(x)}{\phi(x)} = K.\qedhere
+\]
+\end{proof}
+\begin{corollary}
+If $f(x)$ is continuous and differentiable, then
+\[
+  \mathop{L}_{x \doteq \infty} \frac{f(x)}{x} =
+  \mathop{L}_{x \doteq \infty} f'(x).
+\]
+\end{corollary}
+
+The theorem above can be extended by the substitution
+\[
+  z=\frac{1}{x-a}
+\]
+to the case where $x$ approaches a finite value $a$. The approach must
+of course be one-sided.
+
+\begin{theorem}[91]\hypertarget{thm91}{}
+If $f(x)$ and $\phi(x)$ are continuous and differentiable on some
+$V^*(a)$ and $f(x)$ is bounded on every finite interval, while
+$\phi(x)$ is monotonic and
+\begin{enumerate}
+\item[\textnormal{(1)}] $\displaystyle \mathop{L}_{x \doteq a} f(x)=0$,
+$\displaystyle \mathop{L}_{\text{\correction{$x\doteq a$}{$x=a$}}} \phi(x)=0$  or
+
+\item[\textnormal{(2)}] $\displaystyle \mathop{L}_{\text{\correction{$x\doteq a$}{$x=a$}}} \phi(x)=+\infty$
+or  $-\infty$:
+\end{enumerate}
+then if
+\[
+  \mathop{L}_{x \doteq a} \frac{f'(x)}{\phi'(x)}=K,
+\]
+it follows that
+\[
+  \mathop{L}_{x \doteq a} \frac{f(x)}{\phi(x)}
+\]
+exists and is equal to $K$.
+\end{theorem}
+%-----File: 154.png---Folio 142------
+
+\begin{proof}
+If $\displaystyle \mathop{L}_{x\doteq a} \frac{f'(x)}{\phi'(x)}$
+exists, the limit exists when the approach is only on values of
+$x>a$. Consider only such values of $x$. Then if
+\begin{align*}
+  z=\frac{1}{x-a},\ &f(x)=f(a+\frac1z)=F(z)
+\\
+\intertext{and}
+  &\phi(x) = \phi(a+\frac1z) = \Phi(z),
+\end{align*}
+by hypothesis and Theorem~\hyperlink{thm79}{79}, $F'(z)$ and $\Phi'(z)$ exist and
+\begin{align*}
+  &F'(z) = f'(x)\frac{dx}{dz}, \\
+  &\Phi'(z)=\phi'(x)\frac{dx}{dz}.
+\end{align*}
+Hence if
+\[
+  \mathop{L}_{x\doteq a} \frac{f'(x)}{\phi'(x)}=K,
+\]
+then, according to Theorem~\hyperlink{thm38}{38},
+\[
+  \mathop{L}_{x\doteq \infty} \frac{F'(z)}{\Phi'(z)}
+\]
+exists and is equal to $K$.
+Hence, by Theorem~\hyperlink{thm90}{90},
+\[
+  \mathop{L}_{x\doteq \infty} \frac{F(z)}{\Phi(z)}
+\]
+exists and is equal to $K$.
+Hence, by Theorem~\hyperlink{thm38}{38},
+\[
+  \mathop{L}_{x\doteq a} \frac{f(x)}{\phi(x)}
+\]
+exists and is equal to $K$.
+\end{proof}
+
+We have now derived conditions under which we can state a general rule
+for computing an indeterminate form.
+
+Provided $f(x)$ is not zero on every $V^*(a)$, any of the forms \hyperlink{case3}{(3)} to
+\hyperlink{case7}{(7)} can be reduced to
+\hypertarget{a}{\[
+\tag{\textit{a}}
+  \frac{F(x)}{\Phi(x)}
+\]}
+%-----File: 155.png---Folio 143------
+where this is of type~\hyperlink{case1}{(1)} or \hyperlink{case2}{(2)}. Provided $F(x)$ and $\Phi(x)$
+satisfy the conditions of Theorem~\hyperlink{thm91}{91}, the existence of the limit of
+\hyperlink{a}{(a)} depends on the existence of the limit of
+\hypertarget{b}{\[
+  \frac{F'(x)}{\Phi'(x)}.\tag{\textit{b}}
+\]}
+If \hyperlink{b}{(\textit{b})} is indeterminate, and $F'(x)$ and $\Phi'(x)$ satisfy
+the conditions of Theorem~\hyperlink{thm91}{91}, the limit of \hyperlink{b}{(\textit{b})} depends on the
+limit of
+\[
+  \frac{F''(x)}{\Phi''(x)},\tag{\textit{c}}
+\]
+and so on in general. If at each step the conditions of Theorem~\hyperlink{thm91}{91} are
+satisfied and the form is still indeterminate, the limit of
+\[
+  \frac{F^{(n)}(x)}{\Phi^{(n)}(x)}\tag{$n$}
+\]
+depends on the limit of
+\[
+  \frac{F^{(n+1)}(x)}{\Phi^{(n+1)}(x)}.\tag{$n+1$}
+\]
+If ($n$) is indeterminate for all values of $n$, this rule leads to no
+result. If for some value of $n$
+\[
+  \mathop{L}_{x\doteq a}\frac{F^{(n)}(x)}{\Phi^{(n)}(x)}=K,
+\]
+then all the preceding limits exist and are equal to $K$, and so
+\[
+  \mathop{L}_{x\doteq a}\frac{F(x)}{\Phi(x)}=K.
+\]
+The original expression is equal to $K$ or $e^K$ according to the case
+under consideration.
+%-----File: 156.png---Folio 144------
+
+\section{General Theorems on Derivatives.}\hypertarget{chVIIsec7}{}%[7]
+\begin{theorem}[92]\hypertarget{thm92}{}
+If $f(x)$ is continuous and $f'(x)$ exists for every $x$ on an
+interval $\interval{a}{b}$, then $f'(x)$ takes on every value between
+any two of its values.
+\end{theorem}
+
+\begin{proof}
+Consider any two values of $f'(x)$, $f'(x_1)$, and $f'(x_2)$ on the
+interval $\interval{a}{b}$. Consider, further, the function
+$\dfrac{f(x)-f(x_1)}{x-x_1}$ on the interval between $x_1$ and
+$x_2$. Since $\dfrac{f(x)-f(x_1)}{x-x_1}$ is a continuous function of
+$x$ on this interval, it takes on every value between
+$\dfrac{f(x_2)-f(x_1)}{x_2-x_1}$ and $f'(x_1)$, which is its limiting
+value as $x$ approaches $x_1$. Hence, by Theorem~\hyperlink{thm85}{85}, $f'(x)$ takes on
+all values between and including $f'(x_1)$, and
+$\dfrac{f(x_2)-f(x_1)}{x_2-x_1}$ for values of $x$ on the interval
+$\interval{x_1}{x_2}$. By considering in a similar manner the
+function $\dfrac{f(x_2)-f(x)}{x_2-x}$ on the interval $\interval{x_1}{x_2}$, we show that $f'(x)$ takes on all values between
+$\dfrac{f(x_2)-f(x_1)}{x_2-x_1}$ and $f'(x_2)$. Hence $f'(x)$ takes on
+all values between $f'(x_1)$ and $f'(x_2)$.
+\end{proof}
+
+\begin{theorem}[93]\hypertarget{thm93}{}
+If the derivative exists at every point on an interval, including its
+end-points, it does not follow that the derivative is continuous or
+that it takes on its upper and lower bounds.
+\end{theorem}
+
+\begin{proof}
+This is shown by the following example.
+
+The curve shall lie between the $x$-axis and the parabola $y =
+\frac12x^2$. The straight lines of slopes $1, 1\frac12,
+1\frac34,\ldots, 1+\dfrac{2^n-1}{2^n}\ldots$ through the points
+$(\frac12,0), (\frac14,0),\ldots, \left(\dfrac{1}{2^{n+1}},
+0\right),\ldots$, respectively, meet the parabola in points $A_1, A_2,
+A_3,\ldots, A_n,\ldots$ The broken line $A_1\ (\frac12,0)$ $A_2\
+(\frac14, 0)$ $A_3$ \ldots $A_n\ \left(\dfrac{1}{2^n},
+0\right)\ldots\infty$, has an
+%-----File: 157.png---Folio 145------
+\begin{figure}[!htbp]\label{fig19}\hypertarget{fig19}{}
+\centering
+\includegraphics{images/fig19}
+\end{figure}
+infinitude of vertices. In each angle of the broken line consider an
+arc of circle tangent to and terminated by the sides of the angle, the
+points of tangency being one fourth of the distance to the nearest
+vertex. The function whose graph consists of these circular arcs and
+the portions of the broken line between them is continuous and
+differentiable on the interval $\interval{0}{1}$.  Its derivative is
+discontinuous at $x=0$ and has the least upper bound 2, which is never
+reached.
+\end{proof}
+\begin{theorem}[94]\hypertarget{thm94}{}
+If $f'(x)$ exists and is equal to zero for every value of $x$ on the
+interval $\interval{a}{b}$, then $f(x)$ is a constant on that
+interval.
+\end{theorem}
+
+\begin{proof}
+By Theorem~\hyperlink{thm82}{82}, $f(x)$ is continuous. Suppose $f(x)$ not a constant, so
+that for two values of $x$, $x_1$, and $x_2$, $f(x_1) \neq f(x_2)$,
+then, by Theorem~\hyperlink{thm85}{85}, there is a value of $x$, $x = \xi$ between $x_1$
+and $x_2$ such that
+\[
+  f'(\xi) = \frac{f(x_2)-f(x_1)}{x_2-x_1},
+\]
+%-----File: 158.png---Folio 146------
+which is different from zero, whence $f'(x)$ is not zero for every
+value of $x$ on the interval $\interval{a}{b}$. Hence $f(x)$ is a
+constant on $\interval{a}{b}$.
+\end{proof}
+\begin{corollary}
+If $f_1'(x)=f_2'(x)$ and is finite for every value of $x$ on an
+interval $\interval{a}{b}$, then $f_1(x)-f_2(x)$ is a constant on this
+interval.
+\end{corollary}
+
+\begin{theorem}[95]\hypertarget{thm95}{}
+If $f'(x)$ exists and is positive for every value of $x$ on the
+interval $\interval{a}{b}$, then $f(x)$ is monotonic increasing on
+this interval. If $f'(x)$ is negative for every value of $x$ on this
+interval, then $f(x)$ is monotonic decreasing.
+\end{theorem}
+
+\begin{proof}
+If $f'(x)$ is positive for every value of $x$, then it follows from
+Theorem~\hyperlink{thm85}{85}, provided that $f(x)$ is continuous, that the function is
+monotonic increasing, for if there were two values of $x$, $x_1$ and
+$x_2$, such that $f(x_1) \geqq f(x_2)$ while $x_1 < x_2$, then there
+would be a value of $x$, $x = \xi $, between $x_1$ and $x_2$ such that
+\[
+  f'(\xi)=\frac{f(x_2)-f(x_1)}{x_2-x_1}\leqq 0.
+\]
+
+In case $f(x)$ is not supposed continuous, the argument can be made as
+follows: If $f'(x_1)>0$, then, by Theorem~\hyperlink{thm23}{23}, there exists about the
+point $x_1$ a segment \correction{$\overline{(x_1-\delta)\ (x_1 +
+\delta)}$}{$(x_1-\delta)$, $(x_1 + \delta)$}, upon which
+\[
+  \frac{f(x)-f(x_1)}{x-x_1}>0,
+\]
+and hence, if $x>x_1$, $f(x) >f(x_1)$ and if $x<x_1$, $f(x) <
+f(x_1)$. Now about every point of the segment $\overline{a\ b}$ there
+is such a segment.  Let $x'$ and $x''$ be any two points of
+$\interval{a}{b}$ such that $x'<x''$. By Theorem~\hyperlink{thm10}{10}, there is a finite
+set of these segments of lengths $\delta_1 \ldots \delta_n$ which
+include within them every point of the interval $\interval{x'}{x''}$. We thus have a finite set of points, namely, the mid-point and
+points on the overlapping parts of the segments, $x'<x_1<x_2< \ldots
+<x_k<x''$, such that
+%-----File: 159.png---Folio 147------
+\[
+  f(x')<f(x_1)<f(x_2)< \ldots f(x_k)<f(x'').
+\]
+Hence $f(x')<f(x'')$. In a similar manner we prove that the function
+is monotonic decreasing in case $f'(x)$ is negative.
+\end{proof}
+
+\begin{theorem}[96]\hypertarget{thm96}{}
+If a function $f(x)$ is monotonic increasing on an interval $\interval{a}{b}$,
+and if $f'(x)$ exists for every value of $x$ on this
+interval, then there is no point on the interval for which $f'(x)$ is
+negative. That is, $f'(x)$ is either positive or zero for every point
+of $\interval{a}{b}$.
+\end{theorem}
+
+\begin{proof}
+If $f'(x)$ is negative for some value of $x$, say $x_1$, then
+\[
+  \mathop{L}_{x\doteq x_1}\dfrac{f(x)-f(x_1)}{x-x_1}= C, \ \text{a
+  negative number},
+\]
+whence there is a neighborhood of $x_1$ on which $f(x) >f(x_1)$, while
+$x<x_1$, or $f(x_1)>f(x)$, while $x>x_1$, which is contrary to the
+hypothesis that the function is monotonic increasing in the
+neighborhood of $x = x_1$. In the same manner we prove that if the
+function is monotonic decreasing, and if the derivative exists, then
+$f'(x)$ cannot be positive.
+\end{proof}
+
+The following theorem states necessary and sufficient conditions for
+the existence of the progressive and regressive derivatives.
+Conditions for the existence of a derivative proper are obtained by
+adding the condition that the progressive and regressive derivatives
+are equal.
+
+\begin{theorem}[97]\hypertarget{thm97}{}
+If $f(x)$, $x<x_1$, is continuous in some neighborhood of $x=x_1$,
+then a necessary and sufficient condition that $f'(x_1)$ shall exist
+and be finite is that there exists not more than one linear function
+of $x$, $ax+c$, such that $f(x)+ax+c$ vanishes on every neighborhood
+of $x=x_1$.
+\end{theorem}
+
+\begin{proof}(1) \textit{The condition is necessary.} We prove that if $f'(x)$
+exists and is finite, then not more than one function of the form
+$ax+c$ exists such that $f(x)+ax+c$ vanishes on every neighborhood of
+$x=x_1$. If no such function exists, the theorem is verified.  If
+there is one such function, the following argument will show that
+there is only one. Since, by hypothesis,
+%-----File: 160.png---Folio 148------
+\[
+  \mathop{L}_{x\doteq x_1} \frac{f(x)-f(x_1)}{x-x_1}
+\]
+exists, we have, by Theorem~\hyperlink{thm75}{75}, that
+\[
+  \mathop{L}_{x\doteq x_1} \frac{f(x)+ax+c-f(x_1)-ax_1-c}{x-x_1}
+\]
+exists. Let $[x']$ be the subset of the set of values of $x$ on any
+neighborhood of $x=x_1$ such that $f(x')+ax'+c$ vanishes on the set
+$[x']$. By Theorem~\hyperlink{thm41}{41},
+\begin{multline*}
+  \mathop{L}_{x'\doteq x_1}
+  \frac{f(x')+ax'+c-f(x_1)-ax_1-c}{x'-x_1} \\
+= \mathop{L}_{x\doteq x_1}
+  \frac{f(x)+ax+c-f(x_1)-ax_1-c}{x-x_1}=f'(x_1)+a.
+\end{multline*}
+Since $f'(x_1)$ and $a$ are both finite,
+\[
+  \mathop{L}_{x'\doteq x_1}\frac{f(x')+ax'+c'-f(x_1)-ax_1-c}{x'-x_1}
+\]
+is finite. But the numerator of this fraction is a constant,
+$f(x)+ax+c$ being zero on the set $[x']$. Hence
+\[
+  \mathop{L}_{x\doteq x_1} \frac{f(x)+ax+c-f(x_1)-ax_1-c}{x-x_1}=0,
+  \quad \text{or}\quad f'(x_1)+a=0,
+\]
+and, being continuous, $f(x_1)+ax_1+c=0$.
+The numbers $a$ and $c$ are uniquely determined by the equations
+\[
+  \left\{%
+  \begin{aligned}
+  & f'(x_1)+a=0,\\
+  & f(x_1)+ax_1+c=0.
+  \end{aligned}
+  \right.
+\]
+
+(2) \textit{The condition is sufficient.} We are to show that
+%-----File: 161.png---Folio 149------
+\[
+  \mathop{L}_{x\doteq x_1} \frac{f(x)-f(x_1)}{x-x_1}
+\]
+can fail to exist only when there are at least two functions of the
+form $ax+c$ such that $f(x) +ax+c$ vanishes on every neighborhood of
+$x = x_1$. If
+\[
+  \mathop{L}_{x\doteq x_1} \frac{f(x)-f(x_1)}{x-x_1}
+\]
+does not exist, then
+\[
+  \frac{f(x)-f(x_1)}{x-x_1}
+\]
+approaches at least two distinct values $K_1$ and $K_2$. Let
+$K_2<K_1$.  Let $A$ and $B$ be two finite values such that $K_2 < A <
+B < K_1$.  On every neighborhood of $x = x_1$ there are values of $x$
+for which
+\[
+  \frac{f(x)-f(x_1)}{x-x_1}
+\]
+is greater than $B$, and also values of $x$ for which
+\[
+  \frac{f(x)-f(x_1)}{x-x_1}
+\]
+is less than $A$. Hence, since
+\[
+  \frac{f(x)-f(x_1)}{x-x_1}
+\]
+is continuous at every point except possibly $x_1$, in a certain
+neighborhood of $x_1$ there are values of $x$ in every neighborhood of
+$x_1$ for which
+\begin{align*}
+  \frac{f(x)-f(x_1)}{x-x_1} &= A, \\
+  \intertext{or}
+  f(x)-f(x_1) &= A(x-x_1),
+\end{align*}
+which gives
+\[
+  -f(x_1)-A(x-x_1)
+\]
+as one function of the form $ax+c$.
+%-----File: 162.png---Folio 150------
+
+In the same manner we show that $-f(x_1)-B(x-x_1)$ is another function
+$ax+c$, which makes $f(x)+ax+c$ vanish on every neighborhood of
+$x=x_1$.
+\end{proof}
+
+The geometric meaning of this theorem is obvious. If $P$ is a point on
+the curve representing $f(x)$, then a necessary and sufficient
+condition that this curve shall have a tangent at $P$ is that there
+exists not more than one line through $P$ which intersects the curve
+an infinite number of times on any neighborhood of $P$. Compare the
+functions $x \sin \dfrac{1}{x}$ and $x^2 \sin \dfrac{1}{x}$ on
+page~\pageref{oscillp51}.
+
+The earlier mathematicians supposed that every continuous function
+must have a derivative except at particular points.  The first example
+of a function which has no derivative at any point is due to
+\correction{\textsc{Weierstrass}}{\textsc{Weiersrtass}}.\footnote{%
+    For references and remarks see page~\pageref{oscillp51}.}
+The function is\index{Non-differentiable function}\label{nowherediffp150}
+\[
+  f(x) = \sum_{n=0}^\infty b^n \cos (a^n\pi x),
+\]
+where $a$ is an odd integer, $0 < b < 1$ and $ab > 1 + \frac32\pi$.
+%-----File: 163.png---Folio 151------
+
+
+
+
+\chapter{DEFINITE INTEGRALS.}\hypertarget{chapVIII}{}%[VIII]
+
+\section{Definition of the Definite Integral.}\hypertarget{chVIIIsec1}{}%[1]
+
+
+The area of a rectangle the lengths of whose sides are exact multiples
+of the length of the side of a unit square, is the number of squares
+equal to the unit square contained within the rectangle, and is easily
+seen to be equal to the product of the lengths of its base and
+altitude.\footnote{%
+    Of course the units are not necessarily squares; they may be
+    triangles, parallelograms, etc.}
+
+In case the sides of the rectangle and the side of the unit
+square are commensurable, the sides of the rectangle not being
+exact multiples of the side of the square, the rectangle and the
+square are divided into a set of equal squares. The area of the
+rectangle is then defined as the ratio of the number of squares
+in the rectangle to be measured to the number of squares in the
+unit square. Again, the area is equal to the product of the
+base and altitude.
+
+Any figure so related to the unit square that both figures can be
+divided into a finite set of equal squares is said to be commensurable
+with the unit.
+
+The area of a rectangle incommensurable with the unit is defined as
+the least upper bound of the areas of all commensurable rectangles
+contained within it.
+
+It follows directly from the definition of the product of irrational
+numbers that this process gives the area as the product of the base
+and altitude.\footnote{%
+    For the meaning of the length of a segment incommensurable with
+    the unit segment, compare Chapter~\hyperlink{chapII}{II}, page~\pageref{chIIp33}.}
+%-----File: 164.png---Folio 152------
+
+Turning to the figure bounded by the segment $\overline{a\ b}$ (which
+we take on the $x$ axis in a system of rectangular coordinates) the
+graph of a function $y=f(x)$ and the ordinates $x=a$ and $x=b$,
+\begin{figure}[!htpb]\label{fig20}\hypertarget{fig20}{}
+\centering
+\setlength{\unitlength}{0.05\textwidth}
+\begin{picture}(20,8.5)(-2,-1.5)
+\put(-2,0){\line(1,0){20}}
+\path(0,0)(0,4)(2,4)(2,0)
+\path(2,4)(2,6)(5,6)(5,0)
+\dashline{0.25}(1,0)(1,4)
+\dashline{0.25}(3,0)(3,6)
+\qbezier(0,2.5)(0.5,3.3)(1,4)
+\qbezier(1,4)(2,5.4)(3,6)
+\qbezier(3,6)(4.66,7)(7,7)
+\qbezier(7,7)(8,7)(9,6)
+\qbezier(9,6)(12,3)(14,3)
+\qbezier(14,3)(15,3)(16,3.5)
+\qbezier(16,3.5)(17,4)(17.5,6.5)
+\path(14,0)(14,3.5)(17.5,3.5)(17.5,6.5)(17.5,0)
+\dashline{0.25}(16,0)(16,3.5)
+\put(0,-0.25){\makebox(0,0)[tc]{$a$}}
+\put(1,-0.25){\makebox(0,0)[tc]{$\xi_1$}}
+\put(2,-0.25){\makebox(0,0)[tc]{$x_1$}}
+\put(3,-0.25){\makebox(0,0)[tc]{$\xi_2$}}
+\put(5,-0.25){\makebox(0,0)[tc]{$x_2$}}
+\put(14,-0.25){\makebox(0,0)[tc]{$x_{n-1}$}}
+\put(16,-0.25){\makebox(0,0)[tc]{$\xi_n$}}
+\put(17.5,-0.25){\makebox(0,0)[tc]{$b$}}
+\put(8,-1.5){\makebox(0,0)[bc]{\sc Fig.~20}}
+\end{picture}
+\end{figure}
+we obtain as follows an approximation to the common notion of the area
+of such figures.
+
+Let $x_0=a$, $x_1$, $x_2$, $\ldots$, $x_n=b$ be a set of points lying
+in order from $a$ to $b$. Such a set of points is called a partition
+of $\interval{a}{b}$, and is denoted by $\pi$. The intervals
+$\interval{x_0}{x_1}$, $\interval{x_1}{x_2}$, $\ldots$,
+$\interval{x_{n-1}}{x_n}$ are intervals of $\pi$.
+
+Let $x_1-x_0=\Delta_1x$, $x_2-x_1=\Delta_2x$, $\ldots$,
+$x_n-x_{n-1}=\Delta_nx$, and let
+\[
+  \xi_1,\ \xi_2, \ldots,\ \xi_n
+\]
+be a set of points such that $\xi_1$ is on the interval
+$\interval{x_0}{x_1}$, $\xi_2$ is on $\interval{x_1}{x_2} \ldots$, and
+$\xi_n$ is on $\interval{x_{n-1}}{x_n}$.
+Then
+\[
+  f(\xi_1),\ f(\xi_2),\ \ldots,\ f(\xi_n)
+\]
+are the altitudes of a set of rectangles whose combined area is a more
+or less close approximation of the area of our figure.  Denote this
+approximate area by $S$.
+Then
+\[
+S = f(\xi_1)\Delta_1x+f(\xi_2)\Delta_2x+\ldots+f(\xi_n)\Delta_nx
+  = \sum_{k=1}^nf(\xi_k)\Delta_kx.
+\]
+As the greatest $\Delta_k x$ is taken smaller and smaller, the figure
+%-----File: 165.png---Folio 153------
+composed of the rectangles comes nearer to the figure bounded by the
+curve.
+
+In consequence of these geometrical notions we define the area of the
+figure as the limit of $S$ as the $\Delta_kx$'s decrease indefinitely.
+The area $S$ is the definite integral of $f(x)$ from $a$ to $b$. It
+has been tacitly assumed that the graph of $y=f(x)$ is continuous,
+since we do not usually speak of an area being enclosed by a
+discontinuous curve. The definition of the definite integral when
+stated in its general form admits, however, of functions which are
+discontinuous in a great variety of ways.  A more general definition
+of the definite integral is as follows:\index{Definite integral}\index{Integral!definite}
+
+\emph{Let $\interval{a}{b}$ (or $\interval{b}{a}$) be an interval upon
+which a function $f(x)$ is defined, single-valued and bounded. Let
+$\pi_\delta$ stand for any partition of $\interval{a}{b}$ or
+$\interval{b}{a}$ by the points $a=x_0, x_1, x_2,\ldots,x_n = b$ such
+that the numbers $\Delta_1x=x_1-a,
+\Delta_2x=x_2-x_1,\ldots,\Delta_nx=b-x_{n-1}$ are each numerically
+less than or equal to $\delta$.  \correction{Let}{}
+\[
+  \xi_1,\xi_2,\ldots,\xi_n
+\]
+be a set of points on the intervals \correction{$\interval{x_0}{x_1}$}{$\interval{x_0-x_1}$}, $\interval{x_1}{x_2}$,\ldots,
+$\interval{x_{n-1}}{x_n}$ (or if $b<a$, $\interval{x_1}{x_0}$,
+$\interval{x_2}{x_1}$, $\interval{x_3}{x_2}$, \ldots, $\interval{x_n}{x_{n-1}}$) respectively, and let
+\[
+  S_\delta = f(\xi_1)\Delta_1x + f(\xi_2)\Delta_2x + \ldots +
+  f(\xi_n)\Delta_nx = \sum_{k=1}^n f(\xi_k)\Delta_kx.
+\]
+If the many-valued function of $\delta$, $S_\delta$, approaches a
+single limiting value as $\delta$ approaches zero, then
+\[
+  \mathop{L}_{\delta\doteq0}S_\delta=\int_a^bf(x)dx.
+\]}
+
+When we desire to indicate the interval of integration we write
+${}^b_aS_\delta$ and ${}_a^b\pi_\delta$ instead of $S_\delta$ and
+$\pi_\delta$. $a$ and $b$ are called the \index{Limit!of integration}\emph{limits of integration}.
+
+The details of this definition should be carefully noted.
+%-----File: 166.png---Folio 154------
+For every $\delta$ there is an infinite number of different partitions
+$\pi_\delta$, and for every partition there is an infinite set of
+different sets of $\xi_k$, so that for every $\delta$ the function
+$S_\delta$ has an infinite set of values. The graph of the function
+$S_\delta$ is of the type shown in Fig.~\hyperlink{fig21}{21}. Every value of $S_\delta$
+for one $\delta$ is assumed by $S$ for every larger $\delta$. For any
+particular value of $\delta$ the values of $S_\delta$ lie on a
+definite interval $\interval{\underline BS_\delta}{\overline
+BS_\delta}$, whose length never increases as $\delta$ decreases. If
+this interval approaches $0$ as $\delta$ approaches $0$, the required
+limit exists.
+
+\begin{figure}[!htpb]\label{fig21}\hypertarget{fig21}{}
+\centering
+\includegraphics{images/fig21}
+\end{figure}
+
+It is to be noticed that the set of $\pi$'s, $[\pi_\delta]$ includes
+every possible $\pi$ whose largest $\Delta_kx$ is less than
+$\delta$. Thus we cannot obtain the set of all $\pi$'s by sequential
+repartitioning of any given $\pi$, since there are partitions of the
+set $[\pi_\delta]$ which have no partition points in common with any
+given partition. Inattention to this point is perhaps the greatest
+source of error in the development of the notion of a definite
+integral.
+
+\section{Integrability of Functions.}\hypertarget{chVIIIsec2}{}%[2]
+
+The class of integrable functions is very large, including
+nearly all the bounded functions studied in mathematics and
+%-----File: 167.png---Folio 155------
+physics. Even such an arbitrary function as
+\[
+\begin{cases}\label{egp155}
+  y=0 &\text{if $x$ irrational,}\\
+  y=1/n^3&\text{if $x=m/n$,}
+\end{cases}
+\]
+is integrable. (See page~\pageref{p182th127}, Theorem~\hyperlink{thm127}{127}.)
+
+Examples of \index{Non-integrable function}non-integrable functions are $y=1/x$ on the interval
+$\interval{0}{1}$ (where it is not bounded, see page~\pageref{p191}), and the
+function,
+\[
+  \begin{cases}
+  y=0 &\text{if $x$ is irrational and}\\
+  y=1&\text{if $x$ is rational.}
+  \end{cases}
+\]
+
+To determine the conditions of integrability we introduce the concept
+of \index{Integral oscillation}integral oscillation. On any interval $\interval{a}{b}$, $f(x)$ has
+a least upper bound $A$ and a greatest lower bound $B$, between which
+the function varies. If $A-B=\Delta y={}_a^bOf(x)$ is multiplied by
+the length of the interval, $\Delta x=|b-a|$, it gives the area of a
+rectangle, including the graph of $f(x)$. If the interval is
+subdivided by a partition $\pi$, the sum of the products $\Delta
+x\cdot\Delta y$ on the intervals of the partition is called the
+\emph{integral oscillation of $f(x)$ for the partition $\pi$} and is
+denoted by $O_\pi$. If we call $\Delta_ky$ the difference between the
+upper and lower bounds of $f(x)$ on the intervals $\interval{x_{k-1}}{x_k}$, we have
+\[
+  O_\pi = |\Delta_1x|\cdot\Delta_1y + |\Delta_2x|\cdot\Delta_2y +
+  \ldots + |\Delta_nx|\Delta_ny = \sum_{k=1}^n|\Delta_k
+  x|\cdot\Delta_ky.
+\]
+Geometrically $O_\pi$ represents the areas of the rectangles
+$F_1,\ldots,F_n$ (Fig.~\hyperlink{fig22}{22}), and so we expect to find that if the lower
+bound of $O_\pi$ is zero, $f(x)$ is integrable. This proposition,
+which requires some rather delicate argument for its proof, will be
+taken up in \hyperlink{chVIIIsec7}{\S~7}. At present we shall show in a simple manner that
+every continuous and every monotonic function is integrable.
+
+\begin{lemma}[1]\label{lp155}
+If $S_\pi$ and $S_\pi'$ are two sums (formed by using different
+$\xi_k$'s) on the same partition, then
+\[
+  |S_\pi-S_\pi'|\leqq O_\pi.
+\]
+\end{lemma}
+%-----File: 168.png---Folio 156------
+\begin{proof}
+\begin{gather*}
+\begin{aligned}
+  S_\pi &= \sum_{k=1}^n f(\xi_k)\Delta_k x, \\
+  S'_\pi &= \sum_{k=1}^n f(\xi'_k)\Delta_k x,
+  \end{aligned}
+\\
+  |S_\pi-S'_\pi|
+= \left| \sum_{k=1}^n \{ f(\xi_k)-f(\xi'_k)\}\Delta_kx\right|
+\leqq \sum_{k=1}^n|f(\xi_k)-f(\xi'_k)| \cdot|\Delta_k x|.
+\end{gather*}
+But $|f(\xi_k)-f(\xi'_k)| \leqq \Delta_k y$ by the definition of
+$\Delta_k y$. Therefore
+\[
+  |S_\pi-S'_\pi| \leqq \sum_{k=1}^n|\Delta_k x| \cdot \Delta_k y
+\tag{4}\qedhere
+\]
+\end{proof}
+\begin{figure}[!hbtp]\label{fig22}\hypertarget{fig22}{}
+\setlength{\unitlength}{0.035\textwidth}
+\centering
+\begin{picture}(25,25)(-2,-10)
+\put(-2,0){\line(1,0){25}}
+\dashline{0.25}(0,0)(0,12)
+	\put(0,12){\line(1,0){2}}
+	\put(0,12){\line(0,1){3}}
+	\put(0,15){\line(1,0){2}}
+	\put(2,12){\line(0,1){3}}
+	\qbezier(0,12)(0,15)(2,15)
+\dashline{0.25}(2,0)(2,12)
+	\put(2,13.5){\line(1,0){7}}
+	\put(2,12){\line(0,1){3}}
+	\put(2,15){\line(1,0){4}}
+	\put(6,15){\line(0,-1){5}}
+	\qbezier(2,15)(5,15)(6,13.5)
+\dashline{0.25}(6,0)(6,10)
+	\put(6,10){\line(1,0){5}}
+	\put(8,10){\line(0,1){3.5}}
+	\qbezier(6,13.5)(7,10)(8,10)
+\dashline{0.25}(8,0)(8,10)
+	\qbezier(8,10)(8.8,10)(9,13.5)
+\dashline{0.25}(9,0)(9,10)
+	\put(9,10){\line(0,1){3.5}}
+	\put(9,13){\line(1,0){4}}
+	\qbezier(9,10)(9.5,13)(11,13)
+\put(11,13){\line(0,-1){18}}
+	\qbezier(11,13)(11.5,13)(12,12)
+\put(13,13){\line(0,-1){19}}
+	\qbezier(12,-4)(12.2,-4.6)(13,-5)
+\put(11,-5){\line(1,0){2}}
+	\qbezier(13,-5)(15,-6)(16,-6)
+\put(13,-6){\line(1,0){3}}
+\dashline{0.25}(16,0)(16,-6)
+	\qbezier(16,-6)(17,-6)(18,-8)
+	\qbezier(18,-8)(18.5,-6.5)(20,-5.5)
+\put(16,-5.5){\line(1,0){7}}
+\put(16,-8){\line(1,0){4}}
+\put(16,-5.5){\line(0,-1){2.5}}
+\dashline{0.25}(20,0)(20,-2)
+	\put(20,-2){\line(1,0){3}}
+	\put(20,-2){\line(0,-1){6}}
+	\qbezier(20,-5.5)(22.5,-3.5)(23,-2)
+\dashline{0.25}(23,0)(23,-2)
+	\put(23,-2){\line(0,-1){3.5}}
+\put(0,-0.25){\makebox(0,0)[tc]{$a$}}
+\put(1,13.5){\makebox(0,0)[cc]{$F_1$}}
+\put(4,14.25){\makebox(0,0)[cc]{$F_2$}}
+\put(6,11.75){\makebox(0,0)[rc]{$\Delta_3y$}}
+\put(7,10){\makebox(0,0)[tc]{$\Delta_3x$}}
+\put(21.5,-3.75){\makebox(0,0)[cc]{$F_n$}}
+\put(23,0.25){\makebox(0,0)[bc]{$b$}}
+\put(10.5,-10){\makebox(0,0)[bc]{\textsc{Fig.~22}}}
+\end{picture}
+\end{figure}
+
+A \textit{repartition} of a partition $\pi$ is formed by introducing
+new points in $\pi$.
+
+\begin{lemma}[2]\label{lp156}\hypertarget{lem2p156}{}
+If $\pi_1$ is a repartition of $\pi$,
+\[
+  |S_\pi-S_{\pi_1}| \leqq O_\pi.
+\]
+\end{lemma}
+\begin{proof}
+Any interval $\Delta_kx$ of $\pi$ is composed of one or more
+%-----File: 169.png---Folio 157------
+intervals $\Delta'_k x$, $\Delta''_k x$, etc., of $\pi_1$, and these
+contribute to $S_\pi$ the terms
+\hypertarget{eq1p157}{\[
+\tag{1}
+  f(\xi'_k)\Delta'_k x+f(\xi''_k)\Delta''_k x + \ldots
+\]}
+The corresponding term of $S_\pi$ is
+\hypertarget{eq2p157}{\[
+\tag{2}
+  f(\xi_k)\Delta_k x = f(\xi_k)\Delta'_k x + f(\xi_k)\Delta''_k x + \ldots
+\]}
+But since $|f(\xi_k)-f(\xi_{k'})|\leqq\Delta_k y$, the difference
+between \hyperlink{eq1p157}{(1)} and \hyperlink{eq2p157}{(2)} is less than or equal to
+\[
+  \Delta_k y\cdot|\Delta'_k x + \Delta''_k x + \ldots|
+= \Delta_k y\cdot|\Delta_k x|
+\]
+and hence
+\[
+  |S_\pi-S_{\text{\correction{$\pi_1$}{$\pi 1$}}}| \leqq \sum_{k=1}^n \Delta_k
+   y\cdot|\Delta_kx|=O_\pi.\qedhere
+\]
+\end{proof}
+\begin{theorem}[98]\hypertarget{thm98}{}\label{t98p157}
+Every function continuous on $\interval{a}{b}$ is integrable on
+$\interval{a}{b}$.
+\end{theorem}
+
+\begin{proof}
+We have to investigate the existence of the limit
+$\displaystyle\mathop{L}_{\delta\doteq 0} S_\delta$ of the many-valued
+function $S_\delta$ as $\delta\doteq 0$. Since $S_\delta$ approaches
+at least one value as $\delta$ approaches zero (see Theorem~\hyperlink{thm24}{24}), we
+need only to prove that it cannot have more than one value
+approached. Suppose there were two such values, $B$ and $C$,
+$B>C$. Let $\varepsilon=\dfrac{B-C}{4}$. By the definition of value
+approached, for every $\delta$ there must exist an $S$ (which we call
+$S_B$) such that
+\[
+\tag{1}
+  |S_B-B|<\varepsilon
+\]
+and such that the corresponding $\pi_B$ has its largest
+$\Delta_kx<\delta$.  Similarly there must be an $S_C$ such that
+\[
+\tag{2}
+  |S_C-C|<\varepsilon,
+\]
+and such that the corresponding $\pi_C$ has its largest
+$\Delta_kx<\delta$. Let $\pi$ be a partition made up of the points
+both of $\pi_B$ and $\pi_C$, and let $S$ be one of the corresponding
+sums. $\pi$ is a repartition both of $\pi_B$ and $\pi_C$.
+%-----File: 170.png---Folio 158------
+Therefore
+\[
+\tag{3}
+  |S-S_C|\leqq O_{\pi_C}
+\]
+and
+\hypertarget{eq4p158}{\[
+\tag{4}
+  |S-S_B| \leqq O_{\pi_B}.
+\]}
+But since $f(x)$ is continuous, by the theorem of uniform continuity,
+$\delta$ can be so chosen that if any two values of $x$ differ by less
+than $\delta$, the corresponding values of $f(x)$ differ by less than
+$\dfrac{\varepsilon}{|b-a|}$ and hence on the partitions $\pi_B$ and
+$\pi_C$, whose $\Delta_kx$'s are all less than $\delta$, the
+corresponding $\Delta_ky$'s are all less than
+$\dfrac{\varepsilon}{|b-a|}$. So we have (since
+$\displaystyle\sum_{k=1}^n \Delta_kx=b-a$)
+\[
+  O_{\pi_B} = \sum_{k=1}^n|\Delta_kx| \cdot \Delta_ky <
+  \sum_{k=1}^n|\Delta_kx| \cdot \frac{\varepsilon}{|b-a|} =
+  \varepsilon.
+\]
+Hence
+\[
+  O_{\pi_B}<\varepsilon \quad \text{and}\quad O_{\pi_C} < \varepsilon.
+\]
+So we have, since $\varepsilon=\dfrac{B-C}{4}$ and $\delta$ is so
+chosen that whenever $|x'-x''| < \delta$, $|f(x')-f(x'')| <
+\dfrac{\varepsilon}{|b-a|}$:
+\begin{align*}
+  |S_B-B| &< \varepsilon, \\
+  |S_C-C| &< \varepsilon, \\
+  |S_B-S| &< \varepsilon, \\
+  |S_C-S| &< \varepsilon.
+\end{align*}
+From these inequalities it follows that $|B-C|<4\varepsilon$, which
+contradicts the statement that $\varepsilon=\dfrac{B-C}{4}$. Hence the
+hypothesis that $f(x)$ is not integrable is untenable.
+\end{proof}
+\begin{theorem}[99]\hypertarget{thm99}{}
+Every non-oscillating bounded function is integrable.
+\end{theorem}
+
+\begin{proof}
+The proof runs, as in the preceding theorem, to the
+%-----File: 171.png---Folio 159------
+paragraph following \hyperlink{eq4p158}{(4)}. Let $D$ and $d$ be the upper and lower bounds
+of $f(x)$. $\delta$, being arbitrary, can be so chosen that $\delta =
+\dfrac{\varepsilon}{D-d}$.  Then
+\[
+  O_{\pi_B} = \sum_{k=1}^n \Delta_ky\cdot|\Delta_kx| <
+  \sum_{k=1}^n \Delta_ky\cdot\delta,
+\]
+  and since $f(x)$ is non-oscillating,
+\[
+  \sum_{k=1}^n \Delta_ky = D-d.
+\]
+Therefore
+\[
+  O_{\pi_B}<(D-d)\delta=\varepsilon.
+\]
+Similarly $O_{\pi_C}<\varepsilon$. Hence again we have
+\begin{align*}
+  |S_B-B| & < \varepsilon, \\
+  |S_C-C| & < \varepsilon, \\
+  |S_B-S| & < \varepsilon, \\
+  |S_C-S| & < \varepsilon,
+\end{align*}
+and therefore $|B-C|<4\varepsilon$, whereas $\varepsilon$ was assumed
+equal to $\dfrac{B-C}{4}$. Thus the hypothesis of a non-integrable
+non-oscillating function is untenable.
+\end{proof}
+\section{Computation of Definite Integrals.}\hypertarget{chVIIIsec3}{}%[3]
+
+In computing definite integrals it is important to observe that when
+the integral is known to exist the limit can be calculated on any
+properly chosen subset of the $S_\delta$'s. (See Theorem~\hyperlink{thm41}{41}.) So we
+have that if $S_{\delta_1}$, $S_{\delta_2}$, $\ldots$ is any sequence
+of sums such that $\displaystyle\mathop{L}_{n\doteq\infty}\delta_n=0$,
+then
+\[
+  \mathop{L}_{n\doteq\infty} S_{\delta_n} = \int_a^b f(x)dx.
+\]
+
+One case of this kind occurs when $\xi_k$ is taken as an end-point
+%-----File: 172.png---Folio 160------
+of the interval $\interval{x_{k-1}}{x_k}$ and all the $\Delta_kx$'s
+are equal. Then we have
+\[
+\int_a^b f(x)dx =
+  \mathop{L}_{n\doteq\infty} \sum_{k=1}^n f(a+k\Delta x)\Delta x,
+  \text{ where }
+  \Delta x=\frac{b-a}{n}.
+\]
+A simple example of this principle is the proof of the following
+theorem.
+
+\begin{theorem}[100]\hypertarget{thm100}{}
+If $f(x)$ is a constant, $C$, then
+\[
+  \int_a^b Cdx=C(b-a).
+\]
+\end{theorem}
+
+\begin{proof}
+The function $f(x)=C$ is integrable either according to Theorem~\hyperlink{thm98}{98} or
+Theorem~\hyperlink{thm99}{99}. Hence
+\[
+\int_a^b Cdx =
+  \mathop{L}_{n\doteq\infty} \sum_{k=1}^n C\frac{b-a}{n} =
+  \mathop{L}_{n\doteq\infty} n\cdot C\cdot \frac{b-a}{n} =
+  C(b-a).\qedhere
+\]
+\end{proof}
+
+A few other examples follow. In each case the function is known to be
+integrable by the theorems of $\hyperlink{chVIIIsec2}{\S~2}$.
+
+\begin{theorem}[101]\hypertarget{thm101}{}\label{t101p160}
+\[
+  \int_a^b e^xdx=e^b-e^a.
+\]
+\end{theorem}
+
+\begin{proof}
+
+Let
+\begin{align*}
+S_{\Delta x}
+  &= e^a\Delta x + e^{a+\Delta x} \cdot \Delta x +
+     e^{a+2\Delta x}\cdot\Delta x + \ldots +
+     e^{a+(n-1)\Delta x} \cdot \Delta x \\
+  &= e^a \cdot\Delta x[1+e^{\Delta x} +
+     e^{2\Delta x} + \ldots + e^{(n-1)\Delta x}] \\
+  &= e^a\cdot\Delta x\cdot\frac{e^{n\Delta x}-1}{e^{\Delta x}-1} =
+    \frac{e^{b-a}-1}{e^{\Delta x}-1}e^a\cdot\Delta x \\
+  &= (e^b-e^a) \cdot \frac{\Delta x}{e^{\Delta x}-1}.
+\end{align*}
+Whence the result follows since $\displaystyle\mathop{L}_{\Delta
+x\doteq 0} \dfrac{\Delta x}{e^{\Delta x}-1}=1$. (Differentiate
+numerator and denominator with respect to $\Delta x$ according to
+Theorem~\hyperlink{thm90}{90}.\correction{)}{}
+\end{proof}
+%-----File: 173.png---Folio 161------
+
+Instead of arranging the partition-points in an arithmetical
+progression as in the cases above, we may put them in a geometrical
+progression, that is, we let
+\begin{gather*}
+  \left(\frac ba \right)^{\frac1n} = q, \quad \frac ba = q^n,
+\\
+  \Delta_1 x = aq-a, \quad
+  \Delta_2 x = aq^2-aq, \ldots,
+  \Delta_n x = aq^n-aq^{n-1},
+\\
+  \xi_1 = a, \quad \xi_2 = aq, \ldots, \xi_n = aq^{n-1},
+\end{gather*}
+and obtain the formula
+\begin{align*}
+  \int_a^b f(x) dx
+&= \mathop{L}_{q\doteq 1}
+     a(q-1) [f(a) + qf(aq) + \ldots + q^{n-1} f(aq^{n-1})]
+\\
+&= \mathop{L}_{q\doteq 1} a(q-1) \sum\limits_{k=0}^{n-1} q^k f(aq^k).
+\end{align*}
+We apply this scheme to the following.
+
+\begin{theorem}[102]\hypertarget{thm102}{}
+In all cases where $m$ is a whole number $\neq-1$,
+and if $a>0$, $b>0$ for every value of $m \neq-1$,
+\[
+  \int_a^b x^m dx = \frac{b^{m+1}-a^{m+1}}{m+1}.
+\]
+\end{theorem}
+
+\begin{proof}
+\hypertarget{eq1p161}{\begin{gather*}
+  \int_a^b x^m dx
+= \mathop{L}_{q\doteq 1} a(q-1)\sum\limits_{k=0}^{n-1} q^k (aq^k)^m
+\\
+= a^{m+1} \mathop{L}_{q\doteq 1}
+   (q-1) [1 + (q^{m+1})+ (q^{m+1})^2 + \ldots + (q^{m+1})^{n-1}]
+\tag{1}
+\end{gather*}}
+\begin{align*}
+&= a^{m+1}
+   \mathop{L}_{q\doteq 1} (q-1) \frac{(q^{m+1})^n-1}{q^{m+1}-1}
+\\
+&= \mathop{L}_{q\doteq 1}
+     a^{m+1} \{(q^n)^{m+1}-1\} \frac{q-1}{q^{m+1}-1}
+\\
+&= (b^{m+1}-a^{m+1})
+   \mathop{L}_{q\doteq 1} \frac{q-1}{q^{m+1}-1}.
+\end{align*}
+%-----File: 174.png---Folio 162------
+Hence
+\[
+  \int_a^b x^mdx=\frac{b^{m+1}-a^{m+1}}{m+1},
+\]
+since
+\[
+  \mathop{L}_{q\doteq 1} \frac{q-1}{q^{m+1}-1} = \frac{1}{m+1}.\qedhere
+\]
+\end{proof}
+\begin{theorem}[103]\hypertarget{thm103}{}
+\[
+  \int_a^b\frac1xdx = \log b-\log a,\ (0<a<b).
+\]
+\end{theorem}
+
+\begin{proof}
+By equation~\hyperlink{eq1p161}{(1)} in the last theorem, since $q^{m+1}=q^0=1$,
+\[
+  \int_a^b\frac1xdx=\mathop{L}_{n\doteq\infty} n(q-1);
+\]
+but $n=\dfrac{\log\left(\frac ba\right)}{\log q}$, hence
+\[
+\int_a^b\frac1xdx =
+  \mathop{L}_{q\doteq 1} \frac{q-1}{\log q} \cdot \log\left(\frac ba\right) =
+  \log\left(\frac ba\right) = \log b-\log a,
+\]
+since (\hyperlink{chVIIsec6}{\S~6}, Chapter~\hyperlink{chapVII}{VII}) \textsc{l'Hospital}'s rule gives
+\[
+\mathop{L}_{q\doteq 1} \frac{q-1}{\log q} = 1.\qedhere
+\]
+\end{proof}
+
+The following theorem is of frequent use in computing both
+derivatives and integrals.
+
+\begin{theorem}[104]\hypertarget{thm104}{}
+If on an interval $\interval{a}{b}$ two functions $f(x)$ and $F(x)$
+have the property that for every two values of $x$, $x_1$ and $x_2$,
+where $a<x_1<x_2<b$,
+\[
+  f(x_1)(x_2-x_1) \leqq F(x_2)-F(x_1) \leqq f(x_2)(x_2-x_1);
+\]
+or if
+\[
+  f(x_1)(x_2-x_1) \geqq F(x_2)-F(x_1) \geqq f(x_2)(x_2-x_1),
+\]
+then\begin{enumerate}
+\item[\textnormal{(1)}]\hypertarget{concl1}{} if $f(x)$ is continuous,
+\[
+\frac{dF(x)}{dx}=f(x);
+\]
+%-----File: 175.png---Folio 163------
+and \item[\textnormal{(2)}]\hypertarget{concl2}{} whether $f(x)$ is continuous or not,
+\[
+  \int_a^b f(x)dx \text{ exists and is equal to } F(b)-F(a).
+\]
+\end{enumerate}
+\end{theorem}
+
+\begin{proof}
+We consider first the case
+\[
+  f(x_1)(x_2-x_1) \leqq F(x_2)-F(x_1) \leqq f(x_2)(x_2-x_1).
+\]
+This gives
+\[
+  f(x_1) \leqq \frac{F(x_2)-F(x_1)}{x_2-x_1} \leqq f(x_2).
+\]
+Since $f(x)$ is continuous at $x=x_1$,
+$\displaystyle{\mathop{L}_{x_2\doteq x_1}} f(x_2) = f(x_1)$. Hence, by
+Theorem~\hyperlink{thm40}{40} (Corollary~\hyperlink{cor2p82}{2}),
+\[
+  \mathop{L}_{x_2\doteq x_1} \frac{F(x_2)-F(x_1)}{x_2-x_1} = f(x_1),
+\]
+which proves \hyperlink{concl1}{(1)}.
+
+To prove \hyperlink{concl2}{(2)} we observe that $f(x)$ is non-oscillating and therefore
+integrable according to Theorem~\hyperlink{thm99}{99}. On any partition $\pi$ whose
+dividing points are $x_1$, $x_2, \ldots, x_{n-1}$ we have
+\[
+\begin{array}{lll}
+ f(a)(x_1-a) & \leqq F(x_1)-F(a) & \leqq f(x_1)(x_1-a),
+\\
+ f(x_1)(x_2-x_1)
+& \leqq F(x_2)-F(x_1)
+& \leqq f(x_2)(x_2-x_1),
+\\
+ \qquad\cdot & \qquad\cdot \qquad\qquad\cdot & \qquad\cdot  \\
+ \qquad\cdot & \qquad\cdot \qquad\qquad\cdot & \qquad\cdot  \\
+ \qquad\cdot & \qquad\cdot \qquad\qquad\cdot & \qquad\cdot
+\\
+ f(x_{n-1})(b-x_{n-1})
+& \leqq F(b)-F(x_{n-1})
+& \leqq f(b)(b-x_{n-1}),
+\end{array}
+\]
+Adding, we get
+\begin{gather*}
+  f(a)(x_1-a) + f(x_1)(x_2-x_1) + \ldots + f(x_{n-1})(b-x_{n-1})
+\leqq F(b)-F(a)
+\\
+\leqq f(x_1)(x_1-a) + f(x_2)(x_2-x_1) + \ldots + f(b)(b-x_{n-1}).
+\end{gather*}
+But
+\[
+  f(a)(x_1-a) +\ldots + f(x_{n-1})(b-x_{n-1}) \geqq \underline{B}S_\pi
+\]
+and
+\[
+  f(x_1)(x_1-a) + \ldots + f(b)(b-x_{n-1}) \geqq \overline{B}S_\pi.
+\]
+%-----File: 176.png---Folio 164------
+Since this holds for every $\pi$, we have by Theorem~\hyperlink{thm40}{40} that as
+(Theorem~\hyperlink{thm99}{99})
+\begin{gather*}
+ \int_a^b f(x) dx \text{ exists,}
+\\
+ \int_a^b f(x) dx = F(b)-F(a).
+\end{gather*}
+
+The proof in case $ f(x_1)(x_2-x_1) \geqq F(x_2)-F(x_1) \geqq
+f(x_2)(x_2-x_1)$ is identical with the above when we write $\geqq$
+instead of $\leqq$.
+\end{proof}
+\section{Elementary Properties of Definite Integrals.}\hypertarget{chVIIIsec4}{}%[4]
+
+\begin{theorem}[105]\hypertarget{thm105}{}
+If $a<b<c$, and if a bounded function $f(x)$ is integrable from $a$ to
+$c$, then it is integrable from $a$ to $b$ and from $b$ to $c$.
+\end{theorem}
+
+\begin{proof}
+Suppose $f(x)$ not integrable from $a$ to $b$, then by the definition
+of a limit (see Chap.~\hyperlink{chapII}{II}.) there must be a set of values of ${}^b_a
+S_\delta$, $[{}^b_a S_\delta']$, such that
+$\displaystyle\mathop{L}_{\delta \doteq 0} {}^b_a S_\delta' = A$,
+and another set $[{}^b_a S_\delta'']$ such that
+$\displaystyle\mathop{L}_{\delta \doteq 0} {}^b_a S_\delta'' = B$,
+while $A$ and $B$ are distinct. Whether $\displaystyle\int_b^c f(x)
+dx$ exists or not, there must be a set of values of ${}^c_b S_\delta$,
+$[{}^c_b S_\delta']$, such that the limit
+$\displaystyle\mathop{L}_{\delta \doteq 0} {}^c_b S_\delta' =
+C$. Now for every ${}^b_a S_\delta'$ and ${}^c_b S_\delta'$ there
+exists a ${}^c_a S_\delta'$ such that ${}^c_a S_\delta' = {}^b_a
+S_\delta' + {}^c_b S_\delta'$. Therefore $A+C$ is a value
+approached by ${}^c_a {S_\delta}$. By similar reasoning, $B+C$ is a
+value approached by ${}^c_a {S_\delta}$. Hence ${}^c_a S_{\delta}$ has
+two values approached, which is contrary to the hypothesis. Hence
+$\displaystyle\int_a^b \text{\correction{$f$}{}}(x) dx$ must exist. By similar reasoning
+$\displaystyle\int_b^c f(x) dx$ must exist.
+\end{proof}
+
+\begin{theorem}[106]\hypertarget{thm106}{}
+If $a<b<c$ and if a bounded function $f(x)$ is integrable from $a$ to
+$b$ and from $b$ to $c$, then $f(x)$ is integrable from $a$ to $c$ and
+$\displaystyle\int_a^c f(x) dx = \int_a^b f(x) dx + \int_b^c f(x) dx$.
+\end{theorem}
+
+\begin{proof}
+Since $\displaystyle\int_a^b f(x) dx$ and $\displaystyle\int_b^c f(x)
+dx$ exist, we know by Theorem~\hyperlink{thm26}{26} that for every $\varepsilon$ there
+exists a $\delta_c'$ such that for
+%-----File: 177.png---Folio 165------
+${}_a^bS_\delta$ where $\delta \leqq \delta_\varepsilon$,
+\hypertarget{eq1p165}{\[
+  \left| {}_a^bS_\delta-\int_a^b f(x)dx\right| < \frac\varepsilon3,
+\tag{1}
+\]}
+and also a $\delta_\varepsilon''$ such that for every value of
+${}_b^cS_\delta$ where $\delta\leqq\delta_\varepsilon''$,
+\hypertarget{eq2p165}{\[
+\tag{2}
+  \left| {}_b^cS_\delta-\int_b^\text{\correction{$c$}{}} f(x)dx\right|<\frac\varepsilon3.
+\]}
+Now if the upper bound of $f(x)$ on $\interval{a}{c}$ is $M$ and its
+lower bound is $m$, let $\delta_\varepsilon''' =
+\dfrac{\varepsilon}{3(M-m)}$, and let $\delta_\varepsilon$, be smaller
+than the smallest of $\delta_\delta'$, $\delta_\delta''$,
+$\delta_\delta'''$.
+
+Consider any value of ${}_a^cS_\delta$. If the point $b$ is one of the
+points of the partition upon which ${}_a^cS_\delta$ is computed, then
+${}_a^cS_\delta$ is the sum of one value of ${}_a^bS_\delta$ and one
+value of ${}_b^cS_\delta$. If $b$ is not a point of this partition,
+let $\Delta_bx$ be the length of the interval of ${}_a^c\pi_\delta$
+that contains $b$. Then for properly chosen ${}_a^bS_\delta$ and
+${}_b^cS_\delta$
+\hypertarget{eq3p165}{\[
+\tag{3}
+|{}_a^bS_\delta + {}_b^cS_\delta-{}_a^cS_\delta| <
+  \Delta_bx(M-m) < \frac\varepsilon3.
+\]}
+So in every case (whether or not $b$ is a partition-point of
+${}_a^c\pi_\delta$) by combining \hyperlink{eq1p165}{(1)}, \hyperlink{eq2p165}{(2)}, and \hyperlink{eq3p165}{(3)} we obtain the
+result that for every $\varepsilon$ there exists a
+$\delta_\varepsilon$ such that for every ${}_a^cS_{\delta\varepsilon}$
+\[
+\left| {}_a^cS_{\delta\varepsilon}-
+  \int_a^b f(x)dx-\int_b^c f(x)dx \right| < \varepsilon.
+\]
+Therefore
+\[
+\mathop{L}_{\delta\doteq 0} {}_a^cS_\delta =
+  \int_a^bf(x)dx + \int_b^c f(x)dx,
+\]
+which proves the theorem.
+\end{proof}
+\begin{theorem}[107]\hypertarget{thm107}{}
+Provided both integrals exist,\footnote{%
+  That the first integral exists if the second exists is shown in
+  Theorem~\hyperlink{thm135}{135}.}  and $a<b$,
+\[
+  \int_a^b|f(x)|dx \geqq \left| \int_a^b f(x)dx \right|.
+\]
+\end{theorem}
+%-----File: 178.png---Folio 166------
+
+\begin{proof}
+\[
+  \sum|f(\xi_k)|\Delta_kx \geqq \left|\sum f(\xi_k)\Delta_kx\right|.
+\]
+Hence for every $S_\delta|f(x)|$ there is a smaller or equal $S_\delta
+f(x)$, the $\delta$'s being the same. Hence by Corollary~\hyperlink{cor2p82}{2},
+Theorem~\hyperlink{thm40}{40},
+\[
+   \mathop{L}_{\delta\doteq 0} S_\delta|f(x)| \geqq
+  |\mathop{L}_{\delta\doteq 0} S_\delta f(x)|.\qedhere
+\]
+\end{proof}
+
+\begin{theorem}[108]\hypertarget{thm108}{}
+If $\displaystyle\int_a^b f(x)dx$ exists, then $\displaystyle\int_b^a
+f(x)dx$ exists and
+\[
+  \int_a^b f(x)dx =-\int_b^a f(x)dx.
+\]
+\end{theorem}
+
+\begin{proof}
+This is a consequence of the theorem (Corollary~\hyperlink{cor1th27}{1} Theorem~\hyperlink{thm27}{27}) that
+\[
+  \mathop{L}_{x\doteq a} (-f(x)) =-\mathop{L}_{x\doteq a} f(x),
+\]
+for to every $S$ used in defining $\displaystyle\int_a^b f(x)dx$
+corresponds a sum equal to $-S$ which is used in defining
+$\displaystyle\int_b^a f(x)dx$.
+
+Similarly to every $S'$ used in defining $\displaystyle\int_b^a
+f(x)dx$ there corresponds a sum $-S'$ used in defining
+$\displaystyle\int_a^b f(x)dx$. Hence the function $S_\delta$ in the
+definition of $\displaystyle\int_a^b f(x)dx$ is the negative of the
+function $S_\delta$ used in the definition of $\displaystyle\text{\correction{$\int_b^a$}{$\int_a^b$}}
+f(x)dx$. Hence the theorem follows from the theorem quoted.
+\end{proof}
+
+We adjoin the following two theorems, the first of which is an
+immediate consequence of the definition of an integral, and the second
+a corollary of Theorems \hyperlink{thm105}{105}, \hyperlink{thm106}{106}, and \hyperlink{thm108}{108}.
+%-----File: 179.png---Folio 167------
+
+\begin{theorem}[109]\hypertarget{thm109}{}
+$\displaystyle\int_{a+h}^{b+h} f(x-h) dx$ exists and is equal to
+$\displaystyle\int_a^b f(x) dx$, provided the latter integral
+exists.\footnote{%
+  First stated formally by \textsc{H.~Lebesgue}, \emph{Le\c cons sur
+  l'Int\'egration}, Chapter~VII, page~98.}
+\end{theorem}
+
+\begin{theorem}[110]\hypertarget{thm110}{}
+If any two of the following integrals exist, so does the third, and
+\[
+  \int_a^b f(x) dx + \int_b^c f(x) dx = \int_a^{\text{\correction{$c$}{$b$}}} f(x) dx.
+\]
+\end{theorem}
+
+\begin{theorem}[111]\hypertarget{thm111}{}
+If $C$ is any constant and if $f(x)$ is integrable on $\interval{a}{b}$, then $Cf(x)$ is integrable on $\interval{a}{b}$ and
+\[
+  \int_a^b Cf(x) dx = C\int_a^b f(x) dx.
+\]
+\end{theorem}
+
+\begin{proof}
+\[
+  S_\delta =\sum\limits_{k=1}^n f(\xi_k) \Delta_k x
+\]
+is an $S_\delta$ of the set which defines $\displaystyle\int_a^b f(x) dx$ and
+\[
+  S_\delta' =\sum\limits_{k=1}^n Cf(\xi_k)\Delta_k x
+\] 
+is the corresponding $S_\delta$ of the set which defines
+$\displaystyle\int_a^b Cf(x) dx$. Hence our theorem follows
+immediately from Theorem~\hyperlink{thm34}{34}, a special case of which is
+$\displaystyle\mathop{L}_{x\doteq a} Cf(x) =C\mathop{L}_{x\doteq a} f(x)$.
+\end{proof}
+
+\begin{theorem}[112]\hypertarget{thm112}{}
+If $f_1(x)$ and $f_2(x)$ are any two functions each integrable on the
+interval $\interval{a}{b}$, then $f(x) = f_1(x) \pm f_2(x)$ is
+integrable on $\interval{a}{b}$ and
+\[
+  \int_a^b f(x) dx = \int_a^b f_1(x) dx \pm \int_a^b f_2(x) dx.
+\]
+\end{theorem}
+
+\begin{proof}
+The proof depends directly upon the theorem that
+if
+$\displaystyle\mathop{L}_{x\doteq a} \phi_1(x) =b_1$, and
+$\displaystyle\mathop{L}_{x\doteq a} \phi_2(x)=b_2$, then
+$\displaystyle\mathop{L}_{x\doteq a} \text{\correction{$\big($}{}}\phi_1(x) \pm \phi_2(x)\text{\correction{$\big)$}{}} = b_1 \pm b_2$
+(Theorem~\hyperlink{thm34}{34}).
+\end{proof}
+%-----File: 180.png---Folio 168------
+
+\begin{theorem}[113]\hypertarget{thm113}{}
+If $f_1(x)$ and $f_2(x)$ are integrable on $\interval{a}{b}$ and such
+that for every value of $x$ on $\interval{a}{b}$ $f_1(x)\geqq f_2(x)$,
+then
+\[
+  \int_a^b f_1(x)dx \geqq \int_a^b f_2(x)dx.
+\]
+\end{theorem}
+
+\begin{proof}
+Since $S_1$ is always greater than or equal to $S_2$, then, by
+Theorem~\hyperlink{thm34}{34}, $\underset{\delta\doteq 0}{L} S_1 \geqq
+\underset{\delta\doteq 0}{L} S_2$, which proves the theorem.
+\end{proof}
+
+\begin{theorem}[114]\hypertarget{thm114}{}(Maximum-Minimum Theorem.)
+If
+\begin{enumerate}
+\item[\textnormal{(1)}] the product $f_1(x)\cdot f_2(x)$ and the factor $f_1(x)$
+are integrable on $\interval{a}{b}$,
+
+\item[\textnormal{(2)}] $f_1(x)$ is always positive or always negative on
+$\interval{a}{b}$,
+
+\item[\textnormal{(3)}] $M$ and $m$ are the least upper and the greatest lower
+bounds respectively of $f_2(x)$ on $\interval{a}{b}$,
+\end{enumerate}
+then
+\[
+  \text{\correction{$m$}{$\underline{m}$}}\cdot \int_a^b f_1(x)dx
+\leqq \int_a^b f_1(x)\text{\correction{$\cdot$}{}} f_2(x)dx \leqq M\cdot \int_a^b f_1(x)dx,
+\]
+or
+\[
+  \text{\correction{$m$}{$\underline{m}$}}\cdot \int_a^b f_1(x)dx
+\geqq \int_a^b f_1(x)\cdot f_2(x)dx \geqq M\cdot \int_a^b f_1(x)dx.
+\]
+\end{theorem}
+
+\begin{proof}
+By Theorem~\hyperlink{thm111}{111},
+\begin{align*}
+  M\cdot \int_a^b f_1(x)dx &= \int_a^b M\cdot f_1(x)dx
+\\
+\intertext{and}
+  m\cdot \int_a^b f_1(x)dx = \int_a^b m\cdot f_1(x)dx.
+\end{align*}
+But in case $f_1(x)$ is always positive,
+\[
+  m\cdot f_1(x) \leqq f_1(x)\cdot f_2(x) \leqq M\cdot f_1(x).
+\]
+Hence, by the preceding theorem,
+%-----File: 181.png---Folio 169------
+\begin{alignat*}{2}
+  \int_a^b m\cdot f_1(x)dx &\leqq \int_a^b f_1(x)\cdot f_2(x)dx
+&&\leqq \int_a^b M\cdot \text{\correction{$f_1$}{$f$}}(x)dx,\\
+\intertext{and therefore}
+m \cdot \int_a^b f_1(x)dx &\leqq
+  \int_a^b f_1(x) \cdot f_2(x) dx &&\leqq
+  M \cdot \int_a^b f_1(x)dx.
+\end{alignat*}
+If $f_1(x)$ is always negative, it follows in the same manner that
+\[
+m \cdot \int_a^b f_1(x)dx \geqq
+  \int_a^b f_1(x) \cdot f_2(x) dx \geqq
+  M \cdot \int_a^b f_1(x)dx.\qedhere
+\]
+\end{proof}
+
+As an obvious corollary of this theorem we have the Mean-value
+Theorem:
+
+\begin{theorem}[115]\hypertarget{thm115}{}\index{Mean-value theorem!of the integral calculus}
+Under the hypothesis of Theorem~\hyperlink{thm114}{114} there exists a number $K$,
+$\text{\correction{$m$}{$\underline{m}$}} \leqq K \leqq \text{\correction{$M$}{$\overline{M}$}}$, such that
+\[
+\int_a^b f_1(x) \cdot f_2(x) dx =
+  K\int_a^b f_1(x)dx.
+\]
+\end{theorem}
+
+\begin{ncorollary}[1]
+In case $f_2(x)$ is continuous we have a value $\xi$ of $x$ on
+$\interval{a}{b}$ such that
+\[
+\int_a^b f_1(x) \cdot f_2(x)dx =
+  f_2(\xi) \int_a^b f_1(x)dx.
+\]
+\end{ncorollary}
+In case $f_1(x)=1$,
+\[
+  \int_a^b f_1(x)dx = b-a,
+\]
+and the theorem reduces to this:
+
+\begin{theorem}[116]\hypertarget{thm116}{}
+If $f(x)$ is any integrable function on the interval $\interval{a}{b}$, there exists a number $M$ lying between the upper and lower
+bounds of $f(x)$ on $\interval{a}{b}$ such that
+\[
+\int_a^b f(x)dx = M(b-a),
+\]
+and if $f(x)$ is continuous, there is a value $\xi$ of $x$ on
+$\interval{a}{b}$ such that
+\[
+\int_a^b f(x)dx = f(\xi)(b-a).
+\]
+\end{theorem}
+%-----File: 182.png---Folio 170------
+
+In many applications of the integral calculus the expression
+\[
+  \dfrac{\int_a^b f(x)dx}{b-a}
+\]
+represents the notion of
+an average value of the dependent variable $y = f(x)$ as $x$ varies
+from $a$ to $b$. An average of an infinite set of values of $f(x)$ is
+of course to be described only by means of a limiting
+process. Consider a set of points $x_1$, $x_2, \ldots, x_{n-1}$, $x_n
+= b$ on the interval $\interval{a}{b}$ such that
+\[
+  x_1-a = x_2-x_1 = x_3-x_2 = \ldots
+= x_{n-1}-x_{n-2} = b-x_{n-1}.
+\]
+Then
+\[
+  M_n = \frac1n \sum_{k=1}^n f(x_k),
+\]
+and we define the mean value of $f(x)$, ${}_a^bM f(x) =
+\displaystyle{\mathop{L}_{n\doteq\infty}}M_n$ if this limit
+exists. But $x_{k+1}-x_k = \frac{b-a}{n} = \Delta
+x$.
+
+If the definite integral $\displaystyle\int_a^b f(x)dx$ exists, we may
+write
+\[
+  \int_a^b f(x)dx = \underset{\delta\doteq 0}{L} S_\delta,
+\]
+where
+\[
+  S_\delta
+= \sum_{k=1}^n f(x_k) \Delta x
+= \sum_{k=1}^n f(x_k) \frac{b-a}{n}
+= \frac{b-a}{n} \sum_{k=1}^n f(x_k) = (b-a) M_n.
+\]
+Therefore
+\[
+\mathop{L}_{\delta\doteq 0} S_\delta
+= (b-a) \mathop{L}_{n\doteq\infty} M_n.
+\]
+We therefore have the theorem:
+
+\begin{theorem}[117]\hypertarget{thm117}{}
+In case the integral of $f(x)$ exists on the interval $\interval{a}{b}$,
+\[
+  {}_a^bM f(x) = \frac{\displaystyle\int_a^b f(x)dx}{b-a}.
+\]
+\end{theorem}
+
+We note that ${}_a^bM$ is the same as the $K$ which occurs in the
+mean-value theorem, and that the last theorem suggests a simple
+%-----File: 183.png---Folio 171------
+method of approximating the value of a definite integral by
+multiplying the average of a finite number of ordinates by $b-a$.
+
+\section{The Definite Integral as a Function of the Limits of
+Integration.}\hypertarget{chVIIIsec5}{}%[5]
+
+\begin{theorem}[118]\hypertarget{thm118}{}
+If $f(x)$ is integrable on an interval $\interval{a}{b}$, and if $x$
+is any point of $\interval{a}{b}$, $\displaystyle \int_a^xf(x)dx$ is a
+continuous function of $x$.
+\end{theorem}
+
+\begin{proof}
+$\displaystyle \int_a^xf(x)dx$ exists, by Theorem~\hyperlink{thm105}{105}, and by the
+definition of a continuous function we need only to show that
+\[
+  \mathop{L}_{x'\doteq
+  x}\left(\int_a^{x'}f(x)dx-\int_a^xf(x)dx\right)=0.
+\]
+By the theorems of the preceding section,
+\[
+  \int_a^{x'}f(x)dx-\int_a^xf(x)dx=\int_x^{x'}f(x)dx\leqq|_x^{x'}
+  \overline{B}\cdot (x'-x)|\leqq|\overline{B}\cdot (x'-x)|,
+\]
+where $_x^{x'}\overline{B}$ stands for the least upper bound of $f(x)$
+on the interval $\interval{x}{x'}$, and $\overline{B}$ for the least
+upper bound of $f(x)$ on $\interval{a}{b}$. Since $\overline{B}$ is a
+constant, $\overline{B}(x'-x)$ approaches zero as $x'$ approaches $x$,
+and therefore by Theorem~\hyperlink{thm40}{40}, Corollary~\hyperlink{cor4p82}{4}, the conclusion of our
+theorem follows.
+\end{proof}
+
+\begin{theorem}[119]\hypertarget{thm119}{}
+If $f(x)$ is continuous on an interval $\interval{a}{b}$,
+$\displaystyle \int_a^x f(x)dx\ (a<x<b)$ possesses a derivative with
+respect to $x$ such that
+\[
+\frac{d}{dx}\int_a^xf(x)dx=\text{\correction{$f$}{}}(x).
+\]
+\end{theorem}
+
+\begin{proof}
+By the preceding theorem $\displaystyle \int_a^xf(x)dx$ is continuous.
+%-----File: 184.png---Folio 172------
+To form the derivative we investigate the expression
+\hypertarget{eq1p172}{\[
+  \frac{\displaystyle\int_a^{x'}f(x)dx-\int_a^xf(x)dx}{x'-x}
+= \frac{\displaystyle\int_x^{x'}f(x)dx}{x'-x} \tag{1}
+\]}
+as $x'$ approaches $x$.
+
+By Theorem~\hyperlink{thm115}{115} (the mean-value theorem),
+\[
+  \int_x^{x'}f(x)dx=f\left(\xi(x')\right)(x'-x),
+\]
+where $\xi(\text{\correction{$x'$}{$x$}})$ is a value of $x$ between $x$ and $x'$ and is a function
+of $x'$.
+Hence \hyperlink{eq1p172}{(1)} is equal to
+\hypertarget{eq2p172}{\[
+  f(\xi).\tag{2}
+\]}
+But as $x'$ approaches $x$, $\xi$ also approaches $x$ and so, by
+Theorem~\hyperlink{thm39}{39}, as $x'$ approaches $x$, \hyperlink{eq2p172}{(2)} approaches $f(x)$. Therefore
+\[
+  \mathop{L}_{x'\doteq x}
+  \frac{\displaystyle\int_a^{x'}f(x)dx
+      -\displaystyle\int_a^{x }f(x)dx}{x'-x}
+= f(x) = \frac{d}{dx} \int_a^xf(x)dx.\qedhere
+\]
+\end{proof}
+
+Following is a more general statement of Theorem~\hyperlink{thm119}{119}.
+
+\begin{corollary}
+If $f(x)$ is continuous at a point $x_1$ of $\interval{a}{b}$ and
+integrable on $\interval{a}{b}$, then at $x=x_1$
+\[
+  \frac{d}{dx}\int_a^xf(x)dx=f(x).
+\]
+\end{corollary}
+
+The proof is like that of Theorem~\hyperlink{thm112}{112} except that
+\[
+  \int_{x_1}^xf(x)dx=(x-x_1)M(x),
+\]
+and $M(\text{\correction{$x$}{$x_1$}})$ is a value between the upper and lower bounds of
+%-----File: 185.png---Folio 173------
+$f(x)$ on $\interval{x_1}{x}$. But by the continuity of $f(x)$ at \correction{$x_1$}{${}_1$}
+\[
+ \mathop{L}_{x\doteq x_1}M(x)=f(x_1),
+\]
+and hence the conclusion follows as in the theorem.
+
+\begin{theorem}[120]\hypertarget{thm120}{}
+If $f(x)$ is any continuous function on the interval $\interval{a}{b}$, and $F(x)$ any function on this interval such that
+\[
+  \frac{d}{dx}F(x)=f(x),
+\]
+then $F(x)$ differs from $\displaystyle \int_a^xf(x)dx$ at most by an
+additive constant.
+\end{theorem}
+
+\begin{proof}
+Let $\displaystyle F(x) = \int_a^xf(x)dx+\phi(x)$.
+
+Since $F(x)$ and $\displaystyle \int_a^xf(x)dx$ are both
+differentiable,
+\[
+  \frac{d}{dx}F(x)=\frac{d}{dx}\left(\int_a^xf(x)dx+\phi(x)\right)
+= \frac{d}{dx}\left(\int_a^xf(x)dx\right)+\frac{d}{dx}\phi(x).
+\]
+By the preceding theorem
+\[
+  \frac{d}{dx}\int_a^xf(x)dx=f(x).
+\]
+Hence $\dfrac{d}{dx}\phi(x) =0$, whence, by Theorem~\hyperlink{thm94}{94}, $\phi(x)$ is a
+constant.
+\end{proof}
+
+\begin{theorem}[121]\hypertarget{thm121}{}
+If $f(x)$ is a continuous function on an interval $\interval{a}{b}$
+and $F(x)$ is such that
+\[
+  \frac{d}{dx}F(x)=f(x),
+\]
+then
+\[
+  \int_a^bf(x)dx=F(b)-F(a).
+\]
+\end{theorem}
+%-----File: 186.png---Folio 174------
+
+\begin{proof}
+By the last theorem,
+\[
+  \int_a^xf(x)dx =F(x)+c.
+\]
+But
+\[
+  0=\int_a^af(x)dx =F(a)+c.
+\]
+Therefore
+\[
+ -F(a) =c.
+\]
+Whence
+\[
+  \int_a^bf(x)dx =F(b)+c=F(b)-F(a).
+\]
+The symbol $[F(x)]_a^b$ or $|_a^bF(x)$ is frequently used for
+$F(b)-F(a)$.  In these terms the above theorem is expressed by the
+equation
+\[
+  \int_a^bf(x)dx=|_a^bF(x).\qedhere
+\]
+\end{proof}
+
+By this last theorem the theory of definite and indefinite integrals
+is united as far as continuous functions are concerned, and a table of
+derivatives gives a table of integrals. For discontinuous functions
+the correspondence does not in general hold. That is, there are on the
+one hand integrable functions $f(x)$ such that $\displaystyle
+\int_a^xf(x)dx$ is not differentiable with respect to $x$, and on the
+other hand differentiable functions $\phi(x)$ such that $\phi'(x)$ is
+not integrable.\footnote{%
+    For a good discussion of this subject the reader is referred to
+    \textsc{H. Lebesgue}, \textit{Le\c cons sur l'Int\correction{\'e}{e}gration.}}
+
+
+\section{Integration by Parts and by Substitution.}\hypertarget{chVIIIsec6}{}%[6]
+
+The formulas for integration by parts and by substitution are
+ordinarily written as follows:
+\begin{align*}
+  &\int udv = uv-\int v\text{\correction{$d$}{$\underset{\centerdot}{d}$}}u,\\
+  &\int f(y)dy=\int f(y)\cdot \frac{dy}{dx}\cdot dx.
+\end{align*}
+%-----File: 187.png---Folio 175------
+The following theorems state sufficient conditions for their validity.
+
+\begin{theorem}[122]\hypertarget{thm122}{} (Integration by parts.)
+\[
+  \int_a^bf_1(x)\cdot f_2'(x)dx
+= \left[f_1(x)\cdot f_2(x)\right]_a^b
+-\int_a^bf_2(x)\cdot f_1'(x)dx,
+\]
+provided $f_1'(x)$ and $f_2'(x)$ exist and are continuous on the
+interval $\interval{a}{b}$.
+\end{theorem}
+
+\begin{proof}
+By Theorem~\hyperlink{thm75}{75},
+\[
+  \frac{d}{dx}\left(f_1(x)\cdot f_2(x)\right)
+= f_1(x)\cdot f_2'(x)+f_2(x)\cdot f_1'(x).
+\]
+Therefore
+\[
+  \int_a^b\frac{d}{dx}\left(f_1(x)\cdot f_2(x)\right)dx
+= \int_a^bf_1(x)\cdot f_2'(x)dx
++ \int_a^bf_2(x)\cdot f_1'(x)dx.
+\]
+(The integral exists since it follows from the existence and
+continuity of $f_1'(x)$ and $f_2'(x)$ that $f_1(x)$ and $f_2(x)$
+are continuous).  By Theorem~\hyperlink{thm121}{121},
+\[
+  \int_a^b\frac{d}{dx}\left\{f_1(x)\cdot f_2(x)\right\}dx
+= f_1(b)\cdot f_2(b)-f_1(a)\cdot f_2(a).
+\]
+Therefore
+\[
+  \int_a^bf_1(x)\cdot f_2'(x)dx
+= \left[f_1(x)\cdot f_2(x)\right]_a^b
+-\int_a^bf_2(x)\cdot f_1'(x)dx.\qedhere
+\]
+\end{proof}
+
+\begin{theorem}[123]\hypertarget{thm123}{}(Integration by substitution.)\index{Change of variable}
+If $y=\phi(x)$ has a continuous derivative at every point of
+$\interval{a}{b}$ and $f(y)$ is continuous for all values taken by
+$y=\phi(x)$ as $x$ varies from $a$ to $b$,
+\[
+  \int_A^Bf(y)dy=\int_a^bf(y)\frac{dy}{dx}dx,
+\]
+where $A=\phi(a)$, $B=\phi(b)$.
+\end{theorem}
+%-----File: 188.png---Folio 176------
+
+\begin{proof}
+By Theorem~\hyperlink{thm120}{120} and by Theorem~\hyperlink{thm79}{79},
+\[
+  \int_A^{\phi(x)}f(y)dy
+= \int_a^x\frac{d}{dx}\left(\int_A^{\phi(x)}f(y)dy\right)dx+C
+= \int_a^xf(y)\frac{dy}{dx}\cdot dx+C,
+\]
+$C$ being an arbitrary constant. $C$ is determined by letting
+$x=a$. Then if $x=b$ we have
+\[
+  \int_A^Bf(y)dy=\int_a^bf(y)\frac{dy}{dx}\cdot dx.\qedhere
+\]
+\end{proof}
+\begin{theorem}[124]\hypertarget{thm124}{}
+\[
+  \int_a^bf(x)dx=\int_A^Bf\left(\phi(y)\right)\frac{dx}{dy}dy,
+\]
+where $x=\phi(y)$ and $a=\phi(A)$, $b=\phi(B)$; provided that both
+integrals exist, and that $\phi(y)$ is non-oscillating and has a
+finite derivative.
+\end{theorem}
+
+\begin{proof}
+\[
+  \int_a^bf(x)dx
+  = \mathop{L}_{n\doteq \infty}\sum_{k=1}^nf(\xi_k)\Delta_kx
+\tag{1}
+\]
+whenever the least upper bound of $\Delta_kx$ for each $n$ approaches
+zero as $n$ approaches $+\infty$. Now let $\Delta y=\dfrac{B-A}{n}$,
+\begin{gather*}
+  y_k=A+k\cdot \Delta y,
+\\
+  \phi(y_k)-\phi(y_{k-1})=\Delta_kx.
+\end{gather*}
+Hence, by Theorem~\hyperlink{thm85}{85},
+\[
+  \Delta_kx=\phi'(\eta_k)\Delta y,
+\]
+where $\eta_k$ lies between $y_k$ and $y_{k-1}$. Now if $\xi_k=
+\phi(\eta_k)$, it will lie between $\phi(y_k)$ and $\phi(y_{k-1})$;
+moreover the $\Delta_kx$'s are all of the same sign or zero; and since
+the hypothesis makes $\phi(y)$ uniformly continuous, their least upper
+bound approaches zero as $n$ approaches $+\infty$.  Therefore
+\begin{align*}
+  \int_a^bf(x)dx
+  &=\mathop{L}_{n\doteq \infty}\sum_{k=1}^nf(\xi_k)\Delta_kx\\
+  &=\mathop{L}_{n\doteq \infty}\sum_{k=1}^n
+    f\bigl(\phi(\eta_k)\bigr)\cdot \phi'(\eta_k)\cdot \Delta y\\
+  &=\int_A^Bf\big(\phi(y)\text{\correction{$\big)$}{}}\phi'(y)dy,
+\end{align*}
+%-----File: 189.png---Folio 177------
+provided the latter integral exists.
+Hence
+\[
+  \int_a^bf(x)dx
+= \int_A^Bf\left(\phi(y)\right)\cdot \frac{dx}{dy}dy.\qedhere
+\]
+\end{proof}
+\begin{corollary}
+The validity of this theorem remains if $\phi(y)$ has a finite number
+of oscillations.
+\end{corollary}
+
+\begin{proof}
+Suppose the maximum and minimum values of
+$\phi(y)$ are
+\[
+  a_1,a_2,a_3,\ldots,a_n,
+\]
+corresponding to the values of $y$,
+\[
+  A_1,A_2,A_3,\ldots,A_n.
+\]
+Then we have
+\begin{align*}
+   \int_{a }^{b } f(x)dx
+&= \int_{a }^{a_1} f(x)dx
+ + \int_{a_1}^{a_2} f(x)dx+\ldots
+ + \int_{a_n}^{b } f(x)dx
+\\
+&= \int_{A }^{A_1} f\big(\phi(x)\text{\correction{$\big)$}{}} \frac{dx}{dy}dy
+ + \int_{A_1}^{A_2} f\big(\phi(x)\text{\correction{$\big)$}{}} \frac{dx}{dy}dy \ldots
+ + \int_{A_n}^{B } f\big(\phi(x)\text{\correction{$\big)$}{}} \frac{dx}{dy}dy
+\\
+&= \int_{A }^{B } f\big(\phi(x)\text{\correction{$\big)$}{}} \frac{dx}{dy}dy.
+\end{align*}
+The form of this proposition given in Theorem~\hyperlink{thm123}{123} would permit an
+infinitude of oscillations of $\phi(y)$.
+\end{proof}
+
+\section{General Conditions for Integrability.}\hypertarget{chVIIIsec7}{}%[7]
+
+The following lemmas are to be associated with those on pages \pageref{lp155} and
+\pageref{lp156}.
+
+\begin{lemma}[3]\hypertarget{lem3p177}{}
+If $\pi_1$ is a repartition of $\pi$, then for any function bounded on
+$\interval{a}{b}$
+\[
+O_{\pi_1} \leqq O_{\pi}.
+\]
+\end{lemma}
+\begin{proof}
+Any interval $\Delta_k x$ of $\pi$ is composed of one or more
+intervals $\Delta_k' x$, $\Delta_k'' x$, etc., of $\pi_1$, and
+these contribute to $O_{\pi_1}$ the terms
+\hypertarget{eq1p177}{\[
+  \left|\Delta_k'x\right|\Delta_k'y
+  +\left|\Delta_k''x\right|\Delta_k''y + \ldots. \tag{1}
+\]}
+%-----File: 190.png---Folio 178------
+
+The corresponding term of $O_\pi$ is
+\hypertarget{eq2p178}{\[
+  |\Delta_k x|\Delta_k y = |\Delta_k'x|\Delta_k y +
+   |\Delta_k''x|\Delta_k y + \ldots. \tag{2}
+\]}
+
+Since each of $\Delta_k'y$, $\Delta_k''y,$ etc., is less than or
+equal to $\Delta_ky$, \hyperlink{eq1p177}{(1)} is less than or equal to \hyperlink{eq2p178}{(2)}, and hence
+$O_{\pi_1} \leqq O_\pi$.
+\end{proof}
+\begin{lemma}[4]\hypertarget{lem4p178}{}
+If $\pi_0$ is any partition of the interval $\interval{a}{b}$, and
+$\varepsilon_0$ any positive number, then for any bounded function
+there exists a number $\delta_0$ such that for every partition $\pi$
+whose greatest $\Delta$ is less than $\delta_0$
+\[
+  O_{\pi_0} + \varepsilon_0 \geqq O_\pi.
+\]
+\end{lemma}
+\begin{proof}
+We prove the lemma by showing that if $\pi_0$ has $N + 1$ partition
+points $x_0$, $x_1$, $x_2, \ldots$, $x_\text{\correction{$N$}{$n$}}$, an effective choice of
+$\delta_0$ is
+\[
+  \delta_0 = \frac{\varepsilon_0}{R \cdot N},
+\]
+where $R$ is the oscillation of the function on $\interval{a}{b}$.
+
+Of the intervals of $\pi$ there are at most $N-1$ which contain as
+interior points, points of $x_0$, $x_1$, $x_2, \ldots$, $x_N$. Denote
+the lengths of these intervals of $\pi$ by $\Delta_px$, and denote by
+$\Delta_qx$ the lengths of the intervals of $\pi$ which contain as
+interior points no points of $x_0$, $x_1$, $x_2, \ldots$, $x_N$. Then
+\[
+O_\pi = \textstyle
+  \sum\limits_p|\Delta_px| \cdot \Delta_py
++ \sum\limits_q|\Delta_qx| \cdot \Delta_qy.
+\]
+If $\pi'$ is a repartition of $\pi_0$ obtained by introducing the
+points of $\pi$, then
+\[
+  \textstyle\sum\limits_q|\Delta_qx| \cdot \Delta_qy
+\]
+is a subset of the terms whose sum constitutes $O_{\pi'}$. Hence, by
+Lemma~\hyperlink{lem3p177}{3},
+\[
+  \textstyle\sum\limits_q|\Delta_qx| \cdot \Delta_qy
+\leqq O_{\pi'} \leqq O_{\pi_0}.
+\]
+Since
+\[
+  |\Delta_px| \leqq \delta_0 = \frac{\varepsilon_0}{R \cdot N},
+\]
+%-----File: 191.png---Folio 179------
+it follows that
+\[
+  \textstyle\sum\limits_p \left|\Delta_px\right| \cdot \Delta_py \leqq
+  \varepsilon_0.
+\]
+Therefore
+\[
+  O_{\pi_0} + \varepsilon_0 \geqq O_{\pi}.\qedhere
+\]
+\end{proof}
+\begin{lemma}[5]\hypertarget{lem5p179}{}
+If $\pi$ is any partition, $O_{\pi}$ is the least upper bound of the
+expression
+\[
+  S_\pi'-S_\pi'',
+\]
+where $S_\pi'$ and $S_\pi''$ may be any two values of $S_\pi$
+corresponding to different choices of the $\xi$'s.
+\end{lemma}
+
+\begin{proof}
+Without loss of generality we may assume every $\Delta_kx$ positive.
+
+\noindent Then
+\[
+  \overline{B}S_{\pi}-\underline{B}S_{\pi}
+= \overline{B}\left|{S_{\pi}}'-{S_{\pi}}''\right|.
+\mspace{100mu}
+\]
+But
+\begin{align*}
+  \overline{B}S_{\pi}
+&= \overline{B}\left\{%
+   \textstyle\sum\limits_{k=1}^n
+   f(\xi_k)\cdot \Delta_kx \right\}
+ = \textstyle\sum\limits_{k=1}^n \left\{%
+   \overline{B}f(\xi_k) \right\}\Delta_kx\\
+\intertext{and}
+   \underline{B}S_{\pi}
+&= \underline{B}\left\{%
+   \textstyle\sum\limits_{k=1}^n
+   f(\xi_k)\cdot \Delta_kx \right\}
+ = \textstyle\sum\limits_{k=1}^n
+   \left\{ \underline{B}f(\xi_k) \right\}\Delta_kx.
+\end{align*}
+Therefore
+\begin{align*}
+  \overline{B}S_{\pi}-\underline{B}S_{\pi}
+&= \textstyle\sum\limits_{k=1}^n\left[
+   \overline{B}f(\xi_k)-\underline{B}f(\xi_k) \right]\Delta_kx
+\\
+&= \textstyle\sum\limits_{k=1}^n\Delta_ky\cdot \Delta_kx=O_{\pi}.
+\end{align*}
+Therefore
+\[
+  \overline{B}({S_{\pi}}'-{S_{\pi}}'')=O_{\pi}.\qedhere
+\]
+\end{proof}
+
+\begin{theorem}[125]\hypertarget{thm125}{}
+A necessary and sufficient condition that a function $f(x)$, defined,
+single-valued, and bounded on an interval $\interval{a}{b}$ shall be
+integrable on $\interval{a}{b}$, is that the greatest lower bound of
+$O_{\pi}$ for this function shall be zero.
+\end{theorem}
+
+\begin{proof}
+We first show that if $f(x)$ is integrable the lower bound of
+$O_{\pi}$ is zero. By hypothesis,
+\[
+  \int_a^bf(x)dx=\mathop{L}_{\delta\doteq 0}S_{\delta}
+\]
+exists. By Theorem~\hyperlink{thm27}{27}, Chapter~\hyperlink{chapIV}{IV}, this implies that for every $\varepsilon$
+%-----File: 192.png---Folio 180------
+there exists a $\delta_{\varepsilon}$ such that for every
+$\delta_1<\delta_\varepsilon$ and $\delta_2<\delta_\varepsilon$
+\[
+  \left|S_{\delta_1}-S_{\delta_2}\right|<\varepsilon.
+\]
+Hence, if $\pi$ be a partition whose intervals $\Delta_k x$ are all
+less than $\delta_\varepsilon$, we must have
+\[
+  \left|S_\pi'-S_\pi''\right|<\varepsilon
+\]
+for every $S_\pi'$ and $S_\pi''$. By Lemma~\hyperlink{lem5p179}{5} this implies that
+$O_\pi\leqq \varepsilon$.  But if for every $\varepsilon$ there exists
+a $\pi$ such that $O_\pi\leqq \varepsilon$, then
+\[
+  \underline{B}O_\pi=0.
+\]
+
+Secondly, we show that if the lower bound of $O_\pi$ is zero,
+$S_\delta$ converges to a single value,
+\[
+\int_a^bf(x)dx,
+\]
+as $\delta$ approaches zero. Given any positive quantity $\varepsilon$
+there exists a partition $\pi_\varepsilon$, such that
+$O_{\pi_\varepsilon}<\dfrac\varepsilon4$. By Lemma~\hyperlink{lem4p178}{4} there exists a
+$\delta_\varepsilon$ such that for every $\pi$ whose intervals are
+numerically less than $\delta_\varepsilon$
+\[
+  O_\pi\leqq O_{\pi_\varepsilon}+\frac\varepsilon4<\frac\varepsilon2.
+\]
+
+Now let $S_{\pi_\varepsilon'}$ and $S_{\pi_\varepsilon''}$ be any
+two values of $S_{\delta_\varepsilon}$, and let $\pi_\varepsilon'''$
+be the partition composed of the points of both $\pi_\varepsilon'$
+and $\pi_\varepsilon''$. Then for any value of
+${S_{\pi_\varepsilon'''}}$ we have, by Lemma~\hyperlink{lem2p156}{2},
+\begin{align*}
+  \left|S_{\pi_\varepsilon'}-S_{\pi_\varepsilon'''}\right| &\leqq
+  O_{\pi_\varepsilon'}< \frac\varepsilon2,\\
+  \left|S_{\pi_\varepsilon''}-S_{\pi_\varepsilon'''}\right| &\leqq
+  O_{\pi_\varepsilon''}< \frac\varepsilon2.\\
+\intertext{Therefore}
+  \left|S_{\pi_\varepsilon'}-S_{\pi_\varepsilon''}\right|
+  &<\varepsilon.
+\end{align*}
+%-----File: 193.png---Folio 181------
+Hence for every $\varepsilon$ we have a $\delta_\varepsilon$ such that
+for every two values of $S_\delta$, $\delta < \delta_\varepsilon$,
+\[
+  |S_{\pi'_\varepsilon}-S_{\pi''_\varepsilon}|
+< \varepsilon.
+\]
+By Theorem~\hyperlink{thm27}{27}, this implies the existence of $\displaystyle\mathop{L}_{\delta\doteq
+0} S_\delta$.
+\end{proof}
+
+In case the definite integral does not exist it is sometimes desirable
+to use the upper and lower bounds of indeterminateness of $S_\delta$
+as $\delta$ approaches zero. These are denoted respectively by the
+symbols
+$\overline{\displaystyle\int_a^b} f(x)dx$ and
+$\underline{\displaystyle\int_a^b} f(x)dx$\footnote{%
+    For a more extended theory of these integrals,
+    cf.~\textsc{Pierpont}, page~337.}
+and are called the upper\index{Upper!integral} and \index{Lower integral}lower definite integrals of
+$f(x)$\correction{.}{}
+They are both equal to
+\[
+  \int_a^b f(x)dx
+\]
+if and only if the latter integral exists. They are usually defined by
+the equations
+\[
+  \overline{\int_a^b} f(x)dx
+= \underline{B}\overline{S}_\pi,
+\]
+where $\overline{S}_\pi = \displaystyle\sum_{k=1}^n \{ \overline{B}
+f(\xi_k) \} \Delta_k x$ for all partitions of $\pi$, and
+\[
+  \underline{\int_a^b} f(x)dx
+= \overline{B}\underline{S}_\pi,
+\]
+where $\underline{S}_\pi = \displaystyle\sum_{k=1}^n \{ \underline{B}
+f(\xi_k) \} \Delta_k x$ for all partitions of $\pi$.
+
+That $\displaystyle\int_a^b f(x)dx$ exists when the upper and lower
+integrals are equal is evident under this definition, because
+\[
+  O_\pi = \overline{S}_\pi-\underline{S}_\pi,
+\]
+%-----File: 194.png---Folio 182------
+and thus $\underline{B}O_\pi = 0$ if and only if
+\[
+   \overline{\int_a^b} f(x)dx
+= \underline{\int_a^b} f(x)dx.
+\]
+
+For every value of $\delta > 0$ there is an infinite set of partitions
+$\pi$, for which the largest $\Delta_k x$ is less than $\delta$, and
+for each of these there is a value of $O_\pi$. If $O_\delta$ stands
+for any such $O_\pi$, then $O_\delta$ is a many-valued function of
+$\delta$.
+
+\begin{theorem}[126]\hypertarget{thm126}{}
+A necessary and sufficient condition that a function $f(x)$, defined,
+single-valued, and bounded on an interval $\interval{a}{b}$, is
+integrable is that
+\[
+  \mathop{L}_{\delta\doteq 0} O_\delta = 0.
+\]
+\end{theorem}
+
+\begin{proof}\textit{The condition is necessary.}
+
+By Theorem~\hyperlink{thm125}{125} the integrability of $f(x)$ implies $\underline{B}
+O_\pi = 0$.  Hence for every $\varepsilon$ there exists a partition
+$\pi$ such that
+\[
+  O_\pi < \varepsilon.
+\]
+By Lemma~\hyperlink{lem4p178}{4} there exists a $\delta_\varepsilon$ such that for every
+$\pi'$ whose greatest $\Delta x$ is less than $\delta_\varepsilon$
+\[
+  O_{\pi'} < O_\pi + \varepsilon < 2\varepsilon.
+\]
+Hence
+\[
+  \mathop{L}_{\delta\doteq 0} \text{\correction{$O_\delta$}{$O^\delta$}} = 0.
+\]
+
+\textit{The condition is sufficient.}
+
+Since
+\[
+  \mathop{L}_{\delta\doteq 0} \text{\correction{$O_\delta$}{$O^\delta$}} = 0,
+\]
+and  $O_\delta > 0$,
+\[
+  \underline{B} O_\pi = 0.
+\]
+Hence the function is integrable by Theorem~\hyperlink{thm125}{125}.
+\end{proof}
+
+\begin{theorem}[127]\hypertarget{thm127}{}\label{p182th127}
+A necessary and sufficient condition that a function, defined,
+single-valued, and bounded on an interval $\interval{a}{b}$, shall be
+integrable on that interval is that for every pair of positive
+%-----File: 195.png---Folio 183------
+numbers $\sigma$ and $\lambda$ there exists a partition $\pi$ such
+that the sum of the lengths of those intervals on which the
+oscillation of the function is greater than $\sigma$ is less than
+$\lambda$.
+\end{theorem}
+
+\begin{proof}\textit{The condition is necessary.}
+
+If for a given pair of positive numbers $\sigma$ and $\lambda$ there
+exists no $\pi$ such as is required by the theorem, then $O_\pi >
+\sigma\cdot\lambda$ for every $\pi$, which is contrary to the
+conclusion of Theorem~\hyperlink{thm125}{125} that
+\[
+  \underline{B}O_\pi = 0.
+\]
+
+\textit{The condition is sufficient.}
+
+For a given positive $\varepsilon$ choose $\sigma$ and $\lambda$ so
+that
+\[
+\sigma(b-a) < \frac\varepsilon2 \text{ and }
+  \lambda \cdot R < \frac\varepsilon2,
+\]
+where $R$ is the oscillation of the function on $\interval{a}{b}$. Let
+$\pi$ be a partition such that the sum of the lengths of those
+intervals on which the oscillation of the function is greater than
+$\sigma$ is less than $\lambda$. Then the sum of the terms of $O_\pi$
+which occur on these intervals is less than
+\[
+  \lambda \cdot R,
+\]
+and the sum of the terms of $O_\pi$ on the remaining intervals is less
+than
+\[
+  \sigma(b-a).
+\]
+Therefore
+\[
+  O_\pi < \lambda \cdot R + \sigma(b-a) < \varepsilon.
+\]
+Hence
+\[
+  \underline{B}O_\pi = 0,
+\]
+whence by Theorem~\hyperlink{thm125}{125} the integral exists.
+\end{proof}
+
+\begin{definition}\index{Content of a set of points}
+The \textit{content} of a set of points $[x]$ on an interval
+$\interval{a}{b}$ is a number $C[x]$ defined as follows: Let $\pi$ be
+any partition of $\interval{a}{b}$, none of the partition points of
+which are points of $[x]$, and $D_\pi$ the sum of the lengths of those
+intervals of $\pi$
+%-----File: 196.png---Folio 184------
+which contain points of [$x$] as interior points. Then
+\[
+  \underline{B}D_\pi = C[x].
+\]
+
+An important special case is where
+\[
+  C[x]=0.
+\]
+
+It is evident that if a set [$x$] has content zero, for every
+$\varepsilon$ there exists a finite set of segments of lengths
+\[
+  \varepsilon_1,\; \varepsilon_2,\; \varepsilon_3, \ldots,\;
+  \varepsilon_n
+\]
+which contain every point [$x$] and such that
+\[
+  \sum_{i=1}^n \varepsilon_i < \varepsilon.
+\]
+It is also evident that if the sets [$x_1$] and [$x_2$] are of content zero,
+then the set of all $x_1$ and $x_2$ is of content zero.\footnote{%
+  For further discussion of the notion \emph{content} see
+  \textsc{Pierpont}, \textit{Real Functions},
+  Vol.~I, p.~352, and \correction{\textsc{Lebesgue}}{\textsc{Lebesque}}, \textit{Le\c cons sur
+  l'Int\'egration}.}
+\end{definition}
+
+\begin{theorem}[128]\hypertarget{thm128}{}
+A necessary and sufficient condition for the integrability of a
+function $f(x)$ on an interval $\interval{a}{b}$ is that for every
+$\sigma > 0$ the set of points $[x_\sigma]$ at which the oscillation
+of $f(x)$ is greater than or equal to $\sigma$ shall be of content
+zero.\footnote{%
+    Compare the example on page~\pageref{egp155}.}
+\end{theorem}
+
+\begin{proof}
+If at every point of an interval $\interval{c}{d}$ the oscillation of
+$f(x)$ is less than $\sigma$, then about each point of $\interval{c}{d}$ there is a segment upon which the oscillation is less than
+$\sigma$, and hence by Theorem~\hyperlink{thm11}{11}, Chapter~\hyperlink{chapII}{II}, there is a partition of
+$\interval{c}{d}$ upon each interval of which the oscillation of
+$f(x)$ is less than $\sigma$.
+
+Now to prove the condition sufficient we observe that if the content
+of [$x_\sigma$] is zero, there exists for every $\lambda$ a partition
+$\pi_\lambda$,
+such that the sum of the lengths of the intervals containing points of
+[$x_\sigma$] is less than $\lambda$. Moreover we have just seen
+%-----File: 197.png---Folio 185------
+that the intervals which do not contain points on $[x_\sigma]$ can be
+repartitioned into intervals on which the oscillation is less than
+$\sigma$. Hence, by Theorem~\hyperlink{thm127}{127}, the function is integrable.
+
+To prove the condition necessary we note that on every interval
+containing a point, $x_\sigma$, the oscillation of $f(x)$ is greater
+than \correction{or equal to}{or equal to or equal to}
+$\sigma$. Hence, if
+\[
+  C[x_\sigma] > 0,
+\]
+the sum of the intervals upon which the oscillation is greater than or
+equal to $\sigma$ is greater than $C[x_\sigma]$.
+\end{proof}
+\begin{definition}\index{Numerably infinite set}\index{Non-numerably infinite set}
+A set of points is said to be numerable if it is capable of being set
+into one-to-one correspondence with the positive integral numbers. If
+a set $[x]$ is numerable, it can always be indicated by the notation
+$x_1$, $x_2$, $x_3, \ldots$, $x_n, \ldots$, or $\{x_n\}$, but if it is
+not numerable, the notation $\{x_n\}$ cannot be applied with the
+understanding that $n$ is integral.
+\end{definition}
+
+\begin{theorem}[129]\hypertarget{thm129}{}
+A perfect set of points is not numerably infinite.\footnote{%
+  For definition of perfect set see page~\pageref{dp41}.}
+\end{theorem}
+
+\begin{proof}
+Suppose the theorem not true. Then there exists a sequence of points
+$\{x_n\}$ containing every point of a perfect set $[x]$. Let $P_1$ be
+any point of $[x]$, and $\overline{a_1\ b_1}$ a segment containing
+$P_1$. Let $x_{n_1}$ be the first of $\{x_n\}$ within $\overline{a_1\
+b_1}$. Since $x_n$ is a limit point of points of $[x]$, there are
+points of the set other than $P_1$ and $x_{n_1}$ on the segment
+$\overline{a_1\ b_1}$. Let $P_2$ be such a point, and let
+$\overline{a_2\ b_2}$ be a segment within $\overline{a_1\ b_1}$ and
+containing $P_2$ but neither $P_1$ nor $x_{n_1}$. Let $x_{n_2}$ be the
+first point of $\{x_n\}$ within $\overline{a_2\ b_2}$. Proceeding in
+this manner we obtain a sequence of segments $\{\overline{a_i\ b_i}\}$
+such that every segment lies within the preceding and such that every
+segment $\overline{a_i\ b_i}$ contains no point $x_{n_{i-k}}$ of the
+sequence $\{x_n\}$. By the lemma on page~\pageref{lp42}, Chapter~\hyperlink{chapII}{II}, there is a
+point $P$ on every segment of this set. Since there are points of
+$[x]$ on every segment $\overline{a_i\ b_i}$, $P$ is a limit point of
+the set $[x]$. Since $[x]$ is a perfect set, $P$ is a point of
+$[x]$. But if $P$
+%-----File: 198.png---Folio 186------
+were in the sequence $\left\{x_n\right\}$, there would be only a
+finite number of points of $[x]$ preceding $P$, whereas by the
+construction there is an infinitude of such points.
+\end{proof}
+
+\begin{theorem}[130]\hypertarget{thm130}{}
+A numerably infinite set of sets of points each of content zero cannot
+contain every point of any interval.
+\end{theorem}
+
+\begin{proof}
+Let the set of sets be ordered into a sequence $\left\{[x]_n\right\}$.
+We show that on every segment $\overline{a\ b}$ there is at least one
+point not of $\left\{[x]_n\right\}$. Since $[x]_1$ is of content zero,
+there is a segment $\overline{a_1\ b_1}$ contained in $\overline{a\
+b}$ which contains no point of $[x]_1$. Let $[x]_{{n}_1}$ be the first
+set of the sequence which contains a point of $\overline{a_1\
+b_1}$. Since $[x]_{{n}_1}$ is of content zero, there is a segment
+$\overline{a_2\ b_2}$ contained in $\overline{a_1\ b_1}$ which
+contains no point of $[x]_{{n}_1}$. Continuing in this manner we
+obtain a sequence of segments $\overline{a\ b}$, $\overline{a_1\
+b_1},\ldots$, $\overline{a_n\ b_n} \ldots$ such that every segment
+lies within the preceding,
+and such that $\overline{a_n\ b_n}$ contains no point of
+$[x]_1,\ldots$, $[x]_n$. By the lemma on page~\pageref{lp42} there is at least one
+point $P$ on all these segments. Hence $P$ is a point of $\overline{a\
+b}$ and is not a point of any set of $\left\{[x]_n\right\}$.
+\end{proof}
+
+\begin{theorem}[131]\hypertarget{thm131}{}
+The points of discontinuity of an integrable function form at most a
+set consisting of a numerable set of sets, each of content zero.
+\end{theorem}
+
+\begin{proof}
+Let $\sigma_1$, $\sigma_2$, $\sigma_3,\ldots$ be any set of numbers
+such that
+\[
+  \sigma_n>\sigma_{n+1},
+\]
+and
+\[
+  \mathop{L}_{n\doteq \infty}\sigma_n =0.
+\]
+By Theorem~\hyperlink{thm128}{128} the set of points $[x_{\sigma_n}]$ at which the
+oscillation of $f(x)$ is greater than or equal to $\sigma_{n+1}$ and
+less than $\sigma_n$ is of content zero. Since the set of sets
+$\left\{[x_{\sigma_n}]\right\}$ includes all the points of
+discontinuity of $f(x)$, this proves the theorem.
+\end{proof}
+
+\begin{theorem}[132]\hypertarget{thm132}{}
+If a function $f(x)$ is integrable on an interval $\interval{a}{b}$,
+then it is continuous at a set of points which is everywhere dense on
+$\interval{a}{b}$.
+\end{theorem}
+%-----File: 199.png---Folio 187------
+
+\begin{proof}
+If the theorem fails to hold, then there exists an interval
+$\interval{a}{b}$ on which the function is discontinuous at every
+point. By Theorem~\hyperlink{thm131}{131} an integrable function is discontinuous at most
+on a numerably infinite set of sets each of content zero, and by
+Theorem~\hyperlink{thm130}{130} such sets of sets fail to contain every point of any
+interval.
+\end{proof}
+
+\begin{theorem}[133]\hypertarget{thm133}{}
+If
+\[
+  \int_a^X f(x)dx=0
+\]
+for every $X$ of $\interval{a}{b}$, then $f(x) =0$ on a set of points
+everywhere dense on $\interval{a}{b}$, and for every $\sigma>0$ the
+points where $|f(x)|>\sigma$ form a set of content zero.
+\end{theorem}
+
+\begin{proof}
+At every point $X$ where $f(x)$ is continuous, according to the
+corollary of Theorem~\hyperlink{thm119}{119},
+\[
+  \frac{d}{dX}\int_a^X f(x)dx = f(X) = 0,
+\]
+since $\displaystyle\int_a^X f(x)dx$ is a constant. The points of
+continuity of $f(x)$ are everywhere dense, according to
+Theorem~\hyperlink{thm132}{132}. Hence the zero points of $f(x)$ are everywhere dense. At
+a point of discontinuity the oscillation of $f(x)$ is greater than or
+equal to $|f(x)|$.  Hence the points where $|f(x)|>\sigma$ form a set
+of content zero.
+\end{proof}
+
+\begin{theorem}[134]\hypertarget{thm134}{}
+If
+\[
+  \int_a^X f(x)dx = \int_a^X \phi(x)dx
+\]
+for every $X$ of $\interval{a}{b}$, then $f(x) = \phi(x)$ on a set of
+points everywhere dense on $\interval{a}{b}$, and for every $\sigma>0$
+the points where $|f(x)-\phi(x)|>\sigma$ forms a set of content zero.
+\end{theorem}
+
+\begin{proof}
+Apply the theorem above to $f(x)-\phi(x)$.
+\end{proof}
+
+\begin{theorem}[135]\hypertarget{thm135}{}
+If $f(x)$ is integrable from $a$ to $b$, then $|f(x)|$ is integrable
+from $a$ to $b$.\footnote{%
+  The converse theorem is not true; cf.~example given on page~\pageref{egp192}.}
+\end{theorem}
+%-----File: 200.png---Folio 188------
+
+\begin{proof}
+Since
+\[
+\text{\correction{$0$}{$O$}}\leqq O_{\pi}\left|f(x)\right|\leqq O_{\pi}f(x),
+\]
+it follows that $\underline{B}\ O_{\pi}f(x)=0$ implies $\underline{B}\
+O_{\pi}|f(x)|=0$, and hence the integrability of $f(x)$ implies the
+integrability of $|f(x)|$.
+\end{proof}
+
+\begin{theorem}[136]\hypertarget{thm136}{}
+If $f(x)$ and $\phi(x)$ are both integrable on an interval
+$\interval{a}{b}$, then
+\hypertarget{fn1}{\[
+  f(x)\cdot \phi(x) \tag{1}
+\]}
+is integrable on $\interval{a}{b}$; and, provided there is a constant
+$m>0$ such that $|\phi(x)|-m>0$ for $x$ on $\interval{a}{b}$, then
+\hypertarget{fn2}{\[
+  f(x) \div \phi(x) \tag{2}
+\]}
+is integrable on $\interval{a}{b}$.
+\end{theorem}
+
+\begin{proof}
+Since $f(x)$ and $\phi(x)$ are both integrable on $\interval{a}{b}$,
+it follows that for every pair of positive numbers $\sigma$ and
+$\lambda$ there is a partition $\pi_1$ for $f(x)$ and a partition
+$\pi_2$ for $\phi(x)$ such that the sums of the lengths of the
+intervals on which the oscillations of $f(x)$ and $\phi(x)$
+respectively are greater than $\sigma$ are less than $\lambda$. Let
+$\pi$ be the partition consisting of the points of both $\pi_1$ and
+$\pi_2$. Then the sum of the intervals of $\pi$ on which the
+oscillation of either $f(x)$ or $\phi(x)$ is greater than $\sigma$ is
+less than $2\lambda$. Let $M$ be the greater of $\overline{B}|f(x)|$
+and $\overline{B}|\phi(x)|$ on $\interval{a}{b}$.  Then on any
+interval of $\pi$ on which the oscillations of $f(x)$ and $\phi(x)$
+are both less than $\sigma$ the oscillation of $f(x)\cdot \phi(x)$ is
+less than $\sigma M$. Hence the sum of the intervals on which the
+oscillation of $f(x)\cdot \phi(x)$ is greater than $\sigma M$ is less
+than $2\lambda$.  Since $\sigma$ and $\lambda$ may be chosen so that
+$2\lambda$ and $\sigma M$ shall be any pair of preassigned numbers, it
+follows by Theorem~\hyperlink{thm127}{127} that $f(x)\cdot \phi(x)$ is integrable on
+$\interval{a}{b}$.
+
+In view of the argument above it is sufficient for the second
+%-----File: 201.png---Folio 189------
+theorem to prove that $\dfrac{1}{\phi(x)}$ is integrable on
+$\interval{a}{b}$ if $\phi(x)$ is integrable and
+$|\phi(x)|>m$. Consider a partition $\pi$ such that the sum of the
+intervals on which the oscillation of $\phi(x)$ is greater than
+$\sigma$ is less than $\lambda$. Since
+\[
+  \left| \frac{1}{ \phi(x_1) }
+      -\frac{1}{ \phi(x_2) } \right|
+= \frac{\left| \phi(x_1)-\phi(x_2) \right|}
+       {\left| \phi(x_1) \right|\cdot \left| \phi(x_2) \right|},
+\]
+it follows that $\pi$ is such that the sum of the intervals on which
+the oscillation of $\dfrac{1}{\phi(x)}$ is greater than
+$\dfrac{\sigma}{m^2}$ is less than $\lambda$, and $\dfrac{1}{\phi(x)}$
+is integrable according to Theorem~\hyperlink{thm127}{127}.
+\end{proof}
+
+A second proof may be made by comparing the integral
+oscillations of $f(x)$ and $\phi(x)$ with those of the functions \hyperlink{fn1}{(1)} and
+\hyperlink{fn2}{(2)} and applying Theorem~\hyperlink{thm125}{125}.\footnote{%
+  Cf.\ \textsc{Pierpont}, Vol.~I, pp.~346, 347, 348.}
+
+\begin{theorem}[137]\hypertarget{thm137}{}
+If $f(x)$ is an integrable function on an interval $\interval{a}{b}$,
+and if $\phi(y)$ is a continuous function on an interval
+$\interval{\underline{B}f}{\overline{B}f}$, where $\underline{B}f$ and
+$\overline{B}f$ are the lower and upper bounds respectively of $f(x)$
+on $\interval{a}{b}$, then $\phi\{f(x)\}$ is an integrable function of
+$x$ on the interval $\interval{a}{b}$.\footnote{%
+  This theorem is due to \textsc{Du Bois Reymond}. It cannot be
+  modified so as to read ``an integrable function of an integrable
+  function is integrable.''  Cf.\ \textsc{E.~H. Moore}, \textit{Annals
+  of Mathematics}, new series, Vol.~2, 1901, p.~153.}
+\end{theorem}
+
+\begin{proof}
+By Theorem~\hyperlink{thm48}{48} there exists for every $\sigma>0$ a $\delta_{\sigma}$
+such that for $|y_1-y_2|<\delta_{\sigma}$,
+\hypertarget{eq1p189}{\[
+  \left|\phi(y_1)-\phi(y_2)\right|<\sigma. \tag{1}
+\]}
+
+Since $f(x)$ is integrable on $\interval{a}{b}$ it follows by
+Theorem~\hyperlink{thm127}{127} that for every positive number $\lambda$ there is a
+partition $\pi$ such
+%-----File: 202.png---Folio 190------
+that the sum of the intervals on which the oscillation of $f(x)$ is
+greater than $\delta_{\sigma}$ is less than $\lambda$. But by \hyperlink{eq1p189}{(1)} this
+means that the sum of the intervals on which the oscillation of
+$\phi\{f(x)\}$ is greater than $\sigma$ is less than $\lambda$. This,
+by Theorem~\hyperlink{thm127}{127}, proves that $\phi\left\{f(x)\right\}$ is integrable.
+\end{proof}
+%-----File: 203.png---Folio 191------
+
+
+
+\chapter{IMPROPER DEFINITE INTEGRALS.}\hypertarget{chapIX}{}%[IX]
+\index{Improper definite integral}
+\section{The Improper Definite Integral on a Finite Interval.}\hypertarget{chIXsec1}{}%[1]
+
+\label{p191}If $f(x)$ is infinite at one or more points of the interval
+$\interval{a}{b}$, then, whatever may be the other properties of the
+function, the definite integral of $f(x)$ defined in Chapter~\hyperlink{chapVIII}{VIII}
+cannot exist on the interval $\interval{a}{b}$.
+
+\begin{definition}\label{dp192}
+If $\displaystyle \int_x^bf(x)dx$ exists for every $x$, $a<x<b$, and
+if\footnote{%
+  We will understand throughout this chapter that in the expression
+  \[
+    \mathop{L}_{x\doteq a}\int_x^bf(x)dx
+  \]
+  $x$ approaches $a$ on the interval $\interval{a}{b}$.  }
+\[
+  \mathop{L}_{x\doteq a}\int_x^bf(x)dx
+\]
+exists and is finite, $f(x)$ being unbounded on every neighborhood of
+$x=a$, then this limit is the \textit{improper definite integral} on
+the interval $\interval{a}{b}$. If $f(x)$ is unbounded in every
+neighborhood of $x=a$, and also in every neighborhood of $x=b$, but
+bounded on some neighborhood of every other point of the interval
+$\interval{a}{b}$, we consider two intervals $\interval{a}{c}$ and
+$\interval{c}{b}$ where $c$ is any point $a<c<b$. If the improper
+definite integral exists on $\interval{a}{c}$ and also on
+$\interval{c}{b}$, then the sum of these integrals is the improper
+definite integral on $\interval{a}{b}$.
+\end{definition}
+%-----File: 204.png---Folio 192------
+
+This definition can obviously be extended to the case where the
+function is unbounded in the neighborhood of a finite number of
+points. Such points are then considered as partition points, dividing
+the interval $\interval{a}{b}$ into a set of subintervals.  If the
+improper definite integral exists on each of these intervals, their
+sum is the improper definite integral on $\interval{a}{b}$.
+
+\begin{theorem}[138]\hypertarget{thm138}{}
+If $\displaystyle \int_x^bf(x)dx$ exists for every $x$, $a<x<b$, then
+a necessary and sufficient condition that
+\[
+  \mathop{L}_{x\doteq a}\int_x^bf(x)dx
+\]
+shall exist and be finite is that for every $\varepsilon$ there exists
+a ${V_{\varepsilon}}^*(a)$ such that for every two values of $x$,
+$x_1$ and $x_2$, on the interval $\interval{a}{b}$ and on
+${V_{\varepsilon}}^*(a)$
+\[
+  \left|\int_{x_1}^{x_2}f(x)dx\right|<\varepsilon.
+\]
+\end{theorem}
+
+\begin{proof}
+This theorem is a special case of Theorem~\hyperlink{thm27}{27}, since,
+by Theorem~\hyperlink{thm110}{110},
+\[
+  \int_{x_1}^{x_2}f(x)dx=\int_{x_1}^bf(x)dx-\int_{x_2}^bf(x)dx.\qedhere
+\]
+\end{proof}
+
+\begin{theorem}[139]\hypertarget{thm139}{}
+If $\displaystyle \int_x^bf(x)dx$ exists for every $x$, $a<x<b$, and
+if
+\[
+  \mathop{L}_{x\doteq a}\int_x^b\left|f(x)\right|dx
+\]
+is finite, then
+\[
+  \mathop{L}_{x\doteq a}\int_x^bf(x)dx
+\]
+exists and is finite.\footnote{%
+     \label{egp192}The first part of the hypothesis in this theorem is not
+     redundant, as is shown by the following example. Let
+     $f(x)=x^{-\frac12}$ for positive rational values of $x$ and
+     $f(x)=-x^{-\frac12}$ for positive irrational values of $x$. In
+     this case $\displaystyle \mathop{L}_{x\doteq 0}\int_x^b|f(x)|dx$
+     exists and is finite, while $\displaystyle \int_x^bf(x)dx$ does
+     not exist for any value of $x$ on the interval $a~b$, and
+     consequently $\displaystyle \mathop{L}_{x\doteq a}\int_x^bf(x)dx$
+     has no meaning since the limitand does not exist.}
+\end{theorem}
+%-----File: 205.png---Folio 193------
+
+\begin{proof}
+By the necessary condition of Theorem~\hyperlink{thm138}{138} there is a
+${V_{\varepsilon}}^*(a)$ corresponding to any preassigned
+$\varepsilon$ such that for any two values of $x$, $x_1$ and $x_2$,
+which lie on the segment $\interval{a}{b}$ and on
+${V_{\varepsilon}}^*(a)$
+\[
+  \left|\int_{x_1}^{x_2}\left|f(x)\right|dx\right|<\varepsilon.
+\]
+But, by Theorem~\hyperlink{thm107}{107},
+\[
+  \left| \int_{x_1}^{x_2}\left|f(x)\right|dx \right|\geqq
+  \left| \int_{x_1}^{x_2} f(x) dx \right|,
+\]
+since, by the hypothesis and Theorem~\hyperlink{thm105}{105}, $\displaystyle
+\int_{x_1}^{x_2}f(x)dx$ exists.  Hence, by the sufficient condition of
+Theorem~\hyperlink{thm138}{138},
+\[
+  \mathop{L}_{x\doteq a}\int_x^bf(x)dx
+\]
+exists and is finite.
+\end{proof}
+
+\begin{theorem}[140]\hypertarget{thm140}{}
+If $\displaystyle \int_x^bf(x)dx$ exists for every $x$ on the segment
+$\overline{a\ b}$, and if $(x-a)^kf(x)$ is bounded on $V^*(a)$ for
+some value of $k$, $0<k<1$, then
+\[
+  \mathop{L}_{x\doteq a}\int_x^bf(x)dx
+\]
+exists and is finite.
+\end{theorem}
+
+\begin{proof}
+By hypothesis $ (x-a)^k|f(x)|\leqq M$, i.e.,
+\[
+  |f(x)|\leqq \frac{M}{(x-a)^k},
+\]
+%-----File: 206.png---Folio 194------
+where $M$ may be taken greater than one. The proof of the theorem
+consists in showing that for every $\varepsilon$ there exists a
+$\delta_{\varepsilon}$
+such that if $0<x_1-a<\delta_{\varepsilon}$,
+$0<x_2-a<\delta_{\varepsilon}$, $x_1<x_2$, then
+\[
+  \left| \int_{x_1}^{x_2}f(x)dx\right|<\varepsilon.
+\]
+
+By Theorems \hyperlink{thm105}{105} and \hyperlink{thm113}{113},
+\begin{multline*}
+   \left| \int_{x_1}^{x_2}f(x)dx\right| \qqle
+\int_{x_1}^{x_2}\left|f(x)\right|dx \qqle
+\int_{x_1}^{x_2}\frac{M}{(x-a)^k}dx\\
+=\frac{M}{1-k}\left\{(x_2-a)^{1-k}-(x_1-a)^{1-k}\right\}.
+\end{multline*}
+That the last term of this series of inequalities is infinitesimal,
+the reader may verify by choosing
+\[
+  \delta_{\varepsilon}
+= \left(\frac{\varepsilon(1-k)}{M}\right)^{\frac{1}{1-k}}.\qedhere
+\]
+\end{proof}
+
+This theorem may also be proved as a corollary of Theorem~\hyperlink{thm143}{143}.
+
+\begin{corollary}
+If $f(x)$ is integrable on $\interval{x}{b}$ for every $x$ of
+$\interval{a}{b}$, and is of the same or lower order than
+$\dfrac{1}{(x-a)^k}$ for some value of $k$, $0<k<1$, then
+\[
+  \mathop{L}_{\text{\correction{$x\doteq a$}{$x=a$}}}\int_x^bf(x)dx
+\]
+exists and is finite.
+\end{corollary}
+
+\begin{theorem}[141]\hypertarget{thm141}{}
+If for any positive number $m$ and for any $k\geqq 1$ there exists a
+$V^*(a)$ on which $f(x)$ does not change sign, and on which
+$(x-a)^kf(x)>m$ for every $x$, then
+\[
+  \mathop{L}_{x\doteq a}\int_x^bf(x)dx
+\]
+cannot exist and be finite.
+\end{theorem}
+%-----File: 207.png---Folio 195------
+
+\begin{proof}(1) In case
+\[
+  \int_x^b f(x)dx
+\]
+fails to exist for some value of $x$ between $a$ and $b$,
+\[
+  \mathop{L}_{x \doteq a} \int_x^b f(x) dx
+\]
+fails to exist because the limitand function does not exist.
+
+(2) If
+\[
+  \int_x^b f(x)dx
+\]
+exists for every value of $x$ between $a$ and $b$, we proceed as
+follows: Let $\delta<1$ be the length of a $V^*(a)$ on which $f(x)$
+does not change sign, and on which $(x-a)^kf(x)>m$, and let $x_2$ be
+the extremity of this neighborhood, which is greater than $a$. Then
+$|f(x)|>\dfrac{m}{(x-a)^k}>\dfrac{m}{(x_2-a)^k}$ for every $x$ on this
+neighborhood.  Take $x_1$ so that $(x_2-a)^k=2(x_2-x_1)$.
+Then
+\[
+  \left|\int_{x_1}^{x_2}f(x)dx\right| >
+  \frac{m}{(x_2-a)^k} (x_2-x_1) = \tfrac{1}{2}m.
+\]
+Hence, by the necessary condition of Theorem~\hyperlink{thm138}{138},
+\[
+  \mathop{L}_{x\doteq a} \int_x^b f(x) dx
+\]
+cannot exist and be finite.
+\end{proof}
+
+\begin{theorem}[142]\hypertarget{thm142}{}
+If
+\[
+  \mathop{L}_{x\doteq a}\int_x^b f(x)dx
+\]
+exists and is finite and if $f(x)$ approaches infinity monotonically
+as $x\doteq a$ on some $V^*(a)$, then
+\[
+  \mathop{L}_{x\doteq a} (x-a) \cdot f(x) = 0,
+\]
+%-----File: 208.png---Folio 196------
+or in other words $f(x)$ has an infinity of order lower than
+$\dfrac{1}{x-a}$.\footnote{%
+  $\displaystyle\mathop{L}_{x\doteq a} (x-a)\cdot f(x) =0$ is not a
+  sufficient condition for the existence of
+  \[
+    \mathop{L}_{x\doteq a} \int_x^b f(x)dx,
+  \]
+  as is shown by the following example. Consider a set of points
+  $x_1$, $x_2$, $x_3,\ldots$, $x_n,\ldots$ such that $x_n-a =
+  2(x_{n+1}-a)$, $x_1-a$ being unity.
+  Define $f(x_1)=1$, $f(x_2)=\frac43$, $f(x_3)=2,\ldots$,
+  $f(x_n)=\dfrac{2^n}{n+1},\ldots$. Let the function be linear from
+  $f(x_1)$ to $f(x_2)$, from $f(x_2)$ to $f(x_3)$, etc. Then
+  \[
+    \left| \int_{x_1}^{x_2} f(x)dx\right| > \tfrac{1}{2}, \qquad
+    \left| \int_{x_2}^{x_3} f(x)dx\right| > \tfrac{1}{3},
+    \text{ etc.}
+  \]
+  Since these integrals are all of the same sign, their sum for any
+  given number of terms is greater than the sum of the corresponding
+  number of terms in the harmonic series. Also $(x_n-a) \cdot f(x_n) =
+  \dfrac{2}{n+1}$, whence $\displaystyle\mathop{L}_{x\doteq a} (x-a)\cdot f(x)=0$.
+  } %end footnote
+\end{theorem}
+
+\begin{proof}
+By means of Theorem~\hyperlink{thm138}{138} it follows from the hypothesis that for every
+$\varepsilon$ there exists a ${V_\varepsilon}^*(a)$ within $V^*(a)$
+such that for every $x_1$ and $x_2$ on $\interval{a}{b}$, and also on
+${V_\varepsilon}^*(a)$,
+\[
+  \left| \int_{x_1}^{x_2} f(x)dx \right| < \varepsilon.
+\]
+Let $x_2$ be any point of such a neighborhood and let $x_1$ be so
+chosen that
+\[
+  x_1-a=x_2-x_1.
+\]
+Since $x_1$ and $x_2$ are on $V^*(a)$,
+\[
+  f(x_1) > f(x_2).
+\]
+It follows from Theorem~\hyperlink{thm116}{116} that
+\[
+  \left| \int_{x_1}^{x_2} f(x)dx \right| > |f(x_2)| \cdot (x_2-x_1).
+\]
+But
+\[
+  f(x_2) \cdot (x_2-x_1) = \tfrac{1}{2} f(x_2) \cdot (x_2-a).
+\]
+%-----File: 209.png---Folio 197------
+Hence for $x=x_2$,
+\[
+  |f(x)| \cdot (x-a) < 2 \varepsilon.
+\]
+Since $\varepsilon$ is arbitrary, and since $x_2$ is any point in
+$V^*(a)$, it follows that
+\[
+  \mathop{L}_{x\doteq a} f(x)\cdot(x-a)=0.\qedhere
+\]
+\end{proof}
+
+\begin{corollary}
+If
+\[
+  \int_x^b f(x)dx
+\]
+exists for every $x$ between $a$ and $b$, and
+\[
+  \mathop{L}_{x\doteq a} \int_x^b f(x)dx
+\]
+exists and is finite, and if $f(x)$ is entirely positive or entirely
+negative, then zero is a value approached by $(x-a)\cdot f(x)$ as $x$
+approaches $a$.
+\end{corollary}
+
+\begin{proof}
+Consider the case when the function is entirely positive. Suppose zero
+is not a value approached. Then there exists a pair of positive
+numbers $\varepsilon$ and $\delta$ such that for every $x$,
+$x-a<\delta$,
+\[
+  (x-a) \cdot f(x)>\varepsilon.
+\]
+On the interval, $\interval{a}{a+\delta}$, consider the function
+\[
+  \frac{\varepsilon}{x-a}.
+\]
+Since
+\[
+  \int_x^b \frac{\varepsilon}{x-a}dx
+\]
+is a non-oscillating function of $x$, it follows from Theorem~\hyperlink{thm25}{25} that
+\[
+  \mathop{L}_{x\doteq a} \int_x^b \frac{\varepsilon}{x-a}dx
+\]
+exists, and by Theorem~\hyperlink{thm142}{142} this limit must be infinite.
+%-----File: 210.png---Folio 198------
+Since
+\[
+  |f(x)|> \frac{\varepsilon}{x-a}
+\]
+on the neighborhood under consideration, it follows from Theorem~\hyperlink{thm107}{107}
+and Corollary~\hyperlink{cor2p82}{2}; Theorem~\hyperlink{thm40}{40}, that
+\[
+  \mathop{L}_{x\doteq a} \int_x^b f(x)dx
+\]
+exists and is infinite, which is contrary to the hypothesis.
+\end{proof}
+\begin{theorem}[143\footnotemark]\hypertarget{thm143}{}\footnotetext{%
+  This is what Professor \textsc{Moore} in his lectures calls the
+  relative convergence theorem. Theorems~143, 144, 151, 152 in this
+  form are due to him.}
+If
+\begin{enumerate}
+\item[\textnormal{(1)}] $f_1(x)$ and $f_2(x)$ are of the same rank of infinity at
+$x = a$, or if $f_1(x)$ is of lower order than $f_2(x)$,
+
+\item[\textnormal{(2)}] $\displaystyle\int_x^b f_1(x)dx$ and $\displaystyle\int_x^b
+f_2(x)dx$ both exist for every $x$ on the segment $\overline{a\ b}$,
+
+\item[\textnormal{(3)}] There is a neighborhood of $x = a$ on which $f_2(x)$ does
+not change sign,
+\item[\textnormal{(4)}] $\displaystyle{\mathop{L}_{x\doteq a} \int_a^b} f_2(x)dx$
+is finite,\footnote{%
+  We notice that since under the hypothesis $f_2(x)$ does not change
+  sign,
+  \[
+    L \int_x^b f_2(x)dx
+  \]
+  cannot fail to exist either finite or infinite, for it follows from
+  this hypothesis that $\displaystyle\int_x^b f_2(x)dx$ is a
+  non-oscillating function of $x$ and therefore, by Theorem~\hyperlink{thm25}{25} that
+  the limit exists.}
+\end{enumerate}
+then it follows that $\displaystyle{\mathop{L}_{x\doteq a} \int_x^b}
+f_1(x)dx$ exists and is finite.
+\end{theorem}
+%-----File: 211.png---Folio 199------
+
+\begin{proof}
+Since from the hypothesis
+\[
+  \mathop{L}_{x\doteq a} \int_x^b f_2(x)dx
+\]
+exists and is finite, we have by Theorem~\hyperlink{thm138}{138} that for every
+$\varepsilon$ there exists a $V_\varepsilon^*(a)$ such that for every
+$x_1$ and $x_2$ on segment $\overline{a\ b}$ and on
+$V_\varepsilon^*(a)$
+\[
+  \left|\int_{x_1}^{x_2} f_2(x)dx \right|< \varepsilon.
+\]
+Consider $x_1$ and $x_2$ on a neighborhood of $x = a$ for which
+$\left|\dfrac{f_1(x)}{f_2(x)} \right|< M$ and for which $f_2(x)$ does
+not change sign. Then, by Theorem~\hyperlink{thm113}{113},
+\[
+          \left|\int_{x_1}^{x_2} f_1(x)dx \right|
+< M \cdot \left|\int_{x_1}^{x_2} f_2(x)dx \right|
+< M \cdot \varepsilon.
+\]
+Since $M \cdot \varepsilon$ can be made small at will by making
+$\varepsilon$ small, it follows by Theorem~\hyperlink{thm138}{138} that
+\[
+  \mathop{L}_{x\doteq a} \int_x^b f_1(x)dx
+\]
+exists and is finite.
+\end{proof}
+
+An important special case of this theorem is when $f_1(x)$ is of the
+same or lower order of infinity than $f_2(x)$, i.e.,
+$\displaystyle{\mathop{L}_{x\doteq a}} \dfrac{f_1(x)}{f_2(x)} = K$, a
+constant not zero.
+
+The reader should verify for himself that Theorem~\hyperlink{thm140}{140} is a corollary
+of Theorem~\hyperlink{thm143}{143}. The other previous tests for the existence of the
+improper definite integral can all be reduced to special cases of
+Theorem~\hyperlink{thm143}{143}. Cf., for example, the logarithmic test on page~410 of
+\textsc{Pierpont}.
+
+\begin{theorem}[144]\hypertarget{thm144}{}
+If
+\begin{enumerate}
+\item[\textnormal{(1)}] $f_1(x)$ and $f_2(x)$ are of the same rank of infinity at
+$x = a$, or if $f_1(x)$ is of higher order than $f_2(x)$,
+\item[\textnormal{(2)}] $\displaystyle\int_x^b f_1(x)dx$ and $\displaystyle\int_x^b
+f_2(x)dx$ both exist for every $x$ on the segment $\interval{a}{b}$,
+%-----File: 212.png---Folio 200------
+\item[\textnormal{(3)}] There is a neighborhood of $x=a$ on which $f_1(x)$ does not
+change sign,
+\item[\textnormal{(4)}] $\displaystyle \mathop{L}_{x\doteq a}\int_x^bf_2(x)dx$ is
+infinite (see note under Theorem~\hyperlink{thm143}{143}),
+\end{enumerate}
+then $\displaystyle \mathop{L}_{x\doteq a}\int_x^bf_1(x)dx$ exists and is infinite or fails to exist.\footnote{%
+  This is what Professor \textsc{Moore} calls the relative divergence
+  theorem.}
+\end{theorem}
+
+\begin{proof}
+This is a direct consequence of Theorem~\hyperlink{thm143}{143}, since if
+\[
+  \mathop{L}_{x\doteq a}\int_x^bf_1(x)dx,
+\]
+which exists by the foot-note of Theorem~\hyperlink{thm143}{143}, were finite, then
+\[
+  \mathop{L}_{x\doteq a}\int_x^bf_2(x)dx
+\]
+would exist and be finite.
+\end{proof}
+
+\begin{theorem}[145]\hypertarget{thm145}{}
+If for a function $f_1(x)$ which does not change sign in the
+neighborhood of $x=a$ there exists a monotonic function $f_2(x)$
+infinite of the same rank as $f_1(x)$ as $x$ approaches $a$,
+$\displaystyle \int_x^bf_1(x)dx$ and $\displaystyle \int_x^bf_2(x)dx$
+both existing for every $x$ on the segment $\overline{a\ b}$, then a
+necessary condition that $\displaystyle \mathop{L}_{x\doteq
+a}\int_x^bf_1(x)dx$ shall exist and be finite is that
+\[
+  \mathop{L}_{x\doteq a}(x-a)\cdot f_1(x)=0.
+\]
+\end{theorem}
+
+\begin{proof}
+By hypothesis
+\[
+  \mathop{L}_{x\doteq a}\int_x^bf_1(x)dx
+\]
+%-----File: 213.png---Folio 201------
+exists and is finite. Hence, by Theorem~\hyperlink{thm143}{143},
+\[
+  \mathop{L}_{x\doteq a}\int_x^bf_2(x)dx
+\]
+exists and is finite. Therefore, by Theorem~\hyperlink{thm142}{142},
+\[
+  \mathop{L}_{x\doteq a}(x-a)\cdot f_2(x)=0.
+\]
+Since $ \left|\dfrac{f_1(x)}{f_2(x)}\right|$ is bounded as $x$
+approaches $a$, i.e., $|f_1(x)|<M\cdot|f_2(x)|$, we have
+\[
+  (x-a)\cdot|f_1(x)|<M\cdot (x-a)\cdot|f_2(x)|.
+\]
+But
+\[
+  \mathop{L}_{x\doteq a} M \cdot (x-a)\cdot|f_2(x)|=0.
+\]
+Therefore, by Corollary~\hyperlink{cor4p82}{4}, Theorem~\hyperlink{thm40}{40},
+\[
+  \mathop{L}_{x\doteq a}(x-a)\cdot|f_1(x)|=0,
+\]
+or by Corollary~\hyperlink{cor2th27}{2}, Theorem~\hyperlink{thm27}{27},
+\[
+  \mathop{L}_{x\doteq a}(x-a)\cdot f_1(x)=0.\qedhere
+\]
+\end{proof}
+
+\section{The Definite Integral on an Infinite Interval.}\hypertarget{chIXsec2}{}%[2]
+
+The integral over an infinite interval, viz.,
+\[
+  \mathop{L}_{x\doteq \infty}\int_a^xf(x)dx,
+\]
+has properties analogous to those of the improper definite integral on
+a finite interval discussed in the preceding section, and is likewise
+called an improper definite integral.
+
+The following theorems correspond to Theorems \hyperlink{thm138}{138} to \hyperlink{thm145}{145}.
+%-----File: 214.png---Folio 202------
+
+\begin{theorem}[146]\hypertarget{thm146}{}
+If
+\[
+  \int_a^xf(x)dx
+\]
+exists for every $x$, $a<x$, then a necessary and sufficient condition
+that
+\[
+  \mathop{L}_{x\doteq \infty}\int_a^xf(x)dx
+\]
+exists and is finite, is that for every $\varepsilon$ there exists a
+$D_{\varepsilon}$, such that for every two values of $x$, $x_1$ and
+$x_2$, each greater than $D_{\varepsilon}$,
+\[
+  \left|\int_{x_1}^{x_2}f(x)dx\right|<\varepsilon\text{\correction{.}{}}
+\]
+\end{theorem}
+
+\begin{proof}
+The theorem is a direct consequence of Theorems \hyperlink{thm105}{105} and \hyperlink{thm27}{27}.
+\end{proof}
+
+\begin{theorem}[147]\hypertarget{thm147}{}
+If
+\[
+  \int_a^xf(x)dx
+\]
+exists for every $x$ greater than $a$, and if
+\[
+   \mathop{L}_{x\doteq \infty}\int_a^x|f(x)|dx
+\]
+is finite,\footnote{%
+  Note on page~\pageref{egp192} shows that this hypothesis is not redundant.}
+then
+\[
+  \mathop{L}_{x\doteq \infty}\int_a^xf(x)dx
+\]
+exists and is finite.
+\end{theorem}
+
+\begin{proof}
+The proof is like that of Theorem~\hyperlink{thm139}{139}.
+\end{proof}
+
+\begin{theorem}[148]\hypertarget{thm148}{}
+If
+\[
+  \int_a^xf(x)dx
+\]
+exists for every $x$ greater than $a$, and if $(x-a)^k\cdot f(x)$ is
+bounded as $x$ approaches infinity for some $k$, $k>1$, then
+\[
+   \mathop{L}_{x\doteq \infty}\int_a^xf(x)dx
+\]
+exists and is finite.
+\end{theorem}
+%-----File: 215.png---Folio 203------
+
+\begin{proof}
+If in the proof of Theorem~\hyperlink{thm140}{140} we write $D_{\varepsilon}^{1-k}=
+\dfrac{\varepsilon (1-k)}{M}$ instead of $\delta_{\varepsilon}^{1-k}=\dfrac{\varepsilon (1-k)}{M}$, and use
+Theorem~\hyperlink{thm146}{146} instead of 138, the proof of Theorem~\hyperlink{thm140}{140} will apply to
+Theorem~\hyperlink{thm148}{148}.
+\end{proof}
+
+\begin{theorem}[149]\hypertarget{thm149}{}
+If $f(x)$ does not change sign for $x$ greater than
+some fixed number $D$, and if for some positive number $m$ and
+some number $ k\leqq 1$\correction{,}{} $\left|(x-a)^k\cdot f(x)\right|>m$ for every $x$ greater than $D$,
+then
+\[
+  \mathop{L}_{x\doteq \infty}\int_a^xf(x)dx
+\]
+cannot exist and be finite.
+\end{theorem}
+
+\begin{proof}
+By making suitable changes in the proof of Theorem~\hyperlink{thm141}{141} so as to make
+$x_1$ and $x_2$ approach infinity instead of $a$, that proof applies
+to this theorem.
+\end{proof}
+
+\begin{theorem}[150]\hypertarget{thm150}{}
+If
+\[
+  \mathop{L}_{x\doteq \infty}\int_a^xf(x)dx
+\]
+exists and is finite, and if $f(x)$ is monotonic for all values of $x$
+greater than some fixed number, then
+\[
+  \mathop{L}_{x\doteq \infty}(x-a)\cdot f(x)=0.
+\]
+\end{theorem}
+
+\begin{proof}
+By making slight modifications of the proof of Theorem~\hyperlink{thm142}{142}, that proof
+applies to this theorem.
+\end{proof}
+
+\begin{corollary}
+If
+\[
+  \int_a^xf(x)dx
+\]
+exists for every $x$ greater than $a$, and
+\[
+  \mathop{L}_{x\doteq \infty}\int_a^xf(x)dx
+\]
+exists and is finite, and if $f(x)$ does not change sign for $x$
+greater
+%-----File: 216.png---Folio 204------
+than some fixed number, then zero is a value approached by $(x-a)f(x)$
+as $x$ approaches $\infty$.
+\end{corollary}
+The proof is similar to that of the corollary of Theorem~\hyperlink{thm142}{142}.
+
+
+\begin{theorem}[151]\hypertarget{thm151}{}
+If
+\begin{enumerate}
+\item[\textnormal{(1)}] $f_1(x)$ and $f_2(x)$ are infinitesimals of the same rank
+as $x$ approaches $\infty$, or if $f_1(x)$ is of higher order than
+$f_2(x)$,
+
+\item[\textnormal{(2)}] $\displaystyle \int_a^xf_1(x)dx$ and $\displaystyle
+\int_a^xf_2(x)dx$ both exist for every $x$, $a<x$,
+
+\item[\textnormal{(3)}] $f_2(x)$ does not change sign for $x$ greater than some
+fixed number,
+
+\item[\textnormal{(4)}] $\displaystyle \mathop{L}_{x\doteq \infty}\int_a^xf_2(x)dx$
+is finite,
+\end{enumerate}
+then it follows that
+\[
+  \mathop{L}_{x\doteq \infty}\int_a^xf_1(x)dx
+\]
+exists and is finite.\footnote{%
+  See note under Theorem~\hyperlink{thm143}{143}.}
+\end{theorem}
+
+\begin{proof}
+The proof is analogous to that of Theorem~\hyperlink{thm143}{143}.
+\end{proof}
+
+\begin{theorem}[152]\hypertarget{thm152}{}
+If
+\begin{enumerate}
+\item[\textnormal{(1)}] $f_1(x)$ and $f_2(x)$ are infinitesimals of the same rank
+as $x$ approaches infinity, or if $f_1(x)$ is of lower order than
+$f_2(x)$,
+
+\item[\textnormal{(2)}] $\displaystyle \int_a^xf_1(x)dx$ and $\displaystyle
+\int_a^xf_2(x)dx$ both exist for every $x$, $a<x$,
+
+\item[\textnormal{(3)}] $f_1(x)$ does not change sign for $x$ greater than some
+fixed number,
+
+\item[\textnormal{(4)}] $\displaystyle \mathop{L}_{x\doteq \infty}\int_a^xf_2(x)dx$
+is infinite,
+\end{enumerate}
+then
+\[
+  \mathop{L}_{x\doteq \infty}\int_a^xf_1(x)dx
+\]
+exists and is infinite or fails to exist.
+\end{theorem}
+
+Proof like that of Theorem~\hyperlink{thm144}{144}.
+
+\begin{theorem}[153]\hypertarget{thm153}{}
+If for a function $f_1(x)$ which does not change sign in the
+neighborhood of $x=\infty$ there exists a monotonic function $f_2(x)$
+such that $f_1(x)$ and $f_2(x)$ are infinitesimals of the same
+%-----File: 217.png---Folio 205------
+rank as $x$ approaches infinity, $\displaystyle \int_a^xf_1(x)dx$ and
+$\displaystyle \int_a^xf_2(x)dx$ both existing for every $x>a$, then a
+necessary condition that
+\[
+  \mathop{L}_{x\doteq \infty}\int_a^xf_1(x)dx
+\]
+shall exist and be finite is that
+\[
+  \mathop{L}_{x\doteq \infty}(x-a)\cdot f_1(x)=0.
+\]
+\end{theorem}
+
+The proof is like that of Theorem~\hyperlink{thm145}{145}.
+
+\section{Properties of the Simple Improper Definite Integral.}\hypertarget{chIXsec3}{}%[3]
+\index{Improper definite integral!simple}\index{Simple improper definite integral}
+The following definition of the simple improper definite integral is
+equivalent in substance to that given on page~\pageref{dp192}, and in
+form is partly the definition of the general improper definite
+integral given on page~\pageref{s3p210}.
+
+The \label{dp205}definite integral of a function is said to \index{Proper existence of the definite integral at a point}\index{Integral!existing properly at a point}\textit{exist properly
+at a point} $x_1$ or in the neighborhood of this point, on the
+interval $\interval{a}{b}$ if there exists an interval on
+$\interval{a_1}{b_1}$ containing $x_1$ as an interior point (or as an
+end point in case $x_1=a$ or $x_1=b$) such that the proper definite
+integral of $f(x)$ exists on this interval. The integral is said to
+\index{Improper existence of the definite integral}exist improperly at a point $x_1$ on the interval $\interval{a}{b}$ if
+$f(x)$ has an infinite singularity at $x_1$ and there exists an
+interval $\interval{a_1}{b_1}$ on $\interval{a}{b}$ containing $x_1$
+as an interior point (or end point in case $x_1=a$ or $x_1=b$) such
+that the improper definite integral exists on each of the intervals
+$\interval{a_1}{x_1}$ and $\interval{x_1}{b_1}$.
+
+If on an interval $\interval{a}{b}$ the definite integral exists
+properly at every point except a finite number of points, and exists
+improperly at each of these points, then the improper definite
+integral is said to exist simply on the interval $\interval{a}{b}$, or
+the simple improper definite integral is said to exist on
+%-----File: 218.png---Folio 206------
+the interval $\interval{a}{b}$. Let $x_1$, $x_2, \ldots$, $x_n$ be the
+points of $\interval{a}{b}$ at which the integral exists
+improperly. The \emph{simple improper definite integral} on
+$\interval{a}{b}$ is the sum of the improper definite integrals on the
+intervals $\interval{a}{x_1}$, $\interval{x_1}{x_2}, \ldots$,
+$\interval{x_{n-1}}{x_n}$, $\interval{x_n}{b}$.
+
+We denote the simple improper definite integral of $f(x)$ on
+the interval $\interval{a}{b}$ by
+\[
+  \sideset{_S}{_a^b}\int f(x)dx.
+\]
+This symbol is used generically to include the proper as well as the
+improper definite integral.
+
+\begin{theorem}[154]\hypertarget{thm154}{}
+If $a<b<c$, and if two of the three simple improper definite integrals
+\[
+  \sideset{_S}{_a^b}\int f(x)dx, \quad
+  \sideset{_S}{_b^c}\int f(x)dx, \quad \text{and}\quad
+  \sideset{_S}{_a^c}\int f(x)dx
+\]
+exist, then the third exists and
+\[
+  \sideset{_S}{_a^b}\int f(x)dx
++ \sideset{_S}{_b^c}\int f(x)dx
+= \sideset{_S}{_a^c}\int f(x)dx.
+\]
+\end{theorem}
+
+\begin{proof}
+If $b$ is a point at which the integral exists improperly, and if
+\[
+  \sideset{_S}{_a^b}\int f(x)dx \quad \text{and} \quad
+  \sideset{_S}{_b^c}\int f(x)dx
+\]
+both exist, then by the definition of
+\[
+  \sideset{_S}{_a^c}\int f(x)dx
+\]
+the latter exists and is equal to the sum of the two former.
+
+If one of the two integrals, say
+\[
+  \sideset{_S}{_a^b}\int f(x)dx,
+\]
+%-----File: 219.png---Folio 207------
+exists, and if
+\[
+  \sideset{_S}{_a^c}\int f(x) dx
+\]
+exists, then
+\[
+  \sideset{_S}{_b^c}\int f(x) dx
+\]
+exists since only in that case does
+\[
+  \sideset{_S}{_a^c}\int f(x) dx
+\]
+exist. The equation
+\[
+  \sideset{_S}{_a^b}\int f(x) dx
++ \sideset{_S}{_b^c}\int f(x) dx
+= \sideset{_S}{_a^c}\int f(x) dx
+\]
+likewise holds.
+
+If $b$ is a point at which the integral exists \textit{properly}, then
+the theorem follows from the above argument and the definition on
+page~\pageref{dp205}.
+\end{proof}
+\begin{theorem}[155]\hypertarget{thm155}{}
+If
+\[
+  \sideset{_S}{_a^b}\int f(x) dx
+\]
+exists, then
+\[
+  \sideset{_S}{_b^a}\int f(x) dx
+\]
+exists and
+\[
+  \sideset{_S}{_a^b}\int f(x) dx
+=-\sideset{_S}{_b^a}\int f(x) dx.
+\]
+\end{theorem}
+
+\begin{proof}
+In case the integral exists improperly only at one point of the
+interval, then the theorem is an immediate consequence of Theorem~\hyperlink{thm108}{108}
+and Corollary~\hyperlink{cor1th27}{1}, Theorem~\hyperlink{thm27}{27}. (If $\displaystyle\mathop{L}_{x \doteq a}
+f(x)=K$, then $\displaystyle\mathop{L}_{x \doteq a} \{-f(x)\} =-K$.)
+The theorem in the general case follows directly from this case and
+the definition of the simple improper definite integral.
+\end{proof}
+
+\begin{theorem}[156]\hypertarget{thm156}{}
+If $c$ is a constant and if the simple improper
+%-----File: 220.png---Folio 208------
+definite integral of $f(x)$ exists on $\interval{a}{b}$, then the
+simple improper definite integral of $\ c\cdot f(x)$ exists on
+$\interval{a}{b}$ and
+\[
+c \sideset{_S}{_a^b}\int f(x) dx
+= \sideset{_S}{_a^b}\int cf(x) dx.
+\]
+\end{theorem}
+
+\begin{proof}
+The theorem is a direct consequence of Theorems \hyperlink{thm111}{111} and \hyperlink{thm34}{34}.
+\end{proof}
+
+\begin{theorem}[157]\hypertarget{thm157}{}
+If the simple improper definite integrals of $f_1(x)$ and $f_2(x)$
+both exist on $\interval{a}{b}$, then the simple improper definite
+integral of $f_1(x) + f_2(x)$ and of $f_1(x)-f_2(x)$ both exist and
+\[
+   \sideset{_S}{_a^b}\int \{f_1(x) \pm f_2(x)\}dx
+= \sideset{_S}{_a^b}\int f_1(x) dx
+\pm\sideset{_S}{_a^b}\int f_2(x) dx.
+\]
+\end{theorem}
+
+\begin{proof}
+The theorem is a direct consequence of Theorems \hyperlink{thm112}{112} and \hyperlink{thm34}{34}.
+\end{proof}
+
+
+\begin{theorem}[158]\hypertarget{thm158}{}
+If the simple improper definite integrals of $f_1(x)$ and $f_2(x)$
+both exist, and if $f_1(x) \geqq f_2(x)$, then
+\[
+      \sideset{_S}{_a^b}\int f_1(x) dx
+\geqq \sideset{_S}{_a^b}\int f_2(x) dx.
+\]
+\end{theorem}
+
+\begin{proof}
+The theorem is a direct consequence of Theorem~\hyperlink{thm113}{113} and Corollary~\hyperlink{cor2p82}{2},
+Theorem~\hyperlink{thm40}{40}.
+\end{proof}
+
+\begin{theorem}[159]\hypertarget{thm159}{}
+If
+\[
+  \sideset{_S}{_a^b}\int f(x) dx
+\]
+exists, then
+\[
+  \sideset{_S}{_a^x}\int f(x) dx
+\]
+is a continuous function of the limit of integration on the interval
+$\interval{a}{b}$.
+\end{theorem}
+
+\begin{proof}
+If $x$ is a point at which the integral exists properly, the theorem
+is the same as 118. If $x$ is a point at which
+%-----File: 221.png---Folio 209------
+the integral exists improperly, then the theorem follows from Theorems
+\hyperlink{thm138}{138} and \hyperlink{thm27}{27}.
+\end{proof}
+
+\begin{theorem}[160]\hypertarget{thm160}{}\label{t160p209}
+If
+\[
+  \sideset{_S}{_a^b}\int f(x) dx
+\]
+exists, it does not follow that
+\[
+  \sideset{_S}{_a^b}\int|f(x)|dx
+\]
+exists.
+\end{theorem}
+
+\begin{proof}
+Let
+\[
+  x_1,\ x_2,\ x_3, \ldots,\ x_n, \ldots
+\]
+be an infinite sequence of points on $\interval{0}{1}$ in the order
+indicated from 1 towards 0 such that
+\[
+  \int_{x_n}^{x_{n-1}} \frac{dx}{x} = \frac1n.
+\]
+Consider a function $f(x)$ defined as follows:
+\begin{align*}
+  &f(x) = \frac1x \quad\textrm{on}\quad \overline{x_1\ 1},\
+  \overline{x_3\ x_2\vphantom{1}}, \text{ etc.}\\
+  &f(x) =-\frac1x \quad\textrm{on}\quad \interval{x_2}{x_1},\
+  \interval{x_4}{x_3}, \text{ etc.}
+\end{align*}
+Obviously
+\[
+  \mathop{L}_{x\doteq 0} \int_x^1 f(x) dx\footnotemark
+\]
+\footnotetext{%
+  That 0 is a limit point of the sequence of points is obvious since
+  in case this sequence has a limit point greater than zero the proper
+  definite integral of the function $\dfrac{1}{x}$ would fail to exist
+  on some interval $\interval{a}{b}$ where $0<a<b$, which is
+  impossible.}
+exists and is finite since the series $\frac12-\frac13 + \frac14
+\ldots$ is convergent, while
+\[
+  \mathop{L}_{x\doteq 0} \int_x^1 |f(x)|dx
+\]
+is divergent since the harmonic series is divergent.
+\end{proof}
+%-----File: 222.png---Folio 210------
+
+\section{A More General Improper Integral.}\hypertarget{chIXsec4}{}%[4]
+\label{s3p210}
+The problem of defining and studying the properties of the improper
+integral when the set of points of singularity is infinite has been
+treated by many writers.\footnote{%
+  \textsc{A.~Cauchy} and \textsc{B.~Riemann} studied the case of a
+  finite number of singularities in papers which are to be found in
+  these writers' collected works.  The infinite case has been treated
+  by
+
+  \textsc{A.~Harnack}, \textit{Mathematische Annalen}, Vols.~21 and 24
+  (1883--84).
+
+  \textsc{O.~H\"older}, \textit{Mathematische Annalen}, Vol.~24
+  (1884).
+
+  \textsc{C.~Jordan}, \textit{Cours d'Analyse}, Vol.~2 (1894, 2d ed.).
+
+  \textsc{O.~Stolz}, \textit{Grundz\"uge der Differential- und
+  Integralrechnung}, Vol.~3.
+
+  \textsc{A.~Schoenflies}, \textit{Jahresbericht der Deutschen
+  Mathematiker-Vereinigung}, Vol.~8 (1900).
+
+  \textsc{Vall\'ee-Poussin}, \textit{Liouville's Journal}, Ser.~4,
+  Vol.~8 (1892).
+
+  \textsc{E.~H.\ Moore}, \textit{Transactions of the American
+    Mathematical Society}, Vol.~2 (1901).
+
+  \textsc{J.~Pierpont}, \textit{Theory of Functions of Real Variables}
+  (1906).  } %endfootnote
+In this section we give a
+few properties of improper integrals as defined by \textsc{Harnack}
+and \textsc{Moore}.
+
+Denote by $P_0$ any set of points of content zero on $\interval{a}{b}$, and by $P$ the set of all points of $\interval{a}{b}$ not points
+of $P_0$. $P$ and $P_0$ are complementary \correction{subsets}{sub-sets} of $\interval{a}{b}$. Denote by $I$ any finite set of non-overlapping intervals of
+$\interval{a}{b}$ which contain no point of the set $P_0$. The symbol
+$m(I)$ stands for the sum of the lengths of the intervals of $I$. For
+the sake of brevity $D$ will be used for $|a-b|$.
+
+The following conditions are assumed to be satisfied:
+\begin{enumerate}
+\item[(\textit{a})] The definite integral of $f(x)$ exists properly at
+every point of $P$. The sum of the integrals of $f(x)$ on the
+intervals of $I$ is denoted by
+\[
+  \int_{a\; I}^b f(x)dx.
+\]
+\item[(\textit{b})] For every positive $\varepsilon$ there exists a
+positive $\delta_\varepsilon$ such that for any two sets, $I'$ and
+$I''$, of intervals none of which contain any point of $P_0$ and for
+which
+\[
+  |D-m(I' )|<\delta_\varepsilon \quad\text{and}\quad
+  |D-m(I'')|<\delta_\varepsilon,
+\]
+%-----File: 223.png---Folio 211------
+\[
+  \left| \int_{a\;I'}^b f(x)dx-\int_{a\;\text{\correction{$I''$}{$I'$}}}^b f(x)dx \right| <
+  \varepsilon.
+\]
+\end{enumerate}
+
+It follows by Theorem~\hyperlink{thm27}{27} that
+\[
+  \mathop{L}_{m(I)\doteq D} \int_{a\;I}^b f(x)dx
+\]
+exists and is finite. This limit is denoted by
+\[
+  \sideset{_b}{_{a\;P_0}^b}\int f(x)dx
+\]
+and is called the \index{Broad improper definite integral}\textit{broad improper definite integral}\index{Improper definite integral!broad} with
+respect to $P_0$ of the function $f(x)$ on the interval $\interval{a}{b}$.
+
+It is to be noticed that all the points of $P_0$ need not be on
+$\interval{a}{b}$; those which are not on $\interval{a}{b}$ do not
+affect the existence of
+\[
+  \sideset{_b}{_{a\;P_0}^b}\int f(x)dx.
+\]
+Therefore if $f(x)$ is improperly integrable on some sub-interval
+$\interval{a'}{b'}$ of $\interval{a}{b}$, its integral may be denoted
+by
+\[
+  \sideset{_b}{_{a'\;P_0}^{b'}} \int f(x)dx.
+\]
+
+\begin{theorem}[161]\hypertarget{thm161}{}
+If $a<b<c$ and if of the integrals
+\[
+  \sideset{_b}{_{a\;P_0}^b} \int f(x)dx, \
+  \sideset{_b}{_{b\;P_0}^c} \int f(x)dx, \
+  \sideset{_b}{_{a\;P_0}^c} \int f(x)dx,
+\]
+either
+\begin{enumerate}
+\item[(a)] $\quad\displaystyle \sideset{_b}{_{a\;P_0}^b}\int
+f(x)dx$ \quad and $\quad\displaystyle \sideset{_b}{_{b\;P_0}^c}\int
+f(x)dx$ exist, or
+\item[(b)] $\quad\displaystyle \sideset{_b}{_{a\;P_0}^c}\int f(x)dx$
+exists,
+\end{enumerate}
+%-----File: 224.png---Folio 212------
+then all three integrals exist and
+\hypertarget{eq1p212}{\[
+  \text{\correction{$\sideset{_b}{_{a\;P_0}^b}\int$}{$\sideset{}{_{a\;P_0}^b}\int$}} f(x) dx
++ \text{\correction{$\sideset{_b}{_{b\;P_0}^c}\int$}{$\sideset{}{_{b\;P_0}^c}\int$}} f(x) dx
+= \sideset{_b}{_{a\; P_0}^c}\int f(x) dx.
+\tag{1}
+\]}
+\end{theorem}
+
+\begin{proof}
+Every set $I$ of intervals on $\interval{a}{c}$ may be regarded as
+composed of a set $\overline{I}$ on $\interval{a}{b}$ and a set
+$\overline{\overline{I}}$ on $\interval{b}{c}$, while, conversely,
+every pair of sets $\overline{I}$ and $\overline{\overline{I}}$
+constitute a set $I$. Hence
+\[
+  \int_{a\; I}^c f(x) dx
+= \int_{a\; \overline{I}}^b f(x) dx
++ \int_{b\; \overline{\overline{I}}}^c f(x) dx.
+\]
+(Note that both members of this equation are multiple-valued functions
+of $m(I)$ and of $m(\overline{I})$ and
+$m(\overline{\overline{I}})$). The conclusion of our theorem follows
+in case ($a$) from Theorem~\hyperlink{thm34}{34}.
+
+It remains to show that if $\displaystyle\sideset{_b}{_{a_{\;
+P_0}}^c}\int f(x) dx$ exists, then $\displaystyle\sideset{_b}{_{a_{\;
+P_0}}^b}\int f(x) dx$ and $\displaystyle\sideset{_b}{_{b_{\;
+P_0}}^c}\int f(x) dx$ exist, and in that case also equation~\hyperlink{eq1p212}{(1)} holds.
+Suppose that on some sequence of sets $[I]$ one of the two expressions
+$\displaystyle\int_{a\; \overline{I}}^b f(x) dx$ and
+$\displaystyle\int_{b\; \overline{\overline{I}}}^c f(x) dx$, say
+$\displaystyle\int_{a\; \overline{I}}^b f(x) dx$, approaches two
+distinct values as $m(I)$ approaches $D$. Since there is some sequence
+of sets of intervals $\left\{\overline{\overline{I'}}\right\}$ on
+which $\displaystyle\int_{b\; \overline{\overline{I}}}^c f(x)$
+approaches only one value, it follows that on the sequence of sets of
+intervals obtained by associating with each $\overline{I}$ an
+$\overline{\overline{I'}}$ and with each $\overline{\overline{I'}}$ an
+$I'$, $\displaystyle\int_{a\; I}^c f(x) dx$ approaches two distinct
+values as $m(I)\doteq D$, which is contrary to hypothesis.
+
+If $\displaystyle\int_{a\; \overline{I}}^b f(x) dx$ approaches
+infinity, then clearly $\displaystyle\int_{b\;
+\overline{\overline{I}}}^c f(x) dx$ must approach infinity of the
+opposite sign. Hence, by the corollary of Theorem~\hyperlink{thm51}{51}a,
+$\displaystyle\int_{a\; I}^c f(x) dx$ will approach both $+\infty$
+%-----File: 225.png---Folio 213------
+and $-\infty$ as $m(I) \doteq D$, which again contradicts the
+hypothesis that $\sideset{_b}{_{a\; P_0}^c}\int f(x)dx$ exists. The
+equality
+\[
+  \sideset{_b}{_{a\; P_0}^c}\int f(x)dx
+= \sideset{_b}{_{a\; P_0}^b}\int f(x)dx
++ \sideset{_b}{_{b\; P_0}^c}\int f(x)dx
+\]
+now follows from the identity of the limitands
+\[
+  \int_{a\; I }^c f(x)dx \text{ and }
+  \int_{a\; \overline{I} }^b f(x)dx
++ \int_{b\; \overline{\overline{I}}}^c f(x)dx.\qedhere
+\]
+\end{proof}
+\begin{theorem}[162]\hypertarget{thm162}{}
+If
+$\sideset{_b}{_{a\; P_0}^b}\int f(x)dx$ exists, then
+$\sideset{_b}{_{b\; P_0}^a}\int f(x)dx$ exists
+and
+\[
+  \sideset{_b}{_{a\; P_0}^b}\int f(x)dx
+=-\sideset{_b}{_{b\; P_0}^a}\int f(x)dx.
+\]
+\end{theorem}
+
+\begin{proof}
+By Theorem~\hyperlink{thm108}{108}, for every $I$
+\begin{align*}
+  \int_{a\; I}^b f(x)dx
+  &=-\int_{b\; I}^a f(x)dx,
+\\
+\intertext{whence}
+   \sideset{_b}{_{a\; P_0}^b}\int f(x)dx
+  &=-\sideset{_b}{_{b\; P_0}^a}\int f(x)dx.\qedhere
+\end{align*}
+\end{proof}
+
+\begin{theorem}[163]\hypertarget{thm163}{}
+If
+$\sideset{_b}{_{a\; P_0}^b}\int f(x)dx$ exists, then
+$\sideset{_b}{_{a\; P_0}^b}\int c\cdot f(x)dx$ exists
+and
+\[
+         \sideset{_b}{_{a\; P_0}^b}\int c\cdot f(x)dx
+= c\cdot \sideset{_b}{_{a\; P_0}^b}\int f(x)dx.
+\]
+\end{theorem}
+
+\begin{proof}
+This is a direct consequence of Theorems~111 and 34.
+\end{proof}
+
+\begin{theorem}[164]\hypertarget{thm164}{}
+If
+$\sideset{_b}{_{a\; P_0}^b}\int f_1(x)dx$ and
+$\sideset{_b}{_{a\; P_0}^b}\int f_2(x)dx$ both exist,\\
+then
+$\sideset{_b}{_{a\; P_0}^b}\int (f_1(x) \pm f_2(x)) dx$ exists and
+%-----File: 226.png---Folio 214------
+\[
+  \sideset{_b}{_{a\; P_0}^b}\int f_1(x)dx \pm
+  \sideset{_b}{_{a\; P_0}^b}\int f_2(x)dx
+= \sideset{_b}{_{a\; P_0}^b}\int (f_1(x)dx \pm f_2(x)) dx.
+\]
+\end{theorem}
+\begin{proof}
+This is a direct consequence of Theorems~112 and 34.
+\end{proof}
+
+\begin{theorem}[165]\hypertarget{thm165}{}
+If $f_1(x) \geqq f_2(x)$, then
+\[
+      \sideset{_b}{_{a\; P_0}^b}\int f_1(x)dx
+\geqq \sideset{_b}{_{a\; P_0}^b}\int f_2(x)dx,
+\]
+provided these integrals exist.
+\end{theorem}
+
+\begin{proof}
+By Theorems~113 and 40.
+\end{proof}
+
+\begin{theorem}[166]\hypertarget{thm166}{}
+If
+$\sideset{_b}{_{a\; P_0}^b}\int f_1(x)dx$ and
+$\sideset{_b}{_{a\; P_0}^b}\int f_2(x)dx$ both exist,
+\[
+  \sideset{_b}{_{a\; P_0}^b}\int f_1(x) \cdot f_2(x)dx
+\]
+does not in general exist.
+\end{theorem}
+
+\begin{proof}
+Let $f_1(x) = f_2(x) = \dfrac{1}{\sqrt{x}}$. In this case the
+hypothesis of the theorem is verified but the product, $\dfrac{1}{x}$,
+fails to be integrable on the interval $\interval{0}{1}$.
+\end{proof}
+
+\begin{theorem}[167]\hypertarget{thm167}{}
+$\sideset{_b}{_{a\; P_0}^x}\int f(x)dx$ is a continuous function of
+$x$.
+\end{theorem}
+
+\begin{proof}
+If $x$ is a point at which the integral exists properly, the
+continuity follows by Theorem~\hyperlink{thm118}{118}. If $x$ is a point of the set $P_0$,
+then, by Theorem~\hyperlink{thm26}{26}, we need to show that for every $\varepsilon$
+there is a $\delta_\varepsilon$, such that for every interval
+$\interval{a'}{b'}$ containing $x_1$ and of length less than
+$\delta_\varepsilon$, $\left|\sideset{_b}{_{a'\;P_0}^{b'}}\int f(x)dx
+\right|< \varepsilon$. By definition there exists a
+$\delta_\varepsilon$ such that for every $I'$ and $I''$ for which
+$|m(I')-D|< \delta_\varepsilon$ and $|m(I'')-D|< \delta_\varepsilon$,
+\[
+  \left|\int_{a\; I' }^b f(x)dx
+           -\int_{a\; \text{\correction{$I''$}{$I'$}}}^b f(x)dx \right|< \varepsilon.
+\]
+%-----File: 227.png---Folio 215------
+Let $\interval{a'}{b'}$ be an interval containing $x_1$ such that
+\[
+  |a'-b'|< \frac{\delta_\varepsilon}{2}.
+\]
+Let $\overline{I'}$ be any set of intervals not containing any point
+of $P_0$ and containing no point of $\interval{a'}{b'}$, and such that
+$|m(\overline{I'})-D|< \delta_\varepsilon$. Denote by $I_{(a'b')}$ any
+set of non-overlapping intervals on $\interval{a'}{b'}$ containing no
+point of $P_0$, and let $\overline{I''}$ be the set of all intervals
+in $\overline{I'}$ and $I_{(a'b')}$. Then
+\[
+  |m(\overline{I''})-D|< \delta_\varepsilon
+\]
+and
+\[
+  \int_{a\overline{I''}}^b f(x)dx
+= \int_{a\overline{I'}}^b f(x)dx
++ \int_{a'\text{\correction{$I$}{$\overline{I}$}}(a'b')}^{b'} f(x)dx
+\]
+and
+\[
+  \left|\int_{a'\text{\correction{$I$}{$\overline{I}$}}(a'b')}^{b'} f(x)dx \right|
+= \left|\int_{a\overline{I''}}^b f(x)dx \right|
+-\left|\int_{a\overline{I'}}^b f(x)dx \right|.
+\]
+Hence
+\[
+  \left|\sideset{_b}{_{a'\; P_0}^{b'}}\int f(x)dx \right|
+\leqq \varepsilon.\qedhere
+\]
+\end{proof}
+
+\begin{corollary}
+For $x_1$ any point on $\interval{a}{b}$
+\[
+  \mathop{L}_{x\doteq x_1} \sideset{_b}{_{x_1}^x}\int f(x)dx = 0.
+\]
+\end{corollary}
+
+\begin{theorem}[168]\hypertarget{thm168}{}
+If $f(x)$ is integrable with respect to $P_0$, and if $P_1$ is a set
+of points of content zero, then $f(x)$ is integrable with respect to
+the set $P_2$ consisting of all points in $P_0$ and in $P_1$ and
+\[
+  \sideset{_b}{_{a\; P_0}^b}\int f(x)dx
+= \sideset{_b}{_{a\; P_2}^b}\int f(x)dx.
+\]
+\end{theorem}
+
+\begin{proof}
+Obviously the set $P_2$ is of content zero. Any set of intervals $I$
+not containing a point of $P_2$ is also a set $\overline{I}$ not
+%-----File: 228.png---Folio 216------
+containing a point of $P_0$. Hence any value approached by
+$\displaystyle\int_{a\; \overline{I}}^b f(x) dx$ as $m(\overline{I})$
+approaches $D$ is a value approached by $\displaystyle\int_{a\; I}^b
+f(x) dx$ as $m(I)$ approaches $D$. Hence
+$\displaystyle\sideset{_b}{_{a\; P_2}^b}\int f(x)dx$ exists and
+\[
+  \sideset{_b}{_{a\; P_0}^b}\int f(x) dx
+= \sideset{_b}{_{a\; P_2}^b}\int f(x) dx.\qedhere
+\]
+\end{proof}
+
+\begin{theorem}[169]\hypertarget{thm169}{}
+If $f_1(x)$ is integrable with respect to $P_1$ and $f_2(x)$ is
+integrable with respect to $P_2$, then $f_1(x) \pm f_2(x)$ is
+integrable with respect to the set, $P_3$, of all points in $P_1$ and
+$P_2$ and
+\[
+  \sideset{_b}{_{a\; P_1}^b}\int f_\text{\correction{$1$}{}}(x) dx \pm
+  \sideset{_b}{_{a\; P_2}^b}\int f_\text{\correction{$2$}{}}(x) dx =
+  \sideset{_b}{_{a\; P_3}^b}\int (f_1(x) \pm f_2(x)) dx.
+\]
+\end{theorem}
+
+\begin{proof}
+By Theorem~\hyperlink{thm168}{168} each of the functions $f_1(x)$ and $f_2(x)$ is
+integrable with respect to $P_3$, and
+\[
+  \sideset{_b}{_{a\; P_1}^b}\int f_1(x) dx =
+  \sideset{_b}{_{a\; P_3}^b}\int f_1(x) dx,
+\]
+and
+\[
+  \sideset{_b}{_{a\; P_2}^b}\int f_2(x) dx =
+  \sideset{_b}{_{a\; P_3}^b}\int f_2(x) dx,
+\]
+and hence, by Theorem~\hyperlink{thm164}{164}, $f_1(x) \pm f_2(x)$ is integrable with
+respect to $P_3$ and
+\[
+  \sideset{_b}{_{a\; P_1}^b}\int f_1(x) dx \text{\correction{$\pm$}{$+$}}
+  \sideset{_b}{_{a\; P_2}^b}\int f_\text{\correction{$2$}{}}(x) dx =
+  \sideset{_b}{_{a\; P_3}^b}\int (f_\text{\correction{$1$}{}}(x) \pm f_\text{\correction{$2$}{}}(x)) dx.\qedhere
+\]
+\end{proof}
+
+The broad improper definite integral as here defined contains as a
+special case the proper definite integral, the integral in that case
+existing properly at every point of the interval $\interval{a}{b}$.
+It does not, however, contain as a special case the simple improper
+definite integral considered in \hyperlink{chIXsec3}{\S~3}. This may readily be shown by
+means of the function used on page~\pageref{t160p209} to show
+%-----File: 229.png---Folio 217------
+that the simple improper definite integral is not absolutely
+convergent. In the case of this function a sequence of sets of
+intervals $I_a$ may be so chosen that $\displaystyle
+\int_{a\;I_a}^bf(x)dx$ shall approach any value whatever as $m(I_a)$
+approaches $D$.
+
+An improper integral which includes both the simple and the broad
+improper integrals is obtained as follows: Every set $I$ is to be such
+that if $I'$ is its complementary set of segments on $\overline{a\
+b}$, then every segment of $I'$ contains at least one point of
+$P_0$. The limit of $\displaystyle \int_{a\;I}^bf(x)dx$ as $m(I)$
+approaches $D$, if existent, is called the \index{Narrow improper definite integral}\index{Improper definite integral!narrow}narrow improper definite
+integral and is denoted by $\displaystyle
+\sideset{_n}{_{a\;P_0}^b}\int f(x)dx$.
+
+It is evident that if the broad integral exists, then the narrow
+integral also exists. The narrow integral includes the simple improper
+definite integral of the preceding chapter. Hence it follows that the
+broad and the narrow integrals are not equivalent.\footnote{%
+  The narrow integral is so called because it has fewer properties
+  than the broad integral. It exists for a wider class of functions.}
+Theorems \hyperlink{thm161}{161} to \hyperlink{thm167}{167} hold of the narrow integral as well as of the
+broad integral. The proofs are identical with the above except that
+the sets $I$ are limited as in the definition of the narrow
+integral. It may be shown by examples that Theorems \hyperlink{thm168}{168} and \hyperlink{thm169}{169} do not
+hold in the case of the narrow integral.  To show that \hyperlink{thm168}{168} does not
+hold consider the function defined in the proof of Theorem~\hyperlink{thm160}{160}, where
+$P_0$ consists of the point 0.  Let $P_1$ be the $[x_i]$ of that
+example. Then obviously the narrow integral $\displaystyle
+\sideset{_n}{_{_0\;P_2}^1}\int f(x)dx$, where $P_2$ contains all the
+points of $P_1$ and $P_2$, fails to exist. The same example shows that
+Theorem~\hyperlink{thm169}{169} does not hold of the narrow integral.
+%-----File: 230.png---Folio 218------
+
+\section[Existence of Improper Definite Integrals on a Finite Interval]{Special Theorems on the Criteria of Existence of the
+Improper Definite Integral on a Finite Interval.}\hypertarget{chIXsec5}{}%[5]
+
+The examples of this section are intended to give an idea of the
+possible singularities of improperly integrable functions, and to
+indicate the difficulty of obtaining more general criteria of the
+divergence or convergence of the simple improper integral than those
+given in \S\S~\hyperlink{chIXsec1}{1} and \hyperlink{chIXsec2}{2} of this chapter.
+
+\begin{lemma}
+For every function $f_1(x)$ which is unbounded in every neighborhood
+of $x=a$ there is a function $f_2(x)$ which is infinitesimal as $x$
+approaches $a$, such that $f_1(x)\cdot f_2(x)$ is unbounded in every
+neighborhood of $x=a$, and such that
+\[
+  \frac{f_2(x)}{x-a}
+\]
+is monotonic increasing as $x$ approaches $a$.
+\end{lemma}
+
+\begin{proof}
+Since $f_1(x)$ is unbounded in every neighborhood of $x=a$, it follows
+that for every point $x_1$ of the segment $\overline{a\ b}$ there is a
+point $x_2$ on the segment $\overline{a\ x_1}$ such that
+\[
+  |f_1(x_2)| > 2 |f_1(x_1)| > 2M,
+\]
+and such that
+\[
+  (x_2-a)\leqq \textstyle\frac12(x_1-a).
+\]
+
+Let $x_1$, $x_2$, $x_3, \ldots$, $x_n,\ldots$ be a sequence of points
+dense only at $a$ such that
+\[
+  |f_1(x_n)| > 2 |f_1 (x_{n-1})| > 2^{n-1} \cdot M,
+\]
+and such that
+\[
+  |x_n-a| \leqq \textstyle\frac12|x_{n-1}-a|.
+\]
+We define $f_2(x)$ as follows:
+\[
+  f_2(x) =\frac1n\ \text{\textit{on the points $x_1$, $x_2, \ldots$,
+  $x_n,\ldots$}}
+\]
+%-----File: 231.png---Folio 219------
+\textit{and $f_2(x)$ is linear between the points of the sequence
+$x_1$, $x_2$, \ldots, $x_n$, \ldots.} Then there are values of $x$ on
+\correction{$\interval{x_n}{x_{n-1}}$}{$\interval{x_n}{x}_{n-1}$} such that
+\[
+  |f_1(x)|\cdot f_2(x) > {\frac2n}^{n-1} \cdot M,
+\]
+whence $f_1(x)\cdot f_2(x)$ is unbounded in the neighborhood of
+$a$.\footnote{%
+  In case
+  $\displaystyle{\mathop{L}_{x=0}} f_1(x) = \infty$,
+  $f_2(x) = \frac{1}{\sqrt{f_1(x)}}$ or
+  $f_2(x) = \frac{1}{\log f_1(x)}$ would satisfy the
+  requirements of the lemma except that they need not make
+  $\frac{f_2(x)}{x-a}$ monotonic.}
+Obviously $\frac{f_2(x)}{x-a}$ is monotonic increasing as $x$
+approaches $a$.
+\end{proof}
+\begin{theorem}[170]\hypertarget{thm170}{}
+For every function $f_1(x)$ which is unbounded in every neighborhood
+of $x=a$ there exists a non-oscillating function $f_2(x)$ such that
+\[
+  \mathop{L}_{x\doteq a} f_1(x) \int_x^b f_2(x)dx
+\]
+exists and is finite, while
+\[
+  (x-a) \cdot f_1(x) \cdot f_2(x)
+\]
+is unbounded in the neighborhood of $x=a$.
+\end{theorem}
+
+\begin{proof}
+According to the lemma there exists a function $f_3(x)$ such that
+\[
+  \mathop{L}_{x\doteq a} f_3(x) = 0,
+\]
+while $f_3(x)\cdot f_1(x)$ is unbounded and the function
+\[
+  f_4(x) = \frac{f_3(x)}{x-a}
+\]
+is monotonic increasing as $x$ approaches $a$. Since
+\[
+  (x-a) f_4(x) \cdot f_1(x) = f_3(x) \cdot f_1(x),
+\]
+%-----File: 232.png---Folio 220------
+$(x-a)\cdot f_4(x)\cdot f_1(x)$ is unbounded in the neighborhood of
+$x=a$.  Let $x_1, \ldots, x_n, \ldots$ be a sequence of points on
+$\interval{a}{b}$ whose only limit point is $a$, such that
+$f_3(x)\cdot f_1(x)$ is unbounded on this set.  In the sequence
+\hypertarget{seq1}{\[
+  (x_1-a)f_4(x_1),\quad (x_2-a)f_4(x_2),\quad \ldots,\quad
+  (x_n-a)f_4\text{\correction{$(x_n)$,}{$(x)_n$.}}
+\tag{1}
+\]}
+$\displaystyle\mathop{L}_{n\doteq\infty} (x_n-a)f_4(x_n) = 0$, since
+$\ \displaystyle\mathop{L}_{x\doteq a} (x-a)f_4(x ) = 0$.  Hence there
+is a value of $n$, $n_1$, such that
+\[
+  |(x_1-a) f_4(x_1 )|\geqq
+ 2|(x_{n_1}-a) f_4(x_{n_1})|,
+\]
+and another value of $n$, $n_2$ such that
+\[
+  |(x_{n_1}-a) f_4(x_{n_1})|\geqq
+ 2|(x_{n_2}-a) f_4(x_{n_2})|, \text{ etc.,}
+\]
+$n_{m+1}$ being so chosen that
+\[
+  |(x_{n_m}-a) f_4(x_{n_m} )|\geqq
+ 2|(x_{n_{m+1}}-a) f_4(x_{n_{m+1}})|.
+\]
+In this manner we select from the sequence~\hyperlink{seq1}{(1)} a set of terms forming
+the convergent series
+\hypertarget{ser2}{\[
+  (x_1-a)f_4(x_1 )
++ (x_{n_1}-a)f_4(x_{n_1}) + \ldots
++ (x_{n_m}-a)f_4(x_{n_m}) + \ldots.
+\tag{2}
+\]}
+We then obtain a function $f_2(x)$ as follows: For the set of values
+of $x$
+\[
+  x_{n_{m+1}} < x \leqq x_{n_m}, \quad f_2(x) = f_4(x_{n_m}).
+\]
+Then
+\begin{enumerate}
+\item[(1)] $f_2(x)$ is non-oscillating since
+\[
+  f_4(x_{n_m}) < f_4(x_{n_{m+1}}).
+\]
+\item[(2)]
+$(x-a) f_2(x)\cdot f_1(x)$ is unbounded on the set $x_1$,
+ $x_{n_1}$, $x_{n_2}, \ldots, x_{n_m}, \ldots$, since on this set
+\[
+  f_2(x) = f_4(x).
+\]
+%-----File: 233.png---Folio 221------
+\item[(3)] $\qquad\displaystyle\mathop{L}_{x\doteq a} \int_x^b
+f_2(x)dx = \sum_{m=1}^\infty (x_{n_m}-x_{n_{m+1}}) f_4(x_{n_m})$.
+\end{enumerate}
+But the terms of this series are numerically smaller than the
+corresponding terms of the convergent series~\hyperlink{ser2}{(2)}. Hence
+\[
+  \mathop{L}_{x\doteq a} \int_x^b f_2(x)dx
+\]
+exists and is finite.
+\end{proof}
+
+Theorem~\hyperlink{thm170}{170} may be regarded as showing that
+\[
+  \mathop{L}_{x\doteq a} (x-a) f_2(x) = 0
+\]
+is a strong necessary condition that, under the hypothesis of
+Theorem~\hyperlink{thm142}{142},
+\[
+  \mathop{L}_{x\doteq a} \int_x^b f_2(x)dx
+\]
+shall exist and be finite. For, according to Theorem~\hyperlink{thm170}{170}, it is
+impossible to modify the function $(x-a)$ by any factor $f_1(x)$ which
+shall approach infinity so slowly that for every function $f_2(x)$
+where
+\[
+  \mathop{L}_{x\doteq a} \int_x^b f_2(x)dx
+\]
+exists and is finite
+\[
+  \mathop{L}_{x\doteq a} (x-a) f_1(x)\cdot f_2(x) = 0.\footnotemark
+\]
+\footnotetext{%
+  See \textsc{Pringsheim}, Mathematische Annalen, Vol.~37,
+  pp.~591--604 (1890).}
+
+\begin{theorem}[171]\hypertarget{thm171}{}
+For every function $f_1(x)$ defined on the interval $\interval{a}{b}$
+there exists a function $f_2(x)$ such that
+\begin{enumerate}
+\item[\textnormal{(1)}] $f_2(x)$ is continuous and does not change sign on a
+certain neighborhood of $x\doteq a$.
+%-----File: 234.png---Folio 222------
+\item[\textnormal{(2)}] $\displaystyle\mathop{L}_{x\doteq a} \int_x^b f_2(x)dx$
+exists and is finite.
+
+\item[\textnormal{(3)}]\hypertarget{set3}{} For $x$ on a certain set $[x']$
+\[
+  \mathop{L}_{x\doteq a} \frac{f_1(x')}{f_2(x')} = 0.
+\]
+\end{enumerate}
+\end{theorem}
+
+
+\begin{proof}
+Let $x_1'$, $x_2', \ldots, x_n', \ldots$ be a set of points of the
+interval $\interval{a}{b}$ dense only at $a$. Let $B_1$, $B_2$, $B_3,
+\ldots, B_n, \ldots$ be a set of numbers such that
+\[
+  B_n\cdot n|f_1({x'}_n)|\geqq
+  2\cdot B_{n+1} (n+1) |f_1(x'_{n+1})|.\qquad
+  \text{($n = 1, 2, 3,\ldots$)}
+\]
+On the $x$ axis lay off a set of segments $[\sigma_n]$ such that
+$\sigma_n$ is of length $B_n$ and $x_n$ is its middle point. On the
+segments $\sigma_n$ as bases construct isosceles triangles on the
+positive side of the $x$ axis whose altitudes are
+$n\cdot|f_1(x)|$. The measures of areas of these triangles form a
+convergent series. Let $f_3(x)$ be any continuous, monotonic,
+unbounded function such that
+\[
+  \mathop{L}_{x\doteq a} \int_x^b f_3(x)dx
+\]
+exists and is finite. We then define $f_2(x)$ as the function
+represented by the following curve:
+\begin{enumerate}
+\item[(1)] Those parts of the boundaries of the isosceles triangles
+just described which lie above the curve defined by $f_3(x)$.
+\item[(2)] Those parts of the curve defined by $f_3(x)$ which lie
+outside the triangles or on their boundary.
+\end{enumerate}
+Obviously the function so defined has the properties specified in the
+theorem, the points $x_1'$, $x_2', \ldots, x_n', \ldots$ being the set
+$[x']$ specified by \hyperlink{set3}{(3)} of the theorem.
+\end{proof}
+
+Theorem~\hyperlink{thm171}{171} means that from the hypothesis that the improper definite
+integral of $f(x)$ exists on $\interval{a}{b}$ it is impossible to
+obtain any conclusion whatever as to the order of infinity or the rank
+of infinity of $f(x)$ at $x=a$. This is what one would
+%-----File: 235.png---Folio 223------
+expect \textit{a priori}, since the definite integral is a function of
+two parameters, while the necessary condition in terms of boundedness
+would be in terms of only one of these.
+
+\section[Existence of Improper Definite Integrals on the Infinite Interval]{Special Theorems on the Criteria of the Existence of the
+Improper Definite Integral on the Infinite Interval.}\hypertarget{chIXsec6}{}%[6]
+
+\begin{theorem}[172]\hypertarget{thm172}{}
+For every function $f_1(x)$ which is unbounded as $x$ approaches
+$\infty$ there exists a non-oscillating function $f_2(x)$ such that
+\[
+  \mathop{L}_{x\doteq\infty} \int_a^x f_2(x)dx
+\]
+exists and is finite, while $(x-a)f_1(x)\cdot f_2(x)$ is unbounded as
+$x$ approaches $\infty$.
+\end{theorem}
+
+\begin{proof}
+Obviously the lemma of Theorem~\hyperlink{thm170}{170} can be stated so as to apply to the
+case where $x$ approaches $\infty$ instead of $a$.  If then in the
+proof of Theorem~\hyperlink{thm161}{161} the set of points $x_1\ldots x_n\ldots$ is so
+taken that
+\[
+  \mathop{L}_{n\doteq\infty} x_n=\infty
+\]
+instead of $a$, the proof of Theorem~\hyperlink{thm161}{161} applies with the exception
+that $f_2(x)$ is non-oscillating \textit{decreasing} instead of
+non-oscillating \textit{increasing}.
+\end{proof}
+
+\begin{theorem}[173]\hypertarget{thm173}{}
+For every function $f_1(x)$ defined on the interval $\interval{a}{\infty}$
+there exists a function $f_2(x)$ such that
+\begin{enumerate}
+\item[\textnormal{(1)}] $f_2(x)$ is continuous and does not change sign for $x$
+greater than a certain fixed number.
+\item[\textnormal{(2)}]
+\[
+  \mathop{L}_{x\doteq\infty} \int_x^a f_2(x)dx
+\]
+exists and is finite.
+%-----File: 236.png---Folio 224------
+\item[\textnormal{(3)}] For $x$ on a certain set $[x']$
+\[
+  \mathop{L}_{x\doteq\infty} \frac{f_1(x')}{f_2(x')} = 0.
+\]
+\end{enumerate}
+\end{theorem}
+
+\begin{proof}
+Such a function $f_2(x)$ may be defined in a manner analogous to that
+of the proof of Theorem~\hyperlink{thm171}{171}.
+
+The remarks as to the meaning of Theorems \hyperlink{thm170}{170} and \hyperlink{thm171}{171} apply with
+obvious modifications to Theorems \hyperlink{thm172}{172} and \hyperlink{thm173}{173}.
+\end{proof}
+
+\backmatter
+%-----File: 237.png---Folio 225------
+%\chapter*{INDEX}
+{\setlength{\columnsep}{1cm}
+\printindex}
+%-----File: 238.png---Folio 226------
+%-----File: 239.png---Folio 227------
+%-----File: 240.png---Folio 228------
+%[Blank Page]
+%-----File: 241.png---Index 1--------
+\pagestyle{plain}
+\newpage\setcounter{page}{1}
+{\centering
+{\sffamily\Huge
+SHORT-TITLE CATALOGUE\\[0.5ex]}
+\small
+OF THE\\[0.5ex]
+\LARGE
+PUBLICATIONS\\[0.5ex]
+\small
+OF\\
+\Huge
+JOHN WILEY \& SONS,\\[0.5ex]
+\sffamily\sc\large
+New York.\\[2ex]
+\Large\sc
+London: CHAPMAN \& HALL, Limited.\\[0.5ex]
+\rule[0.5ex]{2cm}{.2pt}\\
+\normalfont\normalsize
+ARRANGED UNDER SUBJECTS.\\
+\rule[0.5ex]{2cm}{.2pt}\\}
+
+
+\footnotesize Descriptive circulars sent on application. Books marked
+with an asterisk (*) are sold at \textit{net} prices only. All books
+are bound in cloth unless otherwise stated.
+\bigskip
+
+\begin{center} \rule[0.5ex]{2cm}{.2pt} \end{center}
+\bigskip
+
+
+
+\footnotesize
+\begin{longtable}{@{}l@{ }r@{}}
+
+\multicolumn{2}{c}{\large AGRICULTURE.}\\[1em]
+\nopagebreak
+Armsby's Manual of Cattle-feeding.\dotfill\ldots 12mo, &\$1\ 75\\
+
+\indent Principles of Animal Nutrition.\dotfill\ldots 8vo, &4\ 00\\
+
+Budd and Hansen's American Horticultural Manual:\\
+
+\indent Part I\@. Propagation, Culture, and Improvement.\dotfill\ldots
+12mo, &1\ 50\\
+
+\indent Part II\@. Systematic Pomology.\dotfill 12mo, &1\ 50\\
+
+Downing's Fruits and Fruit-trees of America.\dotfill\ldots 8vo, &5\ 00\\
+
+Elliott's Engineering for Land Drainage.\dotfill\ldots 12mo, &1\ 50\\
+
+\indent Practical Farm Drainage.\dotfill\ldots 12mo, &1\ 00\\
+
+Graves's Forest Mensuration.\dotfill\ldots 8vo, &4\ 00\\
+
+Green's Principles of American Forestry.\dotfill\ldots 12mo, &1\ 50\\
+
+Grotenfelt's Principles of Modern Dairy Practice. (Woll.)\dotfill
+12mo, &2\ 00\\
+
+Kemp's Landscape Gardening.\dotfill\ldots 12mo, &2\ 50\\
+
+Maynard's Landscape Gardening as Applied to Home
+Decoration.\dotfill\ldots 12mo, &1\ 50\\
+
+* McKay and Larsen's Principles and Practice of
+  Butter-making.\dotfill\ldots 8vo, &1\ 50\\
+
+Sanderson's Insects Injurious to Staple Crops.\dotfill\ldots 12mo, &1\ 50\\
+
+\indent Insects Injurious to Garden Crops. (In preparation.)\\
+
+\indent Insects Injuring Fruits. (In preparation.)\\
+
+Stockbridge's Rocks and Soils.\dotfill\ldots 8vo, &2\ 50\\
+
+Winton's Microscopy of Vegetable Foods.\dotfill\ldots 8vo, &7\ 50\\
+
+Woll's Handbook for Farmers and Dairymen.\dotfill\ldots 16mo, &1\ 50\\[3em]
+
+\multicolumn{2}{c}{\large ARCHITECTURE.}\\[1em]
+\nopagebreak
+Baldwin's Steam Heating for Buildings.\dotfill\ldots 12mo, &2\ 50\\
+
+Bashore's Sanitation of a Country House.\dotfill\ldots 12mo. &1\ 00\\
+
+Berg's Buildings and Structures of American Railroads.\dotfill\ldots
+4to, &5\ 00\\
+
+Birkmire's Planning and Construction of American
+Theatres.\dotfill\ldots 8vo, &3\ 00\\
+
+\indent Architectural Iron and Steel.\dotfill\ldots 8vo, &3\ 50\\
+
+\indent Compound Riveted Girders as Applied in
+Buildings.\dotfill\ldots 8vo, &2\ 00\\
+
+\indent Planning and Construction of High Office
+Buildings.\dotfill\ldots 8vo, &3\ 50\\
+
+\indent Skeleton Construction in Buildings.\dotfill\ldots 8vo, &3\ 00\\
+
+Brigg's Modern American School Buildings.\dotfill\ldots 8vo, &4\ 00\\
+
+%-----File: 242.png---Index 2--------
+Carpenter's Heating and Ventilating of Buildings.\dotfill\ldots 8vo,
+&4\ 00\\
+
+Freitag's Architectural Engineering.\dotfill\ldots 8vo, &3\ 50\\
+
+\nopagebreak
+\indent Fireproofing of Steel Buildings.\dotfill\ldots 8vo, &2\ 50\\
+
+French and Ives's Stereotomy.\dotfill\ldots 8vo, &2\ 50\\
+
+Gerhard's Guide to Sanitary House-inspection.\dotfill\ldots 16mo, &1\ 00\\
+
+\nopagebreak
+\indent Theatre Fires and Panics.\dotfill\ldots 12mo, &1\ 50\\
+
+* Greene's Structural Mechanics.\dotfill\ldots 8vo, &2\ 50\\
+
+Holly's Carpenters' and Joiners' Handbook.\dotfill\ldots 18mo, &75\\
+
+Johnson's Statics by Algebraic and Graphic Methods.\dotfill\ldots 8vo,
+&2\ 00\\
+
+Kidder's Architects' and Builders' Pocket-book.\\
+
+\nopagebreak
+\indent\indent Rewritten Edition.\dotfill\ldots 16mo, mor., &5\ 00\\
+
+Merrill's Stones for Building and Decoration.\dotfill\ldots 8vo, &5\ 00\\
+
+\nopagebreak
+\indent Non-metallic Minerals: Their Occurrence and
+Uses.\dotfill\ldots 8vo, &4\ 00\\
+
+Monckton's Stair-building.\dotfill\ldots 4to, &4\ 00\\
+
+Patton's Practical Treatise on Foundations.\dotfill\ldots 8vo, &5\ 00\\
+
+Peabody's Naval Architecture.\dotfill\ldots 8vo, &7\ 50\\
+
+Rice's Concrete-block Manufacture.\dotfill\ldots 8vo, &2\ 00\\
+
+Richey's Handbook for Superintendents of Construction.\dotfill\ldots
+16mo, mor., &4\ 00\\
+
+\makebox[0pt]{\hspace{.5ex} *}\indent Building Mechanics' Ready
+Reference Book. Carpenters'\\
+
+\nopagebreak
+\indent\indent and Woodworkers' Edition.\dotfill\ldots 16mo, morocco,
+&1\ 50\\
+
+Sabin's Industrial and Artistic Technology of Paints and
+Varnish.\dotfill\ldots 8vo, &3\ 00\\
+
+Siebert and Biggin's Modern Stone-cutting and Masonry.\dotfill\ldots
+8vo, &1\ 50\\
+
+Snow's Principal Species of Wood.\dotfill\ldots 8vo, &3\ 50\\
+
+Sondericker's Graphic Statics with Applications to Trusses, Beams,\\
+
+\nopagebreak
+\indent\indent and Arches.\dotfill\ldots 8vo, &2\ 00\\
+
+Towne's Locks and Builders' Hardware.\dotfill\ldots 18mo, morocco, &3\ 00\\
+
+Wait's Engineering and Architectural Jurisprudence.\dotfill\ldots 8vo,
+&6\ 00\\
+
+\nopagebreak
+\hfill Sheep, &6\ 50\\
+
+\indent Law of Operations Preliminary to Construction in Engineering\\
+
+\indent\indent and Architecture.\dotfill\ldots 8vo, &5\ 00\\
+
+\nopagebreak
+\hfill Sheep, &5\ 50\\
+
+\indent Law of Contracts.\dotfill\ldots 8vo, &3\ 00\\
+
+Wood's Rustless Coatings: Corrosion and Electrolysis of Iron\\
+
+\nopagebreak
+\indent\indent and Steel.\dotfill\ldots 8vo, &4\ 00\\
+
+Worcester and Atkinson's Small Hospitals, Establishment and\\
+
+\nopagebreak
+\indent\indent Maintenance, Suggestions for Hospital Architecture,\\
+
+\nopagebreak
+\indent\indent with Plans for a Small Hospital.\dotfill\ldots 12mo, &1\ 25\\
+
+The World's Columbian Exposition of 1893.\dotfill\ldots Large 4to, &1\
+00\\[3em]
+
+\multicolumn{2}{c}{\large ARMY AND NAVY.}\\[1em]
+\nopagebreak
+Bernadou's Smokeless Powder, Nitro-cellulose, and the Theory of the \\
+
+\nopagebreak
+\indent\indent Cellulose Molecule.\dotfill\ldots 12mo, &2\ 50\\
+
+* Bruff's Text-book Ordnance and Gunnery.\dotfill\ldots 8vo, &6\ 00\\
+
+Chase's Screw Propellers and Marine Propulsion.\dotfill\ldots 8vo, &3\ 00\\
+
+Cloke's Gunner's Examiner.\dotfill\ldots 8vo, &1\ 50\\
+
+Craig's Azimuth.\dotfill\ldots 4to, &3\ 50\\
+
+Crehore and Squier's Polarizing Photo-chronograph.\dotfill\ldots 8vo,
+&3\ 00\\
+
+* Davis's Elements of Law.\dotfill\ldots 8vo, &2\ 50\\
+
+\nopagebreak
+\makebox[0pt]{\hspace{.5ex} *}\indent Treatise on the Military Law of
+United States.\dotfill\ldots 8vo, &7\ 00\\
+
+\nopagebreak
+\hfill Sheep, &7\ 50\\
+
+De Brack's Cavalry Outposts Duties. (Carr.)\dotfill\ldots 24mo,
+morocco, &2\ 00\\
+
+Dietz's Soldier's First Aid Handbook.\dotfill\ldots 16mo, morocco, &1\ 25\\
+
+* Dudley's Military Law and the Procedure\\
+
+\nopagebreak
+\indent\indent of Courts-martial.\dotfill\ldots Large 12mo, & 2\ 50\\
+
+Durand's Resistance and Propulsion of Ships.\dotfill\ldots 8vo, &5\ 00\\
+
+* Dyer's Handbook of Light Artillery.\dotfill\ldots 12mo, &3\ 00\\
+
+Eissler's Modern High Explosives.\dotfill\ldots 8vo, &4\ 00\\
+
+* Fiebeger's Text-book on Field Fortification.\dotfill\ldots Small
+  8vo, &2\ 00\\
+
+Hamilton's The Gunner's Catechism.\dotfill\ldots 18mo, &1\ 00\\
+
+* Hoff's Elementary Naval Tactics.\dotfill\ldots 8vo, &1\ 50\\
+
+%-----File: 243.png---Index 3--------
+Ingalls's Handbook of Problems in Direct Fire.\dotfill\ldots 8vo, &4\ 00\\
+
+\nopagebreak
+\makebox[0pt]{\hspace{.5ex} *}\indent Ballistic Tables.\dotfill\ldots
+8vo, &1\ 50\\
+
+* Lyons's Treatise on Electromagnetic Phenomena. Vols.~I.\\
+
+\nopagebreak
+\indent\indent and II.\dotfill\ldots 8vo, each, &6\ 00\\
+
+* Mahan's Permanent Fortifications. (Mercur.)\dotfill 8vo, half
+  morocco, &7\ 50\\
+
+Manual for Courts-martial.\dotfill\ldots 16mo, morocco, &1\ 50\\
+
+* Mercur's Attack of Fortified Places.\dotfill\ldots 12mo, &2\ 00\\
+
+\nopagebreak
+\makebox[0pt]{\hspace{.5ex} *}\indent Elements of the Art of
+War.\dotfill\ldots 8vo, &4\ 00\\
+
+Metcalf's Cost of Manufactures---And the Administration\\
+
+\nopagebreak
+\indent\indent of Workshops.\dotfill\ldots 8vo, &5\ 00\\
+
+\nopagebreak
+\makebox[0pt]{\hspace{.5ex} *}\indent Ordnance and Gunnery. 2
+vols.\dotfill\ldots 12mo, &5 00\\
+
+Murray's Infantry Drill Regulations.\dotfill\ldots 18mo, paper, &10\\
+
+Nixon's Adjutants' Manual.\dotfill\ldots 24mo, &1\ 00\\
+
+Peabody's Naval Architecture.\dotfill\ldots 8vo, &7\ 50\\
+
+* Phelps's Practical Marine Surveying.\dotfill\ldots 8vo, &2\ 50\\
+
+Powell's Army Officer's Examiner.\dotfill\ldots 12mo, &4\ 00\\
+
+Sharpe's Art of Subsisting Armies in War.\dotfill\ldots 18mo, morocco,
+&1\ 50\\
+
+* Tupes and Poole's Manual of Bayonet Exercises and Musketry Fencing.\\
+
+\nopagebreak
+\hfill 24mo, leather, &50\\
+
+* Walke's Lectures on Explosives.\dotfill\ldots 8vo, &4\ 00\\
+
+Weaver's Military Explosives.\dotfill\ldots 8vo, &3\ 00\\
+
+* Wheeler's Siege Operations and Military Mining.\dotfill\ldots 8vo,
+  &2\ 00\\
+
+Winthrop's Abridgment of Military Law.\dotfill\ldots 12mo, &2\ 50\\
+
+Woodhull's Notes on Military Hygiene.\dotfill\ldots 16mo, &1\ 50\\
+
+Young's Simple Elements of Navigation.\dotfill\ldots 16mo, morocco,
+&2\ 00\\[3em]
+
+\multicolumn{2}{c}{\large ASSAYING.}\\[1em]
+\nopagebreak
+Fletcher's Practical Instructions in Quantitative Assaying with\\
+
+\nopagebreak
+\indent\indent the Blowpipe.\dotfill\ldots 12mo, morocco, &1\ 50\\
+
+Furman's Manual of Practical Assaying.\dotfill\ldots 8vo, &3\ 00\\
+
+Lodge's Notes on Assaying and Metallurgical Laboratory\\
+
+\nopagebreak
+\indent\indent Experiments.\dotfill\ldots 8vo,&3\ 00\\
+
+Low's Technical Methods of Ore Analysis.\dotfill\ldots 8vo, &3\ 00\\
+
+Miller's Manual of Assaying.\dotfill\ldots 12mo, &1\ 00\\
+
+\nopagebreak
+\indent Cyanide Process.\dotfill\ldots 12mo, &1\ 00\\
+
+Minet's Production of Aluminum and its Industrial
+Use. (Waldo.)\dotfill\ldots 12mo, &2\ 50\\
+
+O'Driscoll's Notes on the Treatment of Gold Ores.\dotfill\ldots 8vo,
+&2\ 00\\
+
+Ricketts and Miller's Notes on Assaying.\dotfill\ldots 8vo, &3\ 00\\
+
+Robine and Lenglen's Cyanide Industry. (Le Clerc.)\dotfill\ldots 8vo,
+&4\ 00\\
+
+Ulke's Modern Electrolytic Copper Refining.\dotfill\ldots 8vo, &3\ 00\\
+
+Wilson's Cyanide Processes.\dotfill\ldots 12mo, &1\ 50\\
+
+\nopagebreak
+\indent Chlorination Process.\dotfill\ldots 12mo, &1\ 50\\[3em]
+
+\multicolumn{2}{c}{\large ASTRONOMY.}\\[1em]
+\nopagebreak
+Comstock's Field Astronomy for Engineers.\dotfill\ldots 8vo, &2\ 50\\
+
+Craig's Azimuth.\dotfill\ldots 4to, &3\ 50\\
+
+Crandall's Text-book on Geodesy and Least Squares.\dotfill\ldots 8vo,
+& 3\ 00\\
+
+Doolittle's Treatise on Practical Astronomy.\dotfill\ldots 8vo, &4\ 00\\
+
+Gore's Elements of Geodesy.\dotfill\ldots 8vo, &2\ 50\\
+
+Hayford's Text-book of Geodetic Astronomy.\dotfill\ldots 8vo, &3\ 00\\
+
+Merriman's Elements of Precise Surveying and Geodesy.\dotfill\ldots
+8vo, &2\ 50\\
+
+* Michie and Harlow's Practical Astronomy.\dotfill\ldots 8vo, &3\ 00\\
+
+* White's Elements of Theoretical and Descriptive
+  Astronomy.\dotfill\ldots 12mo, &\correction{2}{}\ 00\\[3em]
+
+\multicolumn{2}{c}{\large BOTANY.}\\[1em]
+\nopagebreak
+Davenport's Statistical Methods, with Special Reference\\
+
+\nopagebreak
+\indent\indent to Biological Variation.\dotfill\ldots 16mo, morocco,
+&1\ 25\\
+
+Thom\'e and Bennett's Structural and Physiological
+Botany.\dotfill\ldots 16mo, &2\ 25\\
+
+Westermaier's Compendium of General Botany. (Schneider.)\dotfill 8vo,
+&2\ 00\\[3em]
+%-----File: 244.png---Index 4--------
+\multicolumn{2}{c}{\large CHEMISTRY.}\\[1em] \nopagebreak
+* Abegg's Theory of Electrolytic Dissociation. (Von
+Ende.)\dotfill\ldots 12mo, & 1\ 25 \\
+
+Adriance's Laboratory Calculations and Specific Gravity
+Tables.\dotfill\ldots 12mo, &1\ 25\\
+
+Alexeyeff's General Principles of Organic
+Synthesis. (Matthews.)\dotfill\ldots 8vo, &3\ 00\\
+
+Allen's Tables for Iron Analysis.\dotfill\ldots 8vo, &3\ 00\\
+
+Arnold's Compendium of Chemistry. (Mandel.)\dotfill\ldots Small 8vo,
+&3\ 50\\
+
+Austen's Notes for Chemical Students.\dotfill\ldots 12mo, &1\ 50\\
+
+Bernadou's Smokeless Powder.---Nitro-cellulose, and Theory of\\
+
+\nopagebreak
+\indent\indent the Cellulose Molecule.\dotfill\ldots 12mo, &2\ 50\\
+
+* Browning's Introduction to the Rarer Elements.\dotfill\ldots 8vo, &1\ 50\\
+
+Brush and Penfield's Manual of Determinative Mineralogy.\dotfill\ldots
+8vo, &4\ 00\\
+
+* Claassen's Beet-sugar Manufacture. (Hall and Rolfe.)\dotfill\ldots
+  8vo, &3\ 00\\
+
+Classen's Quantitative Chemical Analysis by\\
+
+\nopagebreak
+\indent\indent Electrolysis. (Boltwood.)\dotfill\ldots 8vo, &3\ 00\\
+
+Cohn's Indicators and Test-papers.\dotfill\ldots 12mo, &2\ 00\\
+
+\nopagebreak
+\indent Tests and Reagents.\dotfill\ldots 8vo, &3\ 00\\
+
+Crafts's Short Course in Qualitative Chemical\\
+
+\nopagebreak
+\indent\indent Analysis. (Schaeffer.)\dotfill\ldots 12mo, &1\ 50\\
+
+* Danneel's Electrochemistry. (Merriam.)\dotfill\ldots 12mo, & 1\ 25\\
+
+Dolezalek's Theory of the Lead Accumulator (Storage Battery).\\
+
+\nopagebreak
+\indent (Von Ende.)\dotfill\ldots 12mo, &2\ 50\\
+
+Drechsel's Chemical Reactions. (Merrill.)\dotfill\ldots 12mo, &1\ 25\\
+
+Duhem's Thermodynamics and Chemistry. (Burgess.)\dotfill\ldots 8vo, &4\ 00\\
+
+Eissler's Modern High Explosives.\dotfill\ldots 8vo, &4\ 00\\
+
+Effront's Enzymes and their Applications. (Prescott.)\dotfill\ldots
+8vo, &3\ 00\\
+
+Erdmann's Introduction to Chemical
+Preparations. (Dunlap.)\dotfill\ldots 12mo, &1\ 25\\
+
+Fletcher's Practical Instructions in Quantitative Assaying with\\
+
+\nopagebreak
+\indent\indent the Blowpipe.\dotfill\ldots 12mo, morocco, &1\ 50\\
+
+Fowler's Sewage Works Analyses.\dotfill\ldots 12mo, &2\ 00\\
+
+Fresenius's Manual of Qualitative Chemical
+Analysis. (Wells.)\dotfill\ldots 8vo, &5\ 00\\
+
+\indent Manual of Qualitative Chemical Analysis. Part I.\\
+
+\nopagebreak
+\indent\indent Descriptive. (Wells.)\dotfill\ldots 8vo, &3\ 00\\
+
+\indent System of Instruction in Quantitative Chemical Analysis. (Cohn.)\\
+
+\nopagebreak
+\indent\indent 2 vols.\dotfill\ldots 8vo, &12\ 50\\
+
+Fuertes's Water and Public Health.\dotfill\ldots 12mo, &1\ 50\\
+
+Furman's Manual of Practical Assaying.\dotfill\ldots 8vo, &3\ 00\\
+
+* Getman's Exercises in Physical Chemistry.\dotfill\ldots 12mo, &2\ 00\\
+
+Gill's Gas and Fuel Analysis for Engineers.\dotfill\ldots 12mo, &1\ 25\\
+
+* Gooch and Browning's Outlines of Qualitative\\
+
+\nopagebreak
+\indent\indent Chemical Analysis.\dotfill\ldots Small 8vo, & 1\ 25\\
+
+Grotenfelt's Principles of Modern Dairy
+Practice. (Woll.)\dotfill\ldots 12mo, &2\ 00\\
+
+Groth's Introduction to Chemical Crystallography
+(Marshall)\dotfill\ldots 12mo, &1\ 25\\
+
+Hammarsten's Text-book of Physiological
+Chemistry. (Mandel.)\dotfill\ldots 8vo, &4\ 00\\
+
+Helm's Principles of Mathematical Chemistry. (Morgan.)\dotfill\ldots
+12mo, &1\ 50\\
+
+Hering's Ready Reference Tables (Conversion Factors)\dotfill\ldots
+16mo, morocco, &2\ 50\\
+
+Hind's Inorganic Chemistry.\dotfill\ldots 8vo, &3\ 00\\
+
+\makebox[0pt]{\hspace{.5ex} *}\indent Laboratory Manual for
+Students.\dotfill\ldots 12mo, &1\ 00\\
+
+Holleman's Text-book of Inorganic Chemistry. (Cooper.)\dotfill\ldots
+8vo, &2\ 50\\
+
+\nopagebreak
+\indent Text-book of Organic Chemistry. (Walker and
+Mott.)\dotfill\ldots 8vo, &2\ 50\\
+
+\makebox[0pt]{\hspace{.5ex} *}\indent Laboratory Manual of Organic
+Chemistry. (Walker.)\dotfill\ldots 12mo, &1\ 00\\
+
+Hopkins's Oil-chemists' Handbook.\dotfill\ldots 8vo, &3\ 00\\
+
+Iddings's Rock Minerals.\dotfill\ldots 8vo, & 5\ 00\\
+
+Jackson's Directions for Laboratory Work in Physiological\\
+
+\nopagebreak
+\indent\indent Chemistry.\dotfill\ldots 8vo, &1\ 25\\
+
+Keep's Cast Iron.\dotfill\ldots 8vo, &2\ 50\\
+
+Ladd's Manual of Quantitative Chemical Analysis.\dotfill\ldots 12mo,
+&1\ 00\\
+
+Landauer's Spectrum Analysis. (Tingle.)\dotfill\ldots 8vo, &3\ 00\\
+
+* Langworthy and Austen. The Occurrence of Aluminium in Vegetable\\
+
+\nopagebreak
+\indent\indent Products, Animal Products, and Natural
+Waters.\dotfill\ldots 8vo, &2\ 00\\
+
+Lassar-Cohn's Application of Some General Reactions to Investigations\\
+
+\nopagebreak
+\indent\indent in Organic Chemistry. (Tingle.)\dotfill\ldots 12mo, &1\ 00\\
+
+Leach's The Inspection and Analysis of Food with Special Reference \\
+
+\nopagebreak
+\indent\indent to State Control.\dotfill\ldots 8vo, &7\ 50\\
+
+L\"{o}b's Electrochemistry of Organic
+Compounds. (Lorenz.)\dotfill\ldots 8vo, &3\ 00\\
+
+%-----File: 245.png---Index 5--------
+Lodge's Notes on Assaying and Metallurgical Laboratory\\
+
+\nopagebreak
+\indent\indent Experiments.\dotfill\ldots 8vo, &3\ 00\\
+
+Low's Technical Method of Ore Analysis.\dotfill\ldots 8vo, &3\ 00\\
+
+Lunge's Techno-chemical Analysis. (Cohn.)\dotfill\ldots 12mo, &1\ 00\\
+
+* McKay and Larsen's Principles and Practice of
+  Butter-making.\dotfill\ldots 8vo, &1\ 50\\
+
+Mandel's Handbook for Bio-chemical Laboratory.\dotfill\ldots 12mo, &1\ 50\\
+
+* Martin's Laboratory Guide to Qualitative Analysis with\\
+
+\nopagebreak
+\indent\indent the Blowpipe.\dotfill\ldots 12mo, &60\\
+
+Mason's Water-supply. (Considered Principally from a Sanitary\\
+
+\nopagebreak
+\indent\indent Standpoint.) 3d Edition, Rewritten.\dotfill\ldots 8vo,
+&4\ 00\\
+
+\nopagebreak
+\indent Examination of Water. (Chemical and
+Bacteriological.)\dotfill\ldots 12mo, &1\ 25\\
+
+Matthew's The Textile Fibres.\dotfill\ldots 8vo, &3\ 50\\
+
+Meyer's Determination of Radicles in Carbon Compounds.\\
+
+\nopagebreak
+\indent\indent (Tingle.)\dotfill\ldots 12mo, &1\ 00\\
+
+Miller's Manual of Assaying.\dotfill\ldots 12mo, &1\ 00\\
+
+\nopagebreak
+\indent Cyanide Process.\dotfill\ldots 12mo, &1\ 00\\
+
+Minet's Production of Aluminum and its Industrial
+Use. (Waldo.)\dotfill\ldots 12mo, &2\ 50\\
+
+Mixter's Elementary Text-book of Chemistry.\dotfill\ldots 12mo, &1\ 50\\
+
+Morgan's An Outline of the Theory of Solutions and its
+Results.\dotfill\ldots 12mo, &1\ 00\\
+
+\nopagebreak
+\indent Elements of Physical Chemistry.\dotfill\ldots 12mo, &3\ 00\\
+
+\nopagebreak
+\indent * Physical Chemistry for Electrical Engineers.\dotfill\ldots
+12mo, &1\ 50\\
+
+Morse's Calculations used in Cane-sugar Factories.\dotfill\ldots 16mo,
+morocco, &1\ 50\\
+
+* Muir's History of Chemical Theories and Laws.\dotfill\ldots 8vo, & 4\ 00\\
+
+Mulliken's General Method for the Identification of Pure\\
+
+\nopagebreak
+\indent\indent Organic Compounds. Vol.~I.\dotfill\ldots Large 8vo, &5\ 00\\
+
+O'Brine's Laboratory Guide in Chemical Analysis.\dotfill\ldots 8vo, &2\ 00\\
+
+O'Driscoll's Notes on the Treatment of Gold Ores.\dotfill\ldots 8vo,
+&2\ 00\\
+
+Ostwald's Conversations on Chemistry. Part
+One. (Ramsey.)\dotfill\ldots 12mo, &1\ 50\\
+
+\nopagebreak
+\phantom{Ostw}\makebox[0pt]{``}\phantom{ald's
+Conve}\makebox[0pt]{``}\phantom{rsations }\makebox[0pt]{\;
+``}\phantom{on Chem}\makebox[0pt]{``}\phantom{istry.} Part
+Two. (Turnbull.)..\dotfill 12mo, &2\ 00\\
+
+* Pauli's Physical Chemistry in the Service of
+  Medicine. (Fischer.)\dotfill\ldots 12mo, & 1\ 25\\
+
+* Penfield's Notes on Determinative Mineralogy and Record\\
+
+\nopagebreak
+\indent\indent of Mineral Tests.\dotfill\ldots 8vo, paper, &50\\
+
+Pictet's The Alkaloids and their Chemical
+Constitution. (Biddle.)\dotfill\ldots 8vo, &5\ 00\\
+
+Pinner's Introduction to Organic Chemistry. (Austen.)\dotfill\ldots
+12mo &1\ 50\\
+
+Poole's Calorific Power of Fuels.\dotfill\ldots 8vo, &3\ 00\\
+
+Prescott and Winslow's Elements of Water Bacteriology, with Special \\
+
+\nopagebreak
+\indent\indent Reference to Sanitary Water Analysis.\dotfill\ldots
+12mo, &1\ 25\\
+
+* Reisig's Guide to Piece-dyeing.\dotfill\ldots 8vo, &25\ 00\\
+
+Richards and Woodman's Air, Water, and Food from\\
+
+\nopagebreak
+\indent\indent a Sanitary Standpoint.\dotfill\ldots 8vo, &2\ 00\\
+
+Ricketts and Russell's Skeleton Notes upon Inorganic Chemistry.\\
+
+\nopagebreak
+\indent\indent (Part I\@. Non-metallic Elements.)\dotfill\ldots 8vo,
+morocco, &75\\
+
+Ricketts and Miller's Notes on Assaying.\dotfill\ldots 8vo, &3\ 00\\
+
+Rideal's Sewage and the Bacterial Purification of
+Sewage.\dotfill\ldots 8vo, &3\ 50\\
+
+\nopagebreak
+\indent Disinfection and the Preservation of Food.\dotfill\ldots 8vo,
+&4\ 00\\
+
+Riggs's Elementary Manual for the Chemical Laboratory.\dotfill\ldots
+8vo, &1\ 25\\
+
+Robine and Lenglen's Cyanide Industry. (Le Clerc.)\dotfill\ldots 8vo,
+&4\ 00\\
+
+Ruddiman's Incompatibilities in Prescriptions.\dotfill\ldots 8vo, &2\ 00\\
+
+\makebox[0pt]{\hspace{.5ex} *}\indent Whys in Pharmacy.\dotfill\ldots
+12mo,&1\ 00\\
+
+Sabin's Industrial and Artistic Technology of Paints and
+Varnish.\dotfill\ldots 8vo, &3\ 00\\
+
+Salkowski's Physiological and Pathological
+Chemistry. (Orndorff.)\dotfill\ldots 8vo, &2\ 50\\
+
+Schimpf's Text-book of Volumetric Analysis.\dotfill\ldots 12mo,&2\ 50\\
+
+\nopagebreak
+\indent Essentials of Volumetric Analysis.\dotfill\ldots 12mo,&1\ 25\\
+
+\makebox[0pt]{\hspace{.5ex} *}\indent Qualitative Chemical
+Analysis.\dotfill\ldots 8vo, &1\ 25\\
+
+Smith's Lecture Notes on Chemistry for Dental Students.\dotfill\ldots
+8vo, &2\ 50\\
+
+Spencer's Handbook for Chemists of Beet-sugar Houses.\dotfill\ldots
+16mo, morocco, &3\ 00\\
+
+\nopagebreak
+
+\indent Handbook for Cane Sugar Manufacturers.\dotfill\ldots 16mo,
+morocco, &3\ 00\\
+
+Stockbridge's Rocks and Soils.\dotfill\ldots 8vo, &2\ 50\\
+
+* Tillman's Elementary Lessons in Heat.\dotfill\ldots 8vo, &1\ 50\\
+
+\makebox[0pt]{\hspace{.5ex} *}\indent Descriptive General
+Chemistry.\dotfill\ldots 8vo, &3\ 00\\
+
+Treadwell's Qualitative Analysis. (Hall.)\dotfill\ldots 8vo, &3\ 00\\
+
+\nopagebreak
+
+\indent Quantitative Analysis. (Hall.)\dotfill\ldots 8vo, &4\ 00\\
+
+Turneaure and Russell's Public Water-supplies.\dotfill\ldots 8vo, &5\ 00\\
+
+%-----File: 246.png---Index 6--------
+
+
+Van Deventer's Physical Chemistry for
+Beginners. (Boltwood.)\dotfill\ldots 12mo, &1\ 50\\
+
+* Walke's Lectures on Explosives.\dotfill\ldots 8vo, &4\ 00\\
+
+Ware's Beet-sugar Manufacture and Refining.\dotfill\ldots Small 8vo,
+cloth, &4\ 00\\
+
+Washington's Manual of the Chemical Analysis of Rocks.\dotfill\ldots
+8vo, &2\ 00\\
+
+Weaver's Military Explosives.\dotfill\ldots 8vo, &3\ 00 \\
+
+Wehrenfennig's Analysis and Softening of Boiler
+Feed-Water.\dotfill\ldots 8vo, &4\ 00 \\
+
+Wells's Laboratory Guide in Qualitative Chemical
+Analysis.\dotfill\ldots 8vo, &1\ 50 \\
+
+\nopagebreak
+
+\indent Short Course in Inorganic Qualitative Chemical Analysis for \\
+
+\nopagebreak
+
+\indent\indent Engineering Students.\dotfill\ldots 12mo, &1\ 50 \\
+
+\nopagebreak
+
+\indent Text-book of Chemical Arithmetic.\dotfill\ldots 12mo, &1\ 25 \\
+
+Whipple's Microscopy of Drinking-water.\dotfill\ldots 8vo, &3\ 50 \\
+
+Wilson's Cyanide Processes.\dotfill\ldots 12mo, &1\ 50 \\
+
+\nopagebreak
+
+\indent Chlorination Process.\dotfill\ldots 12mo, &1\ 50 \\
+
+Winton's Microscopy of Vegetable Foods.\dotfill\ldots 8vo, &7\ 50 \\
+
+Wulling's Elementary Course in Inorganic, Pharmaceutical, and Medical \\
+
+\nopagebreak
+
+\indent\indent Chemistry.\dotfill\ldots 12mo, &2\ 00 \\[3em]
+
+
+
+\multicolumn{2}{c}{\large CIVIL ENGINEERING.}\\[1em]
+
+\nopagebreak
+
+\multicolumn{2}{c}{BRIDGES AND ROOFS\@. HYDRAULICS\@. MATERIALS OF
+ENGINEERING.}\\
+
+\nopagebreak
+
+\multicolumn{2}{c}{RAILWAY ENGINEERING.}\\[1em]
+
+\nopagebreak
+
+Baker's Engineers' Surveying Instruments.\dotfill\ldots 12mo, &3\ 00\\
+
+Bixby's Graphical Computing Table.\dotfill\ldots Paper $19\frac12
+\times 24\frac14$ inches &25 \\
+
+Breed and Hosmer's Principles and Practice of Surveying.\dotfill\ldots
+8vo, & 3\ 00\\
+
+* Burr's Ancient and Modern Engineering and\\
+
+\nopagebreak
+
+\indent\indent the Isthmian Canal.\dotfill\ldots 8vo, &3 50 \\
+
+Comstock's Field Astronomy for Engineers.\dotfill\ldots 8vo, &2 50 \\
+
+Crandall's Text-book on Geodesy and Least Squares.\dotfill\ldots 8vo,
+& 3\ 00\\
+
+Davis's Elevation and Stadia Tables.\dotfill\ldots 8vo, &1 00 \\
+
+Elliott's Engineering for Land Drainage.\dotfill\ldots 12mo, &1 50 \\
+
+\nopagebreak
+
+\indent Practical Farm Drainage.\dotfill\ldots 12mo, &1 00 \\
+
+* Fiebeger's Treatise on Civil Engineering.\dotfill\ldots 8vo, &5 00 \\
+
+Flemer's Phototopographic Methods and Instruments.\dotfill\ldots 8vo,
+&5 00 \\
+
+Folwell's Sewerage. (Designing and Maintenance.)\dotfill\ldots 8vo, &3 00 \\
+
+Freitag's Architectural Engineering. 2d Edition,
+Rewritten.\dotfill\ldots 8vo, &3 50 \\
+
+French and Ives's Stereotomy.\dotfill\ldots 8vo, &2 50 \\
+
+Goodhue's Municipal Improvements.\dotfill\ldots 12mo, &1 75 \\
+
+Gore's Elements of Geodesy.\dotfill\ldots 8vo, &2 50 \\
+
+Hayford's Text-book of Geodetic Astronomy.\dotfill\ldots 8vo, &3 00 \\
+
+Hering's Ready Reference Tables (Conversion Factors)\dotfill\ldots
+16mo, morocco, &2 50 \\
+
+Howe's Retaining Walls for Earth.\dotfill\ldots 12mo, &1 25 \\
+
+* Ives's Adjustments of the Engineer's Transit and
+  Level.\dotfill\ldots 16mo, Bds, &25 \\
+
+Ives and Hilts's Problems in Surveying.\dotfill\ldots 16mo, morocco,
+&1 50 \\
+
+Johnson's (J.~B.) Theory and Practice of Surveying.\dotfill\ldots
+Small 8vo, &4 00 \\
+
+Johnson's (L.~J.) Statics by Algebraic and Graphic
+Methods.\dotfill\ldots 8vo, &2 00\\
+
+Laplace's Philosophical Essay on Probabilities (Truscott\\
+
+\nopagebreak
+
+\indent\indent and Emory.)\dotfill\ldots 12mo, &2 00 \\
+
+Mahan's Treatise on Civil Engineering. (1873.) (Wood.)\dotfill\ldots
+8vo, &5 00 \\
+
+\makebox[0pt]{\hspace{.5ex} *}\indent Descriptive
+Geometry.\dotfill\ldots 8vo, &1 50 \\
+
+Merriman's Elements of Precise Surveying and Geodesy.\dotfill\ldots
+8vo, &2 50 \\
+
+Merriman and Brooks's Handbook for Surveyors.\dotfill\ldots 16mo,
+morocco, &2 00 \\
+
+Nugent's Plane Surveying.\dotfill\ldots 8vo, &3 50 \\
+
+Ogden's Sewer Design.\dotfill\ldots 12mo, &2 00 \\
+
+Parsons's Disposal of Municipal Refuse.\dotfill\ldots 8vo, &2 00 \\
+
+Patton's Treatise on Civil Engineering.\dotfill\ldots 8vo half
+leather, &7 50 \\
+
+Reed's Topographical Drawing and Sketching.\dotfill\ldots 4to, &5 00 \\
+
+Rideal's Sewage and the Bacterial Purification of
+Sewage.\dotfill\ldots 8vo, &3 50 \\
+
+Siebert and Biggin's Modern Stone-cutting and Masonry.\dotfill\ldots
+8vo, &1 50 \\
+
+%-----File: 247.png---Index 7--------
+
+
+Smith's Manual of Topographical Drawing. (McMillan.)\dotfill\ldots
+8vo, &2 50 \\
+
+Sondericker's Graphic Statics, with Applications to Trusses, Beams, \\
+
+\nopagebreak
+
+\indent\indent and Arches.\dotfill\ldots 8vo, &2 00 \\
+
+Taylor and Thompson's Treatise on Concrete, Plain and
+Reinforced.\dotfill\ldots 8vo, & 5\ 00\\
+
+* Trautwine's Civil Engineer's Pocket-book.\dotfill\ldots 16mo,
+  morocco, &5\ 00\\
+
+Venable's Garbage Crematories in America.\dotfill\ldots 8vo, &2\ 00\\
+
+Wait's Engineering and Architectural Jurisprudence.\dotfill\ldots 8vo,
+&6\ 00\\
+
+\nopagebreak
+
+\hfill Sheep, &6\ 50\\
+
+\indent Law of Operations Preliminary to Construction in Engineering\\
+
+\nopagebreak
+
+\indent\indent and Architecture.\dotfill\ldots 8vo, &5\ 00\\
+
+\nopagebreak
+
+\hfill Sheep, &5\ 50\\
+
+\indent Law of Contracts.\dotfill\ldots 8vo, &3\ 00\\
+
+Warren's Stereotomy---Problems in Stone-cutting.\dotfill\ldots 8vo,
+&2\ 50\\
+
+Webb's Problems in the Use and Adjustment\\
+
+\nopagebreak
+
+\indent\indent of Engineering Instruments.\dotfill\ldots 16mo,
+morocco, &1\ 25\\
+
+Wilson's Topographic Surveying.\dotfill\ldots 8vo, &3\ 50\\[2em]
+
+
+
+\multicolumn{2}{c}{BRIDGES AND ROOFS.}\\[1em]
+
+\nopagebreak
+
+Boller's Practical Treatise on the Construction of Iron\\
+
+\nopagebreak
+
+\indent\indent Highway Bridges.\dotfill\ldots 8vo, &2\ 00\\
+
+\makebox[0pt]{\hspace{.5ex} *}\indent Thames River
+Bridge.\dotfill\ldots 4to, paper, &5\ 00\\
+
+Burr's Course on the Stresses in Bridges and Roof Trusses, Arched Ribs,\\
+
+\nopagebreak
+
+\indent\indent and Suspension Bridges.\dotfill\ldots 8vo, &3\ 50\\
+
+Burr and Falk's Influence Lines for Bridge and Roof
+Computations.\dotfill\ldots 8vo, &3\ 00\\
+
+\nopagebreak
+
+\indent Design and Construction of Metallic Bridges.\dotfill\ldots
+8vo, &5\ 00\\
+
+Du Bois's Mechanics of Engineering. Vol.~II.\dotfill\ldots Small 4to,
+&10\ 00\\
+
+Foster's Treatise on Wooden Trestle Bridges.\dotfill\ldots 4to, &5\ 00\\
+
+Fowler's Ordinary Foundations.\dotfill\ldots 8vo, &3\ 50\\
+
+Greene's Roof Trusses.\dotfill\ldots 8vo, &1 25\\
+
+\nopagebreak
+
+\indent Bridge Trusses.\dotfill\ldots 8vo, &2\ 50\\
+
+\nopagebreak
+
+\indent Arches in Wood, Iron, and Stone.\dotfill\ldots 8vo, &2\ 50\\
+
+Howe's Treatise on Arches.\dotfill\ldots 8vo, &4\ 00\\
+
+\nopagebreak
+
+\indent Design of Simple Roof-trusses in Wood and Steel.\dotfill\ldots
+8vo, &2\ 00\\
+
+\nopagebreak
+
+\indent Symmetrical Masonry Arches.\dotfill\ldots 8vo, &2\ 50\\
+
+Johnson, Bryan, and Turneaure's Theory and Practice in the Designing\\
+
+\nopagebreak
+
+\indent\indent of Modern Framed Structures.\dotfill\ldots Small 4to,
+&10\ 00\\
+
+Merriman and Jacoby's Text-book on Roofs and Bridges:\\
+
+\indent Part I\@. Stresses in Simple Trusses.\dotfill\ldots 8vo, &2\ 50\\
+
+\indent Part II\@. Graphic Statics.\dotfill\ldots 8vo, &2\ 50\\
+
+\indent Part III\@. Bridge Design.\dotfill\ldots 8vo, &2\ 50\\
+
+\indent Part IV\@. Higher Structures.\dotfill\ldots 8vo, &2\
+50\\
+Morison's Memphis Bridge.\dotfill\ldots 4to, &10\ 00\\
+
+Waddell's De Pontibus, a Pocket-book for Bridge\\
+
+\nopagebreak
+
+\indent\indent Engineers.\dotfill\ldots 16mo, morocco, &2\ 00\\
+
+\nopagebreak
+
+\indent * Specifications for Steel Bridges.\dotfill\ldots 12mo, &\ 50\\
+
+Wright's Designing of Draw-spans. Two parts in one
+volume.\dotfill\ldots 8vo, &3\ 50\\[2em]
+
+
+
+\multicolumn{2}{c}{HYDRAULICS.}\\[1em]
+
+\nopagebreak
+
+Barnes's Ice Formation.\dotfill\ldots 8vo, &3\ 00\\
+
+Bazin's Experiments upon the Contraction of the Liquid Vein Issuing\\
+
+\nopagebreak
+
+\indent\indent from an Orifice. (Trautwine.)\dotfill\ldots 8vo, &2\ 00\\
+
+Bovey's Treatise on Hydraulics.\dotfill\ldots 8vo, &5\ 00\\
+
+Church's Mechanics of Engineering.\dotfill\ldots 8vo, &6\ 00\\
+
+\nopagebreak
+
+\indent Diagrams of Mean Velocity of Water in Open
+Channels.\dotfill\ldots paper, &1\ 50\\
+
+\nopagebreak
+
+\indent Hydraulic Motors.\dotfill\ldots 8vo, &2\ 00\\
+
+Coffin's Graphical Solution of Hydraulic Problems.\dotfill\ldots 16mo,
+morocco, &2\ 50\\
+
+Flather's Dynamometers, and the Measurement of Power.\dotfill\ldots
+12mo, &3\ 00\\
+
+%-----File: 248.png---Index 8--------
+
+
+Folwell's Water-supply Engineering.\dotfill\ldots 8vo, &4\ 00\\
+
+Frizell's Water-power.\dotfill\ldots 8vo, &5\ 00\\
+
+Fuertes's Water and Public Health.\dotfill\ldots 12mo, &1\ 50\\
+
+\nopagebreak
+
+\indent Water-filtration Works.\dotfill\ldots 12mo, &2\ 50\\
+
+Ganguillet and Kutter's General Formula for the Uniform Flow of Water\\
+
+\nopagebreak
+
+\indent\indent in Rivers and Other Channels. (Hering and
+Trautwine.)\dotfill\ldots 8vo, &4\ 00\\
+
+Hazen's Filtration of Public Water-supply.\dotfill\ldots 8vo, &3\ 00\\
+
+Hazlehurst's Towers and Tanks for Water-works.\dotfill\ldots 8vo, &2\ 50\\
+
+Herschel's 115 Experiments on the Carrying Capacity of Large, \\
+
+\nopagebreak
+
+\indent\indent Riveted, Metal Conduits.\dotfill\ldots 8vo, &2\ 00\\
+
+Mason's Water-supply. (Considered Principally from\\
+
+\nopagebreak
+
+\indent\indent a Sanitary Standpoint.)\dotfill\ldots 8vo, &4\ 00\\
+
+Merriman's Treatise on Hydraulics.\dotfill\ldots 8vo, &5\ 00\\
+
+* Michie's Elements of Analytical Mechanics.\dotfill\ldots 8vo, &4\ 00\\
+
+Schuyler's Reservoirs for Irrigation, Water-power, and Domestic\\
+
+\nopagebreak
+
+\indent\indent Water-supply.\dotfill\ldots Large 8vo, &5\ 00\\
+
+* Thomas and Watt's Improvement of Rivers.\dotfill\ldots 4to, &6\ 00\\
+
+Turneaure and Russell's Public Water-supplies.\dotfill\ldots 8vo, &5\ 00\\
+
+Wegmann's Design and Construction of Dams.\dotfill\ldots 4to, &5\ 00\\
+
+\nopagebreak
+
+\indent Water-supply of the City of New York from 1658 to
+1895.\dotfill\ldots 4to, &10\ 00\\
+
+Whipple's Value of Pure Water.\dotfill\ldots Large 12mo, & 1\ 00\\
+
+Williams and Hazen's Hydraulic Tables.\dotfill\ldots 8vo, &1\ 50\\
+
+Wilson's Irrigation Engineering.\dotfill\ldots Small 8vo, &4\ 00\\
+
+Wolff's Windmill as a Prime Mover.\dotfill\ldots 8vo, &3\ 00\\
+
+Wood's Turbines.\dotfill\ldots 8vo, &2\ 50\\
+
+\nopagebreak
+
+\indent Elements of Analytical Mechanics.\dotfill\ldots 8vo, &3\ 00\\[2em]
+
+
+
+\multicolumn{2}{c}{MATERIALS OF ENGINEERING.}\\[1em]
+
+\nopagebreak
+
+Baker's Treatise on Masonry Construction.\dotfill\ldots 8vo, &5\ 00\\
+
+\nopagebreak
+
+\indent Roads and Pavements.\dotfill\ldots 8vo, &5\ 00\\
+
+Black's United States Public Works.\dotfill\ldots Oblong 4to, &5\ 00\\
+
+* Bovey's Strength of Materials and Theory of
+  Structures.\dotfill\ldots 8vo, &7\ 50\\
+
+Burr's Elasticity and Resistance of the Materials of
+Engineering.\dotfill\ldots 8vo, &7\ 50\\
+
+Byrne's Highway Construction.\dotfill\ldots 8vo, &5\ 00\\
+
+\nopagebreak
+
+\indent Inspection of the Materials and Workmanship Employed\\
+
+\nopagebreak
+
+\indent\indent in Construction.\dotfill\ldots 16mo, &3\ 00\\
+
+Church's Mechanics of Engineering.\dotfill\ldots 8vo, &6\ 00\\
+
+Du Bois's Mechanics of Engineering. Vol.~I.\dotfill\ldots Small 4to,
+&7\ 50\\
+
+* Eckel's Cements, Limes, and Plasters.\dotfill\ldots 8vo, &6\ 00\\
+
+Johnson's Materials of Construction.\dotfill\ldots Large 8vo, &6\ 00\\
+
+Fowler's Ordinary Foundations.\dotfill\ldots 8vo, &3\ 50\\
+
+Graves's Forest Mensuration.\dotfill\ldots 8vo, &4\ 00\\
+
+* Greene's Structural Mechanics.\dotfill\ldots 8vo, &2\ 50\\
+
+Keep's Cast Iron.\dotfill\ldots 8vo, &2\ 50\\
+
+Lanza's Applied Mechanics.\dotfill\ldots 8vo, &7\ 50\\
+
+Marten's Handbook on Testing Materials. (Henning.) 2
+vols.\dotfill\ldots 8vo, &7\ 50\\
+
+Maurer's Technical Mechanics.\dotfill\ldots 8vo, &4\ 00\\
+
+Merrill's Stones for Building and Decoration.\dotfill\ldots 8vo, &5\ 00\\
+
+Merriman's Mechanics of Materials.\dotfill\ldots 8vo, &5\ 00\\
+
+\nopagebreak
+
+\makebox[0pt]{\hspace{.5ex} *}\indent Strength of
+Materials.\dotfill\ldots 12mo, &1\ 00\\
+
+Metcalf's Steel. A Manual for Steel-users.\dotfill\ldots 12mo, &2\ 00\\
+
+Patton's Practical Treatise on Foundations.\dotfill\ldots 8vo, &5\ 00\\
+
+Richardson's Modern Asphalt Pavements.\dotfill\ldots 8vo, &3\ 00\\
+
+Richey's Handbook for Superintendents of Construction.\dotfill\ldots
+16mo, mor., &4\ 00\\
+
+* Ries's Clays: Their Occurrence, Properties, and Uses.\dotfill\ldots
+  8vo, &5\ 00\\
+
+Rockwell's Roads and Pavements in France.\dotfill\ldots 12mo, &11\ 25\\
+
+%-----File: 249.png---Index 9--------
+
+
+Sabin's Industrial and Artistic Technology of Paints and
+Varnish.\dotfill\ldots 8vo, &3\ 00\\
+
+Smith's Materials of Machines.\dotfill\ldots 12mo, &1\ 00\\
+
+Snow's Principal Species of Wood.\dotfill\ldots 8vo, &3\ 50\\
+
+Spalding's Hydraulic Cement.\dotfill\ldots 12mo, &2\ 00\\
+
+\nopagebreak
+
+\indent Text-book on Roads and Pavements.\dotfill\ldots 12mo, &2\ 00\\
+
+Taylor and Thompson's Treatise on Concrete, Plain and
+Reinforced.\dotfill\ldots 8vo, &5\ 00\\
+
+Thurston's Materials of Engineering. 3 Parts.\dotfill\ldots 8vo, &8\ 00\\
+
+\nopagebreak
+
+\indent Part I.\quad Non-metallic Materials of Engineering and
+Metallurgy.\dotfill\ldots 8vo, &2\ 00\\
+
+\nopagebreak
+
+\indent Part II.\quad Iron and Steel.\dotfill\ldots 8vo, &3\ 50\\
+
+\nopagebreak
+
+\indent Part III.\quad A Treatise on Brasses, Bronzes, and Other
+Alloys and\\
+
+\nopagebreak
+
+\indent\indent their Constituents.\dotfill\ldots 8vo, &2\ 50\\
+
+Tillson's Street Pavements and Paving Materials.\dotfill\ldots 8vo, &4\ 00\\
+
+Waddell's De Pontibus (A Pocket-book for\\
+
+\nopagebreak
+
+\indent\indent Bridge Engineers.)\dotfill\ldots 16mo, mor., &2\ 00\\
+
+\nopagebreak
+
+\indent Specifications for Steel Bridges.\dotfill\ldots 12mo, &1\ 25\\
+
+Wood's (De V.) Treatise on the Resistance of Materials, and an Appendix\\
+
+\nopagebreak
+
+\indent\indent on the Preservation of Timber.\dotfill\ldots 8vo, &2\ 00\\
+
+Wood's (De V.) Elements of Analytical Mechanics.\dotfill\ldots 8vo, &3\ 00\\
+
+Wood's (M.~P.) Rustless Coatings: Corrosion and Electrolysis of Iron\\
+
+\nopagebreak
+
+\indent\indent and Steel.\dotfill\ldots 8vo, &4\ 00\\[2em]
+
+
+
+\multicolumn{2}{c}{RAILWAY ENGINEERING.}\\[1em]
+
+\nopagebreak
+
+Andrew's Handbook for Street Railway Engineers.\dotfill\ldots $3\times
+5$ inches, morocco, &1\ 25\\
+
+Berg's Buildings and Structures of American Railroads.\dotfill\ldots
+4to, &5\ 00\\
+
+Brook's Handbook of Street Railroad Location.\dotfill\ldots 16mo,
+morocco, &1\ 50\\
+
+Butt's Civil Engineer's Field-book.\dotfill\ldots 16mo, morocco, &2\ 50\\
+
+Crandall's Transition Curve.\dotfill\ldots 16mo, morocco, &1\ 50\\
+
+\nopagebreak
+
+\indent Railway and Other Earthwork Tables.\dotfill\ldots 8vo, &1\ 50\\
+
+Dawson's ``Engineering'' and Electric Traction\\
+
+\nopagebreak
+
+\indent\indent Pocket-book.\dotfill\ldots 16mo, morocco, &5\ 00\\
+
+Dredge's History ol the Pennsylvania Railroad: (1879)\dotfill\ldots
+Paper, &5\ 00\\
+
+Fisher's Table of Cubic Yards.\dotfill\ldots Cardboard, &\ 25\\
+
+Godwin's Railroad Engineers' Field-book\\
+
+\nopagebreak
+
+\indent\indent and Explorers' Guide.\dotfill\ldots 16mo, mor., &2\ 50\\
+
+Hudson's Tables for Calculating the Cubic Contents of Excavations\\
+
+\nopagebreak
+
+\indent\indent and Embankments.\dotfill\ldots 8vo, &1\ 00\\
+
+Molitor and Beard's Manual for Resident Engineers.\dotfill\ldots 16mo,
+&1\ 00\\
+
+Nagle's Field Manual for Railroad Engineers.\dotfill\ldots 16mo,
+morocco, &3\ 00\\
+
+Philbrick's Field Manual for Engineers.\dotfill\ldots 16mo, morocco,
+&3\ 00\\
+
+Searles's Field Engineering.\dotfill\ldots 16mo, morocco, &3\ 00\\
+
+\nopagebreak
+
+\indent Railroad Spiral.\dotfill\ldots 16mo, morocco, &1\ 50\\
+
+Taylor's Prismoidal Formul\ae{} and Earthwork.\dotfill\ldots 8vo, &1\ 50\\
+
+* Trautwine's Method of Calculating the Cube Contents of Excavations\\
+
+\nopagebreak
+
+\indent\indent and Embankments by the Aid of Diagrams.\dotfill\ldots
+8vo, &2\ 00\\
+
+\indent The Field Practice of Laying Out Circular Curves\\
+
+\nopagebreak
+
+\indent\indent for Railroads.\dotfill\ldots 12mo, morocco, &2\ 50\\
+
+\indent Cross-section Sheet.\dotfill\ldots Paper, &\ 25\\
+
+Webb's Railroad Construction.\dotfill\ldots 16mo, morocco, &5\ 00\\
+
+\indent Economics of Railroad Construction.\dotfill\ldots Large 12mo,
+&2\ 50\\
+
+Wellington's Economic Theory of the Location of
+Railways.\dotfill\ldots Small 8vo, &5\ 00\\[3em]
+
+
+
+\multicolumn{2}{c}{\large DRAWING.}\\[1em]
+
+\nopagebreak
+
+Barr's Kinematics of Machinery.\dotfill\ldots 8vo, &2\ 50\\
+
+* Bartlett's Mechanical Drawing.\dotfill\ldots 8vo, &3\ 00\\
+
+* \phantom{Bart}\makebox[0pt]{``}\phantom{lett's
+  Mech}\makebox[0pt]{``}\phantom{anical
+  Dra}\makebox[0pt]{``}\phantom{wing } Abridged Ed.\dotfill\ldots 8vo,
+  &1\ 50\\
+
+Coolidge's Manual of Drawing.\dotfill\ldots 8vo, paper, &1\ 00\\
+
+%-----File: 250.png---Index 10-------
+
+
+Coolidge and Freeman's Elements of General Drafting\\
+
+\nopagebreak
+
+\indent\indent for Mechanical Engineers.\dotfill\ldots Oblong 4to, &2\ 50\\
+
+Durley's Kinematics of Machines.\dotfill\ldots 8vo, &4\ 00\\
+
+Emch's Introduction to Projective Geometry and its
+Applications.\dotfill\ldots 8vo, &2\ 50\\
+
+Hill's Text-book on Shades and Shadows, and Perspective.\dotfill\ldots
+8vo, &2\ 00\\
+
+Jamison's Elements of Mechanical Drawing.\dotfill\ldots 8vo, &2\ 50\\
+
+\nopagebreak
+
+\indent Advanced Mechanical Drawing.\dotfill\ldots 8vo, &2\ 00\\
+
+Jones's Machine Design:\\
+
+\nopagebreak
+
+\indent Part I.\quad Kinematics of Machinery.\dotfill\ldots 8vo, &1\ 50\\
+
+\nopagebreak
+
+\indent Part II.\quad Form, Strength, and Proportions of
+Parts.\dotfill\ldots 8vo, &3\ 00\\
+
+MacCord's Elements of Descriptive Geometry.\dotfill\ldots 8vo, &3\ 00\\
+
+\nopagebreak
+
+\indent Kinematics; or, Practical Mechanism.\dotfill\ldots 8vo, &5\ 00\\
+
+\nopagebreak
+
+\indent Mechanical Drawing.\dotfill\ldots 4to, &4\ 00\\
+
+\nopagebreak
+
+\indent Velocity Diagrams.\dotfill\ldots 8vo, &1\ 50\\
+
+MacLeod's Descriptive Geometry.\dotfill\ldots Small 8vo, &1\ 50\\
+
+* Mahan's Descriptive Geometry and Stone-cutting.\dotfill\ldots 8vo,
+  &1\ 50\\
+
+\nopagebreak
+
+\indent Industrial Drawing. (Thompson.)\dotfill 8vo, &3\ 50\\
+
+Moyer's Descriptive Geometry.\dotfill\ldots 8vo, &2\ 00\\
+
+Reed's Topographical Drawing and Sketching.\dotfill\ldots 4to, &5\ 00\\
+
+Reid's Course in Mechanical Drawing.\dotfill\ldots 8vo, &2\ 00\\
+
+\nopagebreak
+
+\indent Text-book of Mechanical Drawing and Elementary\\
+
+\nopagebreak
+
+\indent\indent Machine Design.\dotfill\ldots 8vo, &3\ 00\\
+
+Robinson's Principles of Mechanism.\dotfill\ldots 8vo, &3\ 00\\
+
+Schwamb and Merrill's Elements of Mechanism.\dotfill\ldots 8vo, &3\ 00\\
+
+Smith's (R.~S.) Manual of Topographical Drawing. (McMillan.)\dotfill
+8vo, &2\ 50\\
+
+Smith (A.~W.) and Marx's Machine Design.\dotfill\ldots 8vo, &3\ 00\\
+
+* Titsworth's Elements of Mechanical Drawing.\dotfill\ldots Oblong
+  8vo, &1\ 25\\
+
+Warren's Elements of Plane and Solid Free-hand Geometrical\\
+
+\nopagebreak
+
+\indent\indent Drawing.\dotfill\ldots 12mo, &1\ 00\\
+
+\indent Drafting Instruments and Operations.\dotfill\ldots 12mo, &1\ 25\\
+
+\indent Manual of Elementary Projection Drawing.\dotfill\ldots 12mo,
+&1\ 50\\
+
+\indent Manual of Elementary Problems in the Linear Perspective of Form\\
+
+\nopagebreak
+
+\indent\indent and Shadow.\dotfill\ldots 12mo, &1\ 00\\
+
+\indent Plane Problems in Elementary Geometry.\dotfill\ldots 12mo, &1\ 25\\
+
+\indent Primary Geometry.\dotfill\ldots 12mo, &\ 75\\
+
+\indent Elements of Descriptive Geometry, Shadows, and
+Perspective.\dotfill\ldots 8vo, &3\ 50\\
+
+\indent General Problems of Shades and Shadows.\dotfill\ldots 8vo, &3\ 00\\
+
+\indent Elements of Machine Construction and Drawing.\dotfill\ldots
+8vo, &7\ 50\\
+
+\indent Problems, Theorems, and Examples in Descriptive
+Geometry.\dotfill\ldots 8vo, &2\ 50\\
+
+Weisbach's Kinematics and Power of Transmission. (Hermann\\
+
+\nopagebreak
+
+\indent\indent and Klein.)\dotfill\ldots 8vo, &5\ 00\\
+
+Whelpley's Practical Instruction in the Art of Letter
+Engraving.\dotfill\ldots 12mo, &2\ 00\\
+
+Wilson's (H.~M.) Topographic Surveying.\dotfill\ldots 8vo, &3\ 50\\
+
+Wilson's (V.~T.) Free-hand Perspective.\dotfill\ldots 8vo, &2\ 50\\
+
+Wilson's (V.~T.) Free-hand Lettering.\dotfill\ldots 8vo, &1\ 00\\
+
+Woolf's Elementary Course in Descriptive Geometry.\dotfill\ldots Large
+8vo, &3\ 00\\[3em]
+
+
+
+\multicolumn{2}{c}{\large ELECTRICITY AND PHYSICS.}\\[1em]
+
+\nopagebreak
+
+* Abegg's Theory of Electrolytic Dissociation. (Von
+  Ende.)\dotfill\ldots 12mo, & 1\ 25 \\
+
+Anthony and Brackett's Text-book of Physics. (Magie.)\dotfill Small
+8vo, &3\ 00\\
+
+Anthony's Lecture-notes on the Theory of Electrical\\
+
+\nopagebreak
+
+\indent\indent Measurements.\dotfill\ldots 12mo, &1\ 00\\
+
+Benjamin's History of Electricity.\dotfill\ldots 8vo, &3\ 00\\
+
+\nopagebreak
+
+\indent Voltaic Cell.\dotfill\ldots 8vo, &3\ 00\\
+
+Classen's Quantitative Chemical Analysis\\
+
+\nopagebreak
+
+\indent\indent by Electrolysis. (Boltwood.)\dotfill 8vo, &3\ 00\\
+
+* Collins's Manual of Wireless Telegraphy.\dotfill\ldots 12mo, &1\ 50\\
+
+\nopagebreak
+
+\hfill Morocco, &2\ 00\\
+
+Crehore and Squier's Polarizing Photo-chronograph.\dotfill\ldots 8vo,
+&3\ 00\\
+
+* Danneel's Electrochemistry. (Merriam.)\dotfill\ldots 12mo, & 1\ 25 \\
+
+Dawson's ``Engineering'' and Electric Traction\\
+
+\nopagebreak
+
+\indent\indent Pocket-book.\dotfill\ldots 16mo, morocco, &5\ 00\\
+
+%-----File: 251.png---Index 11-------
+
+
+Dolezalek's Theory of the Lead Accumulator (Storage Battery).\\
+
+\nopagebreak
+
+\indent\indent (Von Ende.)\dotfill\ldots 12mo, &2 50\\
+
+Duhem's Thermodynamics and Chemistry. (Burgess.)\dotfill\ldots 8vo, &4\ 00\\
+
+Flather's Dynamometers, and the Measurement of Power.\dotfill\ldots
+12mo, &3\ 00\\
+
+Gilbert's De Magnete. (Mottelay.)\dotfill\ldots 8vo, &2\ 50\\
+
+Hanchett's Alternating Currents Explained.\dotfill\ldots 12mo, &1\ 00\\
+
+Hering's Ready Reference Tables (Conversion Factors)\dotfill\ldots
+16mo, morocco, &2\ 50\\
+
+Holman's Precision of Measurements.\dotfill\ldots 8vo, &2\ 00\\
+
+\nopagebreak
+
+\indent Telescopic Mirror-scale Method, Adjustments, and
+Tests.\dotfill\ldots Large 8vo, & 75\\
+
+Kinzbrunner's Testing of Continuous-current Machines.\dotfill\ldots
+8vo, &2\ 00\\
+
+Landauer's Spectrum Analysis. (Tingle.)\dotfill\ldots 8vo, &3\ 00\\
+
+Le Chatelier's High-temperature Measurements.\\
+
+\nopagebreak
+
+\indent\indent (Boudouard---Burgess.)\dotfill\ldots 12mo, &3\ 00\\
+
+L\"ob's Electrochemistry of Organic Compounds. (Lorenz.)\dotfill\ldots
+8vo, &3 00\\
+
+* Lyons's Treatise on Electromagnetic Phenomena. Vols.~I.\\
+
+\nopagebreak
+
+\indent\indent and II.\dotfill\ldots 8vo, each, &6\ 00\\
+
+* Michie's Elements of Wave Motion Relating to Sound and
+  Light.\dotfill\ldots 8vo, &4\ 00\\
+
+Niaudet's Elementary Treatise on Electric
+Batteries. (Fishback.)\dotfill\ldots 12mo, &\ 50\\
+
+* Parshall and Hobart's Electric Machine Design.\dotfill\ldots 4to,
+  half morocco, &12\ 50\\
+
+Reagan's Locomotives: Simple, Compound, and Electric.\\
+
+\nopagebreak
+
+\indent\indent New Edition.\dotfill\ldots Large 12mo, &2 50\\
+
+* Rosenberg's Electrical Engineering. (Haldane
+  Gee---Kinzbrunner.)\dotfill\ldots 8vo, &1\ 50\\
+
+Ryan, Norris, and Hoxie's Electrical Machinery. Vol.~I.\dotfill\ldots
+8vo, &2\ 50\\
+
+Thurston's Stationary Steam-engines.\dotfill\ldots 8vo, &2\ 50\\
+
+* Tillman's Elementary Lessons in Heat.\dotfill\ldots 8vo, &1\ 50\\
+
+Tory and Pitcher's Manual of Laboratory Physics.\dotfill\ldots Small
+8vo, &2\ 00\\
+
+Ulke's Modern Electrolytic Copper Refining.\dotfill\ldots 8vo, &3\ 00\\[3em]
+
+
+
+\multicolumn{2}{c}{\large LAW.}\\[1em]
+
+\nopagebreak
+
+* Davis's Elements of Law.\dotfill\ldots 8vo, &2\ 50\\
+
+\makebox[0pt]{\hspace{.5ex} *}\indent Treatise on the Military Law of
+United States.\dotfill\ldots 8vo, &7 00\\
+
+\nopagebreak
+
+*\hfill Sheep, &7\ 50\\
+
+* Dudley's Military Law and the Procedure\\
+
+\nopagebreak
+
+\indent\indent of Courts-martial.\dotfill\ldots Large 12mo, & 2\ 50\\
+
+Manual for Courts-martial.\dotfill\ldots 16mo, morocco, &1\ 50\\
+
+Wait's Engineering and Architectural Jurisprudence.\dotfill\ldots 8vo,
+&6 00\\
+
+\nopagebreak
+
+\hfill Sheep, &6\ 50\\
+
+\indent Law of Operations Preliminary to Construction in Engineering\\
+
+\nopagebreak
+
+\indent\indent and Architecture.\dotfill\ldots 8vo, &5\ 00\\
+
+\nopagebreak
+
+\hfill Sheep, &5\ 50\\
+
+\indent Law of Contracts.\dotfill\ldots 8vo, &3\ 00\\
+
+Winthrop's Abridgment of Military Law.\dotfill\ldots 12mo, &2\ 50\\[3em]
+
+
+
+\multicolumn{2}{c}{\large MANUFACTURES.}\\[1em]
+
+\nopagebreak
+
+Bernadou's Smokeless Powder---Nitro-cellulose and Theory of\\
+
+\nopagebreak
+
+\indent\indent the Cellulose Molecule.\dotfill\ldots 12mo, &2\ 50\\
+
+Bolland's Iron Founder.\dotfill\ldots 12mo, &2\ 50\\
+
+\nopagebreak
+
+\indent \correction{``}{}The Iron Founder,'' Supplement.\dotfill\ldots 12mo, &2\ 50\\
+
+\indent Encyclopedia of Founding and Dictionary of Foundry Terms Used\\
+
+\nopagebreak
+
+\indent\indent in the Practice of Moulding.\dotfill\ldots 12mo, &3\ 00\\
+
+* Claassen's Beet-sugar Manufacture. (Hall and Rolfe.)\dotfill\ldots
+  8vo, &3\ 00\\
+
+* Eckel's Cements, Limes, and Plasters.\dotfill\ldots 8vo, &6\ 00\\
+
+Eissler's Modern High Explosives.\dotfill\ldots 8vo, &4\ 00\\
+
+Effront's Enzymes and their Applications. (Prescott.)\dotfill\ldots
+8vo, &3\ 00\\
+
+Fitzgerald's Boston Machinist.\dotfill\ldots 12mo, &1\ 00\\
+
+Ford's Boiler Making for Boiler Makers.\dotfill\ldots 18mo, &1\ 00\\
+
+Hopkin's Oil-chemists' Handbook.\dotfill\ldots 8vo, &3\ 00\\
+
+Keep's Cast Iron.\dotfill\ldots 8vo, &2\ 50\\
+
+%-----File: 252.png---Index 12-------
+
+
+Leach's The Inspection and Analysis of Food with Special Reference\\
+
+\nopagebreak
+
+\indent\indent to State Control.\dotfill\ldots Large 8vo, &7\ 50\\
+
+* McKay and Larsen's Principles and Practice of
+  Butter-making.\dotfill\ldots 8vo, &1\ 50\\
+
+Matthews's The Textile Fibres.\dotfill\ldots 8vo, &3\ 50\\
+
+Metcalf's Steel. A Manual for Steel-users.\dotfill\ldots 12mo, &2\ 00\\
+
+Metcalfe's Cost of Manufactures---And the Administration\\
+
+\nopagebreak
+
+\indent\indent of Workshops.\dotfill\ldots 8vo, &5\ 00\\
+
+Meyer's Modern Locomotive Construction.\dotfill\ldots 4to, &10\ 00\\
+
+Morse's Calculations used in Cane-sugar Factories.\dotfill\ldots 16mo,
+morocco, &1\ 50\\
+
+* Reisig's Guide to Piece-dyeing.\dotfill\ldots 8vo, &25\ 00\\
+
+Rice's Concrete-block Manufacture.\dotfill\ldots 8vo, &2\ 00\\
+
+Sabin's Industrial and Artistic Technology of Paints and
+Varnish.\dotfill\ldots 8vo, &3\ 00\\
+
+Smith's Press-working of Metals.\dotfill\ldots 8vo, &3\ 00\\
+
+Spalding's Hydraulic Cement.\dotfill\ldots 12mo, &2\ 00\\
+
+Spencer's Handbook for Chemists of Beet-sugar Houses.\dotfill\ldots
+16mo, morocco, &3\ 00\\
+
+\nopagebreak
+
+\indent Handbook for Cane Sugar Manufacturers.\dotfill\ldots 16mo,
+morocco, &3\ 00\\
+
+Taylor and Thompson's Treatise on Concrete, Plain and
+Reinforced.\dotfill\ldots 8vo, &5\ 00\\
+
+Thurston's Manual of Steam-boilers, their Designs, Construction\\
+
+\nopagebreak
+
+\indent\indent and Operation.\dotfill\ldots 8vo, &5\ 00\\
+
+* Walke's Lectures on Explosives.\dotfill\ldots 8vo, &4\ 00\\
+
+Ware's Beet-sugar Manufacture and Refining.\dotfill\ldots Small 8vo,
+&4\ 00\\
+
+Weaver's Military Explosives.\dotfill\ldots 8vo, &3\ 00\\
+
+West's American Foundry Practice.\dotfill\ldots 12mo, &2\ 50\\
+
+\nopagebreak
+
+\indent Moulder's Text-book.\dotfill\ldots 12mo, &2\ 50\\
+
+Wolff's Windmill as a Prime Mover.\dotfill\ldots 8vo, &3\ 00\\
+
+Wood's Rustless Coatings: Corrosion and Electrolysis of Iron\\
+
+\nopagebreak
+
+\indent\indent and Steel.\dotfill\ldots 8vo, &4\ 00\\[3em]
+
+
+
+\multicolumn{2}{c}{\large MATHEMATICS.}\\[1em]
+
+\nopagebreak
+
+Baker's Elliptic Functions.\dotfill\ldots 8vo, &1\ 50\\
+
+* Bass's Elements of Differential Calculus.\dotfill\ldots 12mo, &4\ 00\\
+
+Briggs's Elements of Plane Analytic Geometry.\dotfill\ldots 12mo, &1\ 00\\
+
+Compton's Manual of Logarithmic Computations.\dotfill\ldots 12mo, &1\ 50\\
+
+Davis's Introduction to the Logic of Algebra.\dotfill\ldots 8vo, &1\ 50\\
+
+* Dickson's College Algebra.\dotfill\ldots Large 12mo, &1\ 50\\
+
+\makebox[0pt]{\hspace{.5ex} *}\indent Introduction to the Theory of
+Algebraic Equations.\dotfill\ldots Large 12mo, &1\ 25\\
+
+Emch's Introduction to Projective Geometry and its
+Applications.\dotfill\ldots 8vo, &2\ 50\\
+
+Halsted's Elements of Geometry.\dotfill\ldots 8vo, &1\ 75\\
+
+\nopagebreak
+
+\indent Elementary Synthetic Geometry.\dotfill\ldots 8vo, &1\ 50\\
+
+\nopagebreak
+
+\indent Rational Geometry.\dotfill\ldots 12mo, &1 75\\
+
+* Johnson's (J.~B.) Three-place Logarithmic Tables:\\
+
+\nopagebreak
+
+\indent\indent Vest-pocket size.\dotfill\ldots paper, &\ 15\\
+
+\nopagebreak
+
+ \hfill 100 copies for &5\ 00\\
+
+\nopagebreak
+
+*\hfill Mounted on heavy cardboard, $8\times10$ inches, &\ 25\\
+
+\nopagebreak
+
+ \hfill 10 copies for &2\ 00\\
+
+Johnson's (W.~W.) Elementary Treatise on Differential\\
+
+\nopagebreak
+
+\indent\indent Calculus.\dotfill\ldots Small 8vo, &3\ 00\\
+
+\nopagebreak
+
+\indent Elementary Treatise on the Integral Calculus.\dotfill\ldots
+Small 8vo, &1\ 50\\
+
+Johnson's (W.~W.) Curve Tracing in Cartesian
+Co-ordinates.\dotfill\ldots 12mo, &1\ 00\\
+
+Johnson's (W.~W.) Treatise on Ordinary and Partial Differential\\
+
+\nopagebreak
+
+\indent\indent Equations.\dotfill\ldots Small 8vo, &3\ 50\\
+
+Johnson's (W.~W.) Theory of Errors and the Method of\\
+
+\nopagebreak
+
+\indent\indent Least Squares.\dotfill\ldots 12mo, &1\ 50\\
+
+* Johnson's (W.~W.) Theoretical Mechanics,.\dotfill\ldots 12mo, &3\ 00\\
+
+Laplace's Philosophical Essay on Probabilities. (Truscott\\
+
+\nopagebreak
+
+\indent\indent and Emory.)\dotfill\ldots 12mo, &2\ 00\\
+
+* Ludlow and Bass. Elements of Trigonometry and Logarithmic\\
+
+\nopagebreak
+
+\indent\indent and Other Tables.\dotfill\ldots 8vo, &3\ 00\\
+
+\nopagebreak
+
+\indent Trigonometry and Tables published separately.\dotfill\ldots
+Each, &2\ 00\\
+
+* Ludlow's Logarithmic and Trigonometric Tables.\dotfill\ldots 8vo, &1\ 00\\
+
+Manning's Irrational Numbers and their Representation by Sequences\\
+
+\nopagebreak
+
+\indent\indent and Series.\dotfill\ldots 12mo &1\ 25\\
+
+%-----File: 253.png---Index 13-------
+
+
+Mathematical Monographs. Edited by Mansfield Merriman and Robert \\
+
+\indent\indent S.~Woodward.\dotfill\ldots Octavo, each &1\ 00\\
+
+\indent
+
+\begin{minipage}{.8\textwidth}
+
+No.~1. History of Modern Mathematics, by David Eugene Smith.\quad
+
+No.~2. Synthetic Projective Geometry, by George Bruce Halsted.\quad
+
+No.~3. Determinants, by Laenas Gifford Weld.\quad No.~4. Hyperbolic
+
+Functions, by James McMahon.\quad No.~5. Harmonic Functions,
+
+by William E. Byerly.\quad No.~6. Grassmann's Space Analysis,
+
+by Edward W. Hyde.\quad No.~7. Probability and Theory of Errors,
+
+by Robert S. Woodward.\quad No.~8. Vector Analysis and Quaternions,
+
+by Alexander Macfarlane.\quad No.~9. Differential Equations, by
+
+William Woolsey Johnson.\quad No.~10. The Solution of Equations,
+
+by Mansfield Merriman.\quad No.~11. Functions of a Complex Variable,
+
+by Thomas S. Fiske.
+
+\end{minipage}\\
+
+Maurer's Technical Mechanics.\dotfill\ldots 8vo, &4\ 00\\
+
+Merriman's Method of Least Squares.\dotfill\ldots 8vo, &2\ 00\\
+
+Rice and Johnson's Elementary Treatise on the Differential\\
+
+\nopagebreak
+
+\indent\indent Calculus.\dotfill\ldots Sm. 8vo, &3\ 00\\
+
+\indent Differential and Integral Calculus. 2 vols.\ in
+one.\dotfill\ldots Small 8vo, &2\ 50\\
+
+* Veblen and Lennes's Introduction to the Real Infinitesimal Analysis\\
+
+\indent\indent of One Variable.\dotfill\ldots 8vo, & 2\ 00\\
+
+Wood's Elements of Co-ordinate Geometry.\dotfill\ldots 8vo, &2\ 00\\
+
+\indent Trigonometry: Analytical, Plane, and Spherical.\dotfill\ldots
+12mo, &1\ 00\\[3em]
+
+
+
+\multicolumn{2}{c}{\large MECHANICAL ENGINEERING.}\\[1em]
+
+\nopagebreak
+
+\multicolumn{2}{c}{MATERIALS OF ENGINEERING, STEAM-ENGINES AND BOILERS.}\\[1em]
+
+\nopagebreak
+
+Bacon's Forge Practice.\dotfill\ldots 12mo, &1\ 50\\
+
+Baldwin's Steam Heating for Buildings.\dotfill\ldots 12mo, &2\ 50\\
+
+Barr's Kinematics of Machinery.\dotfill\ldots 8vo, &2\ 50\\
+
+* Bartlett's Mechanical Drawing.\dotfill\ldots 8vo, &3\ 00\\
+
+* \phantom{Bart}\makebox[0pt]{``}\phantom{lett's
+  Mech}\makebox[0pt]{``}\phantom{anical
+  Dra}\makebox[0pt]{``}\phantom{wing } Abridged Ed.\dotfill\ldots 8vo,
+  &1\ 50\\
+
+Benjamin's Wrinkles and Recipes.\dotfill\ldots 12mo, &2\ 00\\
+
+Carpenter's Experimental Engineering.\dotfill\ldots 8vo, &6\ 00\\
+
+\indent Heating and Ventilating Buildings.\dotfill\ldots 8vo, &4\ 00\\
+
+Clerk's Gas and Oil Engine.\dotfill\ldots Small 8vo, &4\ 00\\
+
+Coolidge's Manual of Drawing.\dotfill\ldots 8vo, paper, &1\ 00\\
+
+Coolidge and Freeman's Elements of General Drafting\\
+
+\indent\indent for Mechanical Engineers.\dotfill\ldots Oblong 4to, &2\ 50\\
+
+Cromwell's Treatise on Toothed Gearing.\dotfill\ldots 12mo, &1\ 50\\
+
+\indent Treatise on Belts and Pulleys.\dotfill\ldots 12mo, &1\ 50\\
+
+Durley's Kinematics of Machines.\dotfill\ldots 8vo, &4\ 00\\
+
+Flather's Dynamometers and the Measurement of Power.\dotfill\ldots
+12mo, &3\ 00\\
+
+\indent Rope Driving.\dotfill\ldots 12mo, &2\ 00\\
+
+Gill's Gas and Fuel Analysis for Engineers.\dotfill\ldots 12mo, &1\ 25\\
+
+Hall's Car Lubrication.\dotfill\ldots 12mo, &1\ 00\\
+
+Hering's Ready Reference Tables (Conversion Factors)\dotfill\ldots
+16mo, morocco, &2\ 50\\
+
+Hutton's The Gas Engine.\dotfill\ldots 8vo, &5\ 00\\
+
+Jamison's Mechanical Drawing.\dotfill\ldots 8vo, &2 50\\
+
+Jones's Machine Design:\\
+
+\indent Part I.\quad Kinematics of Machinery.\dotfill\ldots 8vo, &1\
+50\\
+
+\indent Part II.\quad Form, Strength, and Proportions of
+Parts.\dotfill\ldots 8vo, &3\ 00\\
+
+Kent's Mechanical Engineers' Pocket-book.\dotfill\ldots 16mo, morocco,
+&5\ 00\\
+
+Kerr's Power and Power Transmission.\dotfill\ldots 8vo, &2\ 00\\
+
+Leonard's Machine Shop, Tools, and Methods.\dotfill\ldots 8vo, &4\ 00\\
+
+* Lorenz's Modern Refrigerating Machinery. (Pope, Haven,\\
+
+\indent\indent and Dean.)\dotfill\ldots 8vo, &4\ 00\\
+
+MacCord's Kinematics; or Practical Mechanism.\dotfill\ldots 8vo, &5\ 00\\
+
+\indent Mechanical Drawing.\dotfill\ldots 4to, &4\ 00\\
+
+\indent Velocity Diagrams.\dotfill\ldots 8vo, &1\ 50\\
+
+%-----File: 254.png---Index 14-------
+
+
+MacFarland's Standard Reduction Factors for Gases.\dotfill\ldots 8vo,
+&1\ 50\\
+
+Mahan's Industrial Drawing. (Thompson.)\dotfill\ldots 8vo, &3\ 50\\
+
+Poole's Calorific Power of Fuels.\dotfill\ldots 8vo, &3\ 00\\
+
+Reid's Course in Mechanical Drawing.\dotfill\ldots 8vo, &2\ 00\\
+
+\indent Text-book of Mechanical Drawing and Elementary\\
+
+\nopagebreak
+
+\indent\indent Machine Design.\dotfill\ldots 8vo, &3\ 00\\
+
+Richard's Compressed Air.\dotfill\ldots 12mo, &1 50\\
+
+Robinson's Principles of Mechanism.\dotfill\ldots 8vo, &3\ 00\\
+
+Schwamb and Merrill's Elements of Mechanism.\dotfill\ldots 8vo, &3\ 00\\
+
+Smith's (O.) Press-working of Metals.\dotfill\ldots 8vo, &3\ 00\\
+
+Smith (A.~W.) and Marx's Machine Design.\dotfill\ldots 8vo, &3\ 00\\
+
+Thurston's Treatise on Friction and Lost Work in Machinery\\
+
+\nopagebreak
+
+\indent\indent and Mill Work.\dotfill\ldots 8vo, &3\ 00\\
+
+\indent Animal as a Machine and Prime Motor, and the Laws\\
+
+\nopagebreak
+
+\indent\indent of Energetics.\dotfill\ldots 12mo, &1\ 00\\
+
+Tillson's Complete Automobile Instructor.\dotfill\ldots 16mo, & 1\ 50\\
+
+\nopagebreak
+
+\hfill Morocco, & 2\ 00\\
+
+Warren's Elements of Machine Construction and Drawing.\dotfill\ldots
+8vo, &7\ 50\\
+
+Weisbach's Kinematics and the Power of Transmission. \\
+
+\nopagebreak
+
+\indent\indent (Herrmann---Klein.)\dotfill\ldots 8vo, &5\ 00\\
+
+\indent Machinery of Transmission and
+Governors. (Herrmann---Klein.)\dotfill\ldots 8vo, &5\ 00\\
+
+Wolff's Windmill as a Prime Mover.\dotfill\ldots 8vo, &3\ 00\\
+
+Wood's Turbines.\dotfill\ldots 8vo, &2\ 50\\[2em]
+
+
+
+\multicolumn{2}{c}{MATERIALS OF ENGINEERING.}\\[1em]
+
+\nopagebreak
+
+* Bovey's Strength of Materials and Theory of
+  Structures.\dotfill\ldots 8vo, &7\ 50\\
+
+Burr's Elasticity and Resistance of the Materials of Engineering.\\
+
+\nopagebreak
+
+\indent\indent 6th Edition. Reset.\dotfill\ldots 8vo, &7\ 50\\
+
+Church's Mechanics of Engineering.\dotfill\ldots 8vo, &6\ 00\\
+
+* Greene's Structural Mechanics.\dotfill\ldots 8vo, &2\ 50\\
+
+Johnson's Materials of Construction.\dotfill\ldots 8vo, &6\ 00\\
+
+Keep's Cast Iron.\dotfill\ldots 8vo, &2\ 50\\
+
+Lanza's Applied Mechanics.\dotfill\ldots 8vo, &7\ 50\\
+
+Martens's Handbook on Testing Materials. (Henning.)\dotfill\ldots 8vo,
+&7\ 50\\
+
+Maurer's Technical Mechanics.\dotfill\ldots 8vo, &4\ 00\\
+
+Merriman's Mechanics of Materials.\dotfill\ldots 8vo, &5\ 00\\
+
+\makebox[0pt]{\hspace{.5ex} *}\indent Strength of
+Materials.\dotfill\ldots 12mo, &1\ 00\\
+
+Metcalf's Steel. A Manual for Steel-users.\dotfill\ldots 12mo, &2\ 00\\
+
+Sabin's Industrial and Artistic Technology of Paints and
+Varnish.\dotfill\ldots 8vo, &3\ 00\\
+
+Smith's Materials of Machines.\dotfill\ldots 12mo, &1\ 00\\
+
+Thurston's Materials of Engineering.\dotfill\ldots 3 vols., 8vo, &8\ 00\\
+
+\indent Part II.\quad Iron and Steel.\dotfill\ldots 8vo, &3\ 50\\
+
+\indent Part III.\quad A Treatise on Brasses, Bronzes, and Other
+Alloys and\\
+
+\nopagebreak
+
+\indent\indent their Constituents.\dotfill\ldots 8vo, &2\ 50\\
+
+Wood's (De V.) Treatise on the Resistance of Materials and an Appendix\\
+
+\nopagebreak
+
+\indent\indent on the Preservation of Timber.\dotfill\ldots 8vo, &2\ 00\\
+
+\indent Elements of Analytical Mechanics.\dotfill\ldots 8vo, &3\ 00\\
+
+Wood's (M.~P.) Rustless Coatings: Corrosion and Electrolysis of Iron\\
+
+\indent\indent and Steel.\dotfill\ldots 8vo, &4\ 00\\[2em]
+
+
+
+\multicolumn{2}{c}{STEAM-ENGINES AND BOILERS.}\\[1em]
+
+\nopagebreak
+
+Berry's Temperature-entropy Diagram.\dotfill\ldots 12mo, &1\ 25\\
+
+Carnot's Reflections on the Motive Power of Heat
+(Thurston.)\dotfill\ldots 12mo, &1\ 50\\
+
+Dawson's ``Engineering'' and Electric Traction
+Pocket-book.\dotfill\ldots 16mo mor., &5\ 00\\
+
+Ford's Boiler Making for Boiler Makers.\dotfill\ldots 18mo, &1\ 00\\
+
+Goss's Locomotive Sparks.\dotfill\ldots 8vo, &2\ 00\\
+
+\indent Locomotive Performance.\dotfill\ldots 8vo, & 5\ 00\\
+
+Hemenway's Indicator Practice and Steam-engine Economy.\dotfill\ldots
+12mo, &2\ 00\\
+
+%-----File: 255.png---Index 15-------
+
+
+Hutton's Mechanical Engineering of Power Plants.\dotfill\ldots 8vo, &5\ 00\\
+
+\indent Heat and Heat-engines.\dotfill\ldots 8vo, &5\ 00\\
+
+Kent's Steam boiler Economy.\dotfill\ldots 8vo, &4\ 00\\
+
+Kneass's Practice and Theory of the Injector.\dotfill\ldots 8vo, &1\ 50\\
+
+MacCord's Slide-valves.\dotfill\ldots 8vo, &2\ 00\\
+
+Meyer's Modern Locomotive Construction.\dotfill\ldots 4to, &10\ 00\\
+
+Peabody's Manual of the Steam-engine Indicator.\dotfill\ldots 12mo, &1\ 50\\
+
+\indent Tables of the Properties of Saturated Steam and Other
+Vapors.\dotfill\ldots 8vo, &1\ 00\\
+
+\indent Thermodynamics of the Steam-engine and Other
+Heat-engines.\dotfill\ldots 8vo, &5\ 00\\
+
+\indent Valve-gears for Steam-engines.\dotfill\ldots 8vo, &2\ 50\\
+
+Peabody and Miller's Steam-boilers.\dotfill\ldots 8vo, &4\ 00\\
+
+Pray's Twenty Years with the Indicator.\dotfill\ldots Large 8vo, &2\ 50\\
+
+Pupin's Thermodynamics of Reversible Cycles in Gases and\\
+
+\nopagebreak
+
+\indent\indent Saturated Vapors. (Osterberg.)\dotfill\ldots 12mo, &1\ 25\\
+
+Reagan's Locomotives: Simple, Compound,\\
+
+\nopagebreak
+
+\indent\indent and Electric.\dotfill\ldots Large 12mo, &2\ 50\\
+
+Rontgen's Principles of Thermodynamics. (Du Bois.)\dotfill\ldots 8vo,
+&5\ 00\\
+
+Sinclair's Locomotive Engine Running and Management.\dotfill\ldots
+12mo, &2\ 00\\
+
+Smart's Handbook of Engineering Laboratory Practice.\dotfill\ldots
+12mo, &2\ 50\\
+
+Snow's Steam-boiler Practice.\dotfill\ldots 8vo, &3\ 00\\
+
+Spangler's Valve-gears.\dotfill\ldots 8vo, &2\ 50\\
+
+\indent Notes on Thermodynamics.\dotfill\ldots 12mo, &1\ 00\\
+
+Spangler, Greene, and Marshall's Elements of
+Steam-engineering.\dotfill\ldots 8vo, &3\ 00\\
+
+Thomas's Steam-turbines.\dotfill\ldots 8vo, &3\ 50\\
+
+Thurston's Handy Tables.\dotfill\ldots 8vo, &1\ 50\\
+
+\indent Manual of the Steam-engine.\dotfill\ldots 2 vols., 8vo, &10\ 00\\
+
+\indent Part I.\quad History, Structure, and Theory.\dotfill\ldots
+8vo, &6\ 00\\
+
+\indent Part II.\quad Design, Construction, and
+Operation.\dotfill\ldots 8vo, &6\ 00\\
+
+\indent Handbook of Engine and Boiler Trials, and the Use of the Indicator\\
+
+\nopagebreak
+
+\indent\indent and the Prony Brake.\dotfill\ldots 8vo, &5\ 00\\
+
+\indent Stationary Steam-engines.\dotfill\ldots 8vo, &2\ 50\\
+
+\indent Steam-boiler Explosions in Theory and in
+Practice.\dotfill\ldots 12mo, &1\ 50\\
+
+\indent Manual of Steam-boilers, their Designs, Construction,\\
+
+\nopagebreak
+
+\indent\indent and Operation.\dotfill\ldots 8vo, &5\ 00\\
+
+\correction{Wehrenfennigs's}{Wehrenfenning's} Analysis and Softening of Boiler\\
+
+\nopagebreak
+
+\indent\indent Feed-water (Patterson)\dotfill\ldots 8vo, &4\ 00\\
+
+Weisbach's Heat, Steam, and Steam-engines. (Du Bois.)\dotfill\ldots
+8vo, &5\ 00\\
+
+Whitham's Steam-engine Design.\dotfill\ldots 8vo, &5\ 00\\
+
+Wood's Thermodynamics, Heat Motors,\\
+
+\nopagebreak
+
+\indent\indent and Refrigerating Machines.\dotfill\ldots 8vo, &4\ 00\\[3em]
+
+
+
+\multicolumn{2}{c}{\large MECHANICS AND MACHINERY.}\\[1em]
+
+\nopagebreak
+
+Barr's Kinematics of Machinery.\dotfill\ldots 8vo, &2\ 50\\
+
+* Bovey's Strength of Materials and Theory of
+  Structures.\dotfill\ldots 8vo, &7\ 50\\
+
+Chase's The Art of Pattern-making.\dotfill\ldots 12mo, &2\ 50\\
+
+Church's Mechanics of Engineering.\dotfill\ldots 8vo, &6\ 00\\
+
+\indent Notes and Examples in Mechanics.\dotfill\ldots 8vo, &2\ 00\\
+
+Compton's First Lessons in Metal-working.\dotfill\ldots 12mo, &1\ 50\\
+
+Compton and De Groodt's The Speed Lathe.\dotfill\ldots 12mo, &1\ 50\\
+
+Cromwell's Treatise on Toothed Gearing.\dotfill\ldots 12mo, &1\ 50\\
+
+\indent Treatise on Belts and Pulleys.\dotfill\ldots 12mo, &1\ 50\\
+
+Dana's Text-book of Elementary Mechanics for Colleges\\
+
+\nopagebreak
+
+\indent\indent and Schools.\dotfill\ldots 12mo, & 1\ 50\\
+
+Dingey's Machinery Pattern Making.\dotfill\ldots 12mo, &2\ 00\\
+
+Dredge's Record of the Transportation Exhibits Building of\\
+
+\nopagebreak
+
+\indent\indent the World's Columbian Exposition of 1893.\dotfill\ldots
+4to half morocco, &5\ 00\\
+
+Du Bois's Elementary Principles of Mechanics:\\
+
+\indent Vol.~\phantom{I}I.\quad Kinematics.\dotfill\ldots 8vo. &3\ 50\\
+
+\indent Vol.~II.\quad Statics.\dotfill\ldots 8vo, &4\ 00\\
+
+\indent Mechanics of Engineering. Vol.~I.\dotfill\ldots Small 4to, &7\ 50\\
+
+\indent \phantom{Mechanics of Engineering. }Vol.~II.\dotfill\ldots
+Small 4to, &10\ 00\\
+
+Durley's Kinematics of Machines.\dotfill\ldots 8vo. &4\ 00\\
+
+%-----File: 256.png---Index 16-------
+
+
+Fitzgerald's Boston Machinist.\dotfill\ldots 16mo, & 1\ 00\\
+
+Flather's Dynamometers, and the Measurement of Power.\dotfill\ldots
+12mo, & 3\ 00\\
+
+\indent Rope Driving.\dotfill\ldots 12mo, & 2\ 00\\
+
+Goss's Locomotive Sparks.\dotfill\ldots 8vo, & 2\ 00\\
+
+Locomotive Performance.\dotfill\ldots 8vo, & 5\ 00\\
+
+\correction{}{\indent}* Greene's Structural Mechanics.\dotfill\ldots 8vo, & 2\ 50\\
+
+Hall's Car Lubrication.\dotfill\ldots 12mo, & 1\ 00 \\
+
+Holly's Art of Saw Filing.\dotfill\ldots 18mo, & \ 75\\
+
+James's Kinematics of a Point and the Rational Mechanics\\
+
+\nopagebreak
+
+\indent\indent of a Particle.\dotfill\ldots Small 8vo, & 2\ 00\\
+
+* Johnson's (W.~W.) Theoretical Mechanics.\dotfill\ldots 12mo, & 3\ 00\\
+
+Johnson's (L.~J.) Statics by Graphic and Algebraic
+Methods.\dotfill\ldots 8vo, & 2\ 00\\
+
+Jones's Machine Design:\\
+
+\indent Part~\phantom{I}I.\quad Kinematics of Machinery.\dotfill\ldots
+8vo, & 1\ 50\\
+
+\indent Part~II.\quad Form, Strength, and Proportions of
+Parts.\dotfill\ldots 8vo, & 3\ 00\\
+
+Kerr's Power and Power Transmission.\dotfill\ldots 8vo, & 2\ 00\\
+
+Lanza's Applied Mechanics.\dotfill\ldots 8vo, & 7\ 50\\
+
+Leonard's Machine Shop, Tools, and Methods.\dotfill\ldots 8vo, & 4\ 00\\
+
+* Lorenz's Modern Refrigerating Machinery. (Pope, Haven,\\
+
+\nopagebreak
+
+\indent\indent and Dean.)\dotfill\ldots 8vo, & 4\ 00\\
+
+MacCord's Kinematics; or, Practical Mechanism.\dotfill\ldots 8vo, & 5\ 00\\
+
+\indent Velocity Diagrams.\dotfill\ldots 8vo, & 1\ 50\\
+
+* Martin's Text Book on Mechanics, Vol.~I, Statics.\dotfill\ldots
+  12mo, & 1\ 25\\
+
+Maurer's Technical Mechanics.\dotfill\ldots 8vo, & 4\ 00\\
+
+Merriman's Mechanics of Materials.\dotfill\ldots 8vo, & 5\ 00\\
+
+\makebox[0pt]{\hspace{.5ex} *}\indent Elements of
+Mechanics.\dotfill\ldots 12mo, & 1\ 00\\
+
+* Michie's Elements of Analytical Mechanics.\dotfill\ldots 8vo, & 4\ 00\\
+
+* Parshall and Hobart's Electric Machine Design.\dotfill\ldots 4to,
+  half morocco, & 12\ 50\\
+
+Reagan's Locomotives: Simple, Compound,\\
+
+\nopagebreak
+
+\indent\indent and Electric. New Edition.\dotfill\ldots Large 12mo, &
+3\ 00\\
+
+Reid's Course in Mechanical Drawing.\dotfill\ldots 8vo, & 2\ 00\\
+
+\indent Text-book of Mechanical Drawing and Elementary\\
+
+\nopagebreak
+
+\indent\indent Machine Design.\dotfill\ldots 8vo, & 3\ 00\\
+
+Richards's Compressed Air.\dotfill\ldots 12mo, & 1\ 50\\
+
+Robinson's Principles of Mechanism.\dotfill\ldots 8vo, & 3\ 00\\
+
+Ryan, Norris, and Hoxie's Electrical Machinery. Vol.~I.\dotfill\ldots
+8vo, & 2\ 50\\
+
+Sanborn's Mechanics: Problems.\dotfill\ldots Large 12mo, & 1\ 50\\
+
+Schwamb and Merrill's Elements of Mechanism.\dotfill\ldots 8vo, & 3\ 00\\
+
+Sinclair's Locomotive-engine Running and Management.\dotfill\ldots
+12mo, & 2\ 00\\
+
+Smith's (O.) Press-working of Metals.\dotfill\ldots 8vo, & 3\ 00\\
+
+Smith's (A.~W.) Materials of Machines.\dotfill\ldots 12mo, & 1\ 00\\
+
+Smith (A.~W.) and Marx's Machine Design.\dotfill\ldots 8vo, & 3\ 00\\
+
+Spangler, Greene, and Marshall's Elements of
+Steam-engineering.\dotfill\ldots 8vo, & 3\ 00\\
+
+Thurston's Treatise on Friction and Lost Work in Machinery\\
+
+\nopagebreak
+
+\indent\indent and Mill Work.\dotfill\ldots 8vo, & 3\ 00\\
+
+\indent Animal as a Machine and Prime Motor, and the Laws\\
+
+\nopagebreak
+
+\indent\indent of Energetics.\dotfill\ldots 12mo, & 1\ 00\\
+
+Tillson's Complete Automobile Instructor.\dotfill\ldots 16mo, & 1\ 50\\
+
+\hfill Morocco, & 2\ 00\\
+
+Warren's Elements of Machine Construction and Drawing.\dotfill\ldots
+8vo, & 7\ 50\\
+
+Weisbach's Kinematics and Power of Transmission.\\
+
+\nopagebreak
+
+\indent\indent (Herrmann---Klein.)\dotfill\ldots 8vo, & 5\ 00\\
+
+\indent Machinery of Transmission and Governors.\\
+
+\nopagebreak
+
+\indent\indent (Herrmann---Klein.)\dotfill\ldots 8vo, & 5\ 00\\
+
+Wood's Elements of Analytical Mechanics.\dotfill\ldots 8vo, & 3\ 00\\
+
+\indent Principles of Elementary Mechanics.\dotfill\ldots 12mo, & 1\ 25\\
+
+\indent Turbines.\dotfill\ldots 8vo, & 2\ 50\\
+
+The World's Columbian Exposition of 1893.\dotfill\ldots 4to, & 1\ 00\\[3em]
+
+
+
+\multicolumn{2}{c}{\large MEDICAL.}\\[1em]
+
+\nopagebreak
+
+De Fursac's Manual of Psychiatry. (Rosanoff and
+Collins.)\dotfill\ldots Large 12mo, & 2\ 50\\
+
+Ehrlich's Collected Studies on Immunity. (Bolduan.)\dotfill\ldots 8vo,
+& 6\ 00\\
+
+Hammarsten's Text-book on Physiological
+Chemistry. (Mandel.)\dotfill\ldots 8vo, & 4\ 00\\
+
+%-----File: 257.png---Index 17-------
+
+
+Lassar-Cohn's Practical Urinary Analysis. (Lorenz.)\dotfill\ldots
+12mo, & 1\ 00\\
+
+* Pauli's Physical Chemistry in the Service\\
+
+\nopagebreak
+
+\indent\indent of Medicine. (Fischer.)\dotfill\ldots 12mo, & 1\ 25\\
+
+* Pozzi-Escot's The Toxins and Venoms and\\
+
+\nopagebreak
+
+\indent\indent their Antibodies. (Cohn.)\dotfill\ldots 12mo, & 1\ 00\\
+
+Rostoski's Serum Diagnosis. (Bolduan.)\dotfill\ldots 12mo, & 1\ 00\\
+
+Salkowski's Physiological and Pathological
+Chemistry. (Orndorff.)\dotfill\ldots 8vo, & 2\ 50\\
+
+* Satterlee's Outlines of Human Embryology.\dotfill\ldots 12mo, & 1\ 25\\
+
+Steel's Treatise on the Diseases of the Dog.\dotfill\ldots 8vo, & 3\ 50\\
+
+Von Behring's Suppression of Tuberculosis. (Bolduan.)\dotfill\ldots
+12mo, & 1\ 00\\
+
+Wassermann's Immune Sera: H\ae{}molysis, Cytotoxins,\\
+
+\nopagebreak
+
+\indent\indent and Precipitins.\\
+
+\indent\indent (Bolduan.)\dotfill\ldots 12mo, cloth, & 1\ 00\\
+
+Woodhull's Notes on Military Hygiene.\dotfill\ldots 16mo, & 1\ 50\\
+
+\makebox[0pt]{\hspace{.5ex} *}\indent Personal Hygiene.\dotfill\ldots
+12mo, & 1\ 00\\
+
+Wulling's An Elementary Course in Inorganic Pharmaceutical\\
+
+\indent\indent and Medical Chemistry.\dotfill\ldots 12mo, & 2\ 00\\[3em]
+
+
+
+\multicolumn{2}{c}{\large METALLURGY.}\\[1em]
+
+\nopagebreak
+
+Egleston's Metallurgy of Silver, Gold, and Mercury:\\
+
+\indent Vol.~\phantom{I}I.\quad Silver.\dotfill\ldots 8vo, & 7\ 50\\
+
+\indent Vol.~II.\quad Gold and Mercury.\dotfill\ldots 8vo, & 7\ 50\\
+
+Goesel's Minerals and Metals: A Reference Book.\dotfill\ldots 16mo,
+mor. & 3\ 00\\
+
+* Iles's Lead-smelting.\dotfill\ldots 12mo, & 2\ 50\\
+
+Keep's Cast Iron.\dotfill\ldots 8vo, & 2\ 50\\
+
+Kunhardt's Practice of Ore Dressing in Europe.\dotfill\ldots 8vo, & 1\ 50\\
+
+Le Chatelier's High-temperature Measurements.\\
+
+\nopagebreak
+
+\indent\indent (Boudouard---Burgess.)\dotfill\ldots 12mo, & 3\ 00\\
+
+Metcalf's Steel. A Manual for Steel-users.\dotfill\ldots 12mo, & 2\ 00\\
+
+Miller's Cyanide Process.\dotfill\ldots 12mo, & 1\ 00\\
+
+Minet's Production of Aluminum and its Industrial
+Use. (Waldo.)\dotfill\ldots 12mo, & 2\ 50\\
+
+Robine and Lenglen's Cyanide Industry. (Le Clerc.)\dotfill\ldots 8vo,
+& 4\ 00\\
+
+Smith's Materials of Machines.\dotfill\ldots 12mo, & 1\ 00\\
+
+Thurston's Materials of Engineering. In Three Parts.\dotfill\ldots
+8vo, & 8\ 00\\
+
+\indent Part~II.\quad Iron and Steel.\dotfill\ldots 8vo, & 3\ 50\\
+
+\indent Part~III.\quad A Treatise on Brasses, Bronzes, and Other Alloys\\
+
+\indent\indent and their Constituents.\dotfill\ldots 8vo, & 2\ 50\\
+
+Ulke's Modern Electrolytic Copper Refining.\dotfill\ldots 8vo, & 3\
+00\\[3em]
+
+
+
+\multicolumn{2}{c}{\large MINERALOGY.}\\[1em]
+
+\nopagebreak
+
+Barringer's Description of Minerals\\
+
+\nopagebreak
+
+\indent\indent of Commercial Value.\dotfill\ldots Oblong, morocco, & 2\ 50\\
+
+Boyd's Resources of Southwest Virginia.\dotfill\ldots 8vo, & 3\ 00\\
+
+\indent Map of Southwest \correction{Virginia}{Virignia}.\dotfill\ldots Pocket-book form, &
+2\ 00\\
+
+Brush's Manual of Determinative Mineralogy. (Penfield.)\dotfill\ldots
+8vo, & 4\ 00\\
+
+Chester's Catalogue of Minerals.\dotfill\ldots 8vo, paper, & 1\ 00\\
+
+\hfill Cloth, & 1\ 25\\
+
+\indent Dictionary of the Names of Minerals.\dotfill\ldots 8vo, & 3\ 50\\
+
+Dana's System of Mineralogy.\dotfill\ldots Large 8vo, half leather, &
+12\ 50\\
+
+\indent First Appendix to Dana's New ``System of
+Mineralogy.''\dotfill\ldots Large 8vo, & 1\ 00\\
+
+\indent Text-book of Mineralogy.\dotfill\ldots 8vo, & 4\ 00\\
+
+\indent Minerals and How to Study Them.\dotfill\ldots 12mo, & 1\ 50\\
+
+\indent Catalogue of American Localities of Minerals.\dotfill\ldots
+Large 8vo, & 1\ 00\\
+
+\indent Manual of Mineralogy and Petrography.\dotfill\ldots 12mo, & 2\ 00\\
+
+Douglas's Untechnical Addresses on Technical Subjects.\dotfill\ldots
+12mo, & 1\ 00\\
+
+Eakle's Mineral Tables.\dotfill\ldots 8vo, & 1\ 25\\
+
+Egleston's Catalogue of Minerals and Synonyms.\dotfill\ldots 8vo, & 2\ 50\\
+
+Goesel's Minerals and Metals: A Reference Book.\dotfill\ldots 16mo,
+mor. & 3\ 00\\
+
+Groth's Introduction to Chemical Crystallography
+(Marshall)\dotfill\ldots 12mo, & 1\ 25\\
+
+%-----File: 258.png---Index 18-------
+
+
+Iddings's Rock Minerals.\dotfill\ldots 8vo, & 5\ 00\\
+
+Merrill's Non-metallic Minerals: Their Occurrence and
+Uses.\dotfill\ldots 8vo, & 4\ 00\\
+
+* Penfield's Notes on Determinative Mineralogy and Record\\
+
+\nopagebreak
+
+\indent\indent of Mineral Tests.\dotfill\ldots 8vo, paper, & \ 50\\
+
+* Richards's Synopsis of Mineral Characters.\dotfill\ldots 12mo,
+  morocco, & 1\ 25\\
+
+* Ries's Clays: Their Occurrence, Properties, and Uses.\dotfill\ldots
+  8vo, & 5 00 \\
+
+Rosenbusch's Microscopical Physiography of\\
+
+\nopagebreak
+
+\indent\indent the Rock-making Minerals. (Iddings.)\dotfill\ldots 8vo,
+& 5\ 00\\
+
+* Tillman's Text-book of Important Minerals and Rocks.\dotfill\ldots
+  8vo, & 2\ 00\\[3em]
+
+
+
+\multicolumn{2}{c}{\large MINING.}\\[1em]
+
+\nopagebreak
+
+Boyd's Resources of Southwest Virginia.\dotfill\ldots 8vo, & 3\ 00\\
+
+\indent Map of Southwest Virginia.\dotfill\ldots Pocket-book form, & 2\ 00\\
+
+Douglas's Untechnical Addresses on Technical Subjects.\dotfill\ldots
+12mo, & 1\ 00\\
+
+Eissler's Modern High Explosives.\dotfill\ldots 8vo, & 4\ 00\\
+
+Goesel's Minerals and Metals: A Reference Book.\dotfill\ldots 16mo,
+mor. & 3\ 00\\
+
+Goodyear's Coal-mines of the Western Coast of the United
+States.\dotfill\ldots 12mo, & 2\ 50\\
+
+Ihlseng's Manual of Mining.\dotfill\ldots 8vo, & 5\ 00\\
+
+* Iles's Lead-smelting.\dotfill\ldots 12mo, & 2\ 50\\
+
+Kunhardt's Practice of Ore Dressing In Europe.\dotfill\ldots 8vo, & 1\ 50\\
+
+Miller's Cyanide Process.\dotfill\ldots 12mo, & 1\ 00\\
+
+O'Driscoll's Notes on the Treatment of Gold Ores.\dotfill\ldots 8vo, &
+2\ 00\\
+
+Robine and Lenglen's Cyanide Industry. (Le Clerc.)\dotfill\ldots 8vo,
+& 4\ 00\\
+
+* Walke's Lectures on Explosives.\dotfill\ldots 8vo, & 4\ 00\\
+
+Weaver's Military Explosives.\dotfill\ldots 8vo, & 3\ 00\\
+
+Wilson's Cyanide Processes.\dotfill\ldots 12mo, & 1\ 50\\
+
+\indent Chlorination Process.\dotfill\ldots 12mo, & 1\ 50\\
+
+\indent Hydraulic and Placer Mining.\dotfill\ldots 12mo, & 2\ 00\\
+
+\indent Treatise on Practical and Theoretical Mine
+Ventilation.\dotfill\ldots 12mo, & 1\ 25\\[3em]
+
+
+
+\multicolumn{2}{c}{\large SANITARY SCIENCE.}\\[1em]
+
+\nopagebreak
+
+Bashore's Sanitation of a Country House.\dotfill\ldots 12mo, & 1\ 00\\
+
+\makebox[0pt]{\hspace{.5ex} *}\indent Outlines of Practical
+Sanitation.\dotfill\ldots 12mo, & 1\ 25\\
+
+Folwell's Sewerage. (Designing, Construction, and
+Maintenance.)\dotfill\ldots 8vo, & 3\ 00\\
+
+\indent Water-supply Engineering.\dotfill\ldots 8vo, & 4\ 00\\
+
+Fowler's Sewage Works Analyses.\dotfill\ldots 12mo, & 2\ 00\\
+
+Fuertes's Water and Public Health.\dotfill\ldots 12mo, & 1\ 50\\
+
+\indent Water-filtration Works.\dotfill\ldots 12mo, & 2\ 50\\
+
+Gerhard's Guide to Sanitary House-inspection.\dotfill\ldots 16mo, & 1\ 00\\
+
+Hazen's Filtration of Public Water-supplies.\dotfill\ldots 8vo, & 3\ 00\\
+
+Leach's The Inspection and Analysis of Food with Special Reference\\
+
+\indent\indent to State Control.\dotfill\ldots 8vo, & 7\ 50\\
+
+Mason's Water-supply (Considered principally from\\
+
+\nopagebreak
+
+\indent\indent a Sanitary Standpoint)\dotfill\ldots 8vo, & 4\ 00\\
+
+\indent Examination of Water. (Chemical and
+Bacteriological.)\dotfill\ldots 12mo, & 1\ 25\\
+
+* Merriman's Elements of Sanitary Engineering.\dotfill\ldots 8vo, & 2\ 00\\
+
+Ogden's Sewer Design.\dotfill\ldots 12mo, & 2\ 00\\
+
+Prescott and Winslow's Elements of Water Bacteriology, with Special\\
+
+\indent\indent Reference to Sanitary Water Analysis.\dotfill\ldots
+12mo, & 1\ 25\\
+
+* Price's Handbook on Sanitation.\dotfill\ldots 12mo, & 1\ 50\\
+
+Richards's Cost of Food. A Study in Dietaries.\dotfill\ldots 12mo, & 1\ 00\\
+
+\indent Cost of Living as Modified by Sanitary Science.\dotfill\ldots
+12mo, & 1\ 00\\
+
+\indent Cost of Shelter.\dotfill\ldots 12mo, & 1\ 00\\
+
+%-----File: 259.png---Index 19-------
+
+
+Richards and Woodman's Air, Water, and Food from a Sanitary\\
+
+\nopagebreak
+
+\indent\indent Standpoint.\dotfill\ldots 8vo, &2\ 00\\
+
+* Richards and Williams's The Dietary Computer.\dotfill\ldots 8vo, &1\ 50\\
+
+Rideal's Sewage and Bacterial Purification of Sewage.\dotfill\ldots
+8vo, &4\ 00\\
+
+Turneaure and Russell's Public Water-supplies.\dotfill\ldots 8vo, &5\ 00\\
+
+Von Behring's Suppression of Tuberculosis. (Bolduan.)\dotfill 12mo, &1\ 00\\
+
+Whipple's Microscopy of Drinking-water.\dotfill\ldots 8vo, &3\ 50\\
+
+Winton's Microscopy of Vegetable Foods.\dotfill\ldots 8vo, &7\ 50\\
+
+Woodhull's Notes on Military Hygiene.\dotfill\ldots 16mo, &1\ 50\\
+
+\makebox[0pt]{\hspace{.5ex} *}\indent Personal Hygiene.\dotfill\ldots
+12mo, &1\ 00\\[3em]
+
+
+
+\multicolumn{2}{c}{\large MISCELLANEOUS.}\\[1em]
+
+\nopagebreak
+
+Emmons's Geological Guide-book of the Rocky Mountain Excursion\\
+
+\nopagebreak
+
+\indent\indent of the International Congress of
+Geologists.\dotfill\ldots Large 8vo, &1\ 50\\
+
+Ferrel's Popular Treatise on the Winds.\dotfill\ldots 8vo, &4\ 00\\
+
+Gannett's Statistical Abstract of the World.\dotfill\ldots 24mo, & \ 75\\
+
+Haines's American Railway Management.\dotfill\ldots 12mo, &2\ 50\\
+
+Ricketts's History of Rensselaer Polytechnic Institute,\\
+
+\nopagebreak
+
+\indent\indent 1824--1894.\dotfill\ldots Small 8vo, &3\ 00\\
+
+Rotherham's Emphasized New Testament.\dotfill\ldots Large 8vo, &3\ 00\\
+
+The World's Columbian Exposition of 1893.\dotfill\ldots 4to, &1 00\\
+
+Winslow's Elements of Applied Microscopy.\dotfill\ldots 12mo, &1 50\\[3em]
+
+
+
+\multicolumn{2}{c}{\large HEBREW AND CHALDEE TEXT-BOOKS.}\\[1em]
+
+\nopagebreak
+
+Green's Elementary Hebrew Grammar.\dotfill\ldots 12mo, &1 25\\
+
+\indent Hebrew Chrestomathy..\dotfill 8vo, &2 00\\
+
+Gesenius's Hebrew and Chaldee Lexicon to the Old Testament\\
+
+\nopagebreak
+
+\indent\indent Scriptures. (Tregelles.)\dotfill\ldots Small 4to, half
+morocco, &5 00\\
+
+Letteris's Hebrew Bible.\dotfill\ldots 8vo, &2 25\\
+
+
+\end{longtable}
+
+\newpage
+
+\small
+\pagenumbering{gobble}
+\begin{verbatim}
+
+End of Project Gutenberg's Introduction to Infinitesimal Analysis
+by Oswald Veblen and N. J. Lennes
+
+*** END OF THIS PROJECT GUTENBERG EBOOK INFINITESIMAL ANALYSIS  ***
+
+*** This file should be named 18741-t.tex or 18741-t.zip ***
+*** or                    18741-pdf.pdf or 18741-pdf.pdf ***
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+[18] [19] [20] [21] [22
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+]
+Chapter 2.
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+Chapter 3.
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+
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+
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+
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+ <use images/fig11.pdf>
+[41] [42 <images/fig11.pdf>] [43] [44] [45] [46
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+]
+Chapter 4.
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+File: images/fig13.pdf Graphic file (type pdf)
+
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+[55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68
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+]
+Chapter 5.
+[69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80
+
+]
+Chapter 6.
+[81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92]
+Chapter 7.
+[93
+
+] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106]
+[107] [108] [109] [110] [111] [112] [113] [114]
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+,
+ []
+
+[117] [118] [119] [120]
+Chapter 8.
+[121
+
+] [122] <images/fig21.pdf, id=1873, 372.39125pt x 250.9375pt>
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+
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+[135] [136] [137] [138] [139] [140] [141] [142] [143] [144] [145] [146]
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+
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diff --git a/18741-t/images/fig05.pdf b/18741-t/images/fig05.pdf
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diff --git a/LICENSE.txt b/LICENSE.txt
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--- /dev/null
+++ b/LICENSE.txt
@@ -0,0 +1,11 @@
+This eBook, including all associated images, markup, improvements,
+metadata, and any other content or labor, has been confirmed to be
+in the PUBLIC DOMAIN IN THE UNITED STATES.
+
+Procedures for determining public domain status are described in
+the "Copyright How-To" at https://www.gutenberg.org.
+
+No investigation has been made concerning possible copyrights in
+jurisdictions other than the United States. Anyone seeking to utilize
+this eBook outside of the United States should confirm copyright
+status under the laws that apply to them.
diff --git a/README.md b/README.md
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+++ b/README.md
@@ -0,0 +1,2 @@
+Project Gutenberg (https://www.gutenberg.org) public repository for
+eBook #18741 (https://www.gutenberg.org/ebooks/18741)