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\begin{document}
\title{Corrections to The Pleasures of Counting}
\author{T.~W.~K\"{o}rner}
\maketitle
The following pages contain
the list of corrections and additions
for the first reprinting. I should like to thank
Robin Chapman of Exeter (first mistake spotted),
Professor Cassels, Dr Garling, Dr Osborne,
Professor Barenblatt,
and Prof J. Taylor (all of Cambridge),
Harold N. Ward (University of Virginia),
Adam Atkinson, Paul Shallhorn and Pat Henry
for spotting various problems and
\emph{for telling me about them}.
I would particularly like to thank
Douglas Quadling and Dr Francis Clarke
(Swansea) for long lists of useful comments.
\newpage
{\sf Side note on page 54}
\noindent
Some bats have a mechanism for making themselves
temporarily deaf when emitting sound bursts.
(See~[260], a book evidently written by a man
who has never seen a bat he did not love.)
They use high frequency sound because their prey
is small.
%Index,
{\sf Side note on page 190}
\noindent
According to Arnol'd in~[264],
Kolmogorov started as a student of history.
His first paper, written when he was seventeen,
concerned the mediaeval tax records in Novgorod.
After he had presented his conclusions
to a seminar he asked the historian in charge
whether he agreed with them. `Young man,'
the professor said, `in history we need
at least five proofs for any conclusion.'
Next day, Kolmogorov switched to mathematics.
{\sf Side note on page 221}
When we talked about oxygen absorption we assumed
that the area for absorption $A$ was related to
the volume $V$ of an animal in the same way
as the surface area of a sphere is related to
its volume so that $A\propto V^{2/3}$. But we
noted on page 218 that lungs look very `fractal'.
If the surface of the lung has
`Richardson number' different
from the surface of a sphere then
we would have $A\propto V^{\beta}$ with
$\beta\neq 2/3$ and the slope of the
mouse-elephant curve becomes easier to explain.
(See pages 342-5 of [261].)
{\sf Side note on page 392}
\noindent
However, most of those whose opinions I respect
reject Penrose's views on the nature of human thought
`not because they are crazy,
but because they are not crazy enough'.
{\sf Side note on page 501}
\noindent
(Added in second printing.)
If you wish to write complex formulae,
I strongly recommend Gr\"{a}tzer's
\emph{Math Into \LaTeX}~[263].
{\sf Side note on page 505}
\noindent
Hacker lore says that two heads are faster than one
and three heads slightly faster than two. Thereafter,
adding manpower to a software project actually makes it slower.
The only way to speed up a project is to use cleverer
people. Hence the \emph{three Knuth rule}. If three
Knuths working together cannot do it, it cannot be done.
%index
\newpage
{\sf Page 53,line before (i) add} (The restrictions on $\theta_{1}$
are rather arbitrary, but some restrictions are needed.)
{\sf Page 53, near middle of written matter. Replace
$\angle BCX$ by $\angle ACX$.}
{\sf Page 53, 4 lines up. Replace $a=AX\cos\theta_{1}$
by $AX=a\sin\theta_{1}$.}
{\sf Page 53, last displayed formula}
\[a\sin\theta_{1}=2r\sin\frac{\theta_{2}-\theta_{1}}{1}\]
{\sf Page 59 middle. Replace} \emph{for $1\leq i\leq n$
and guess that that $a_{m+1}$ will be close to}
\[T_{n}(a_{m-n+1},a_{m-n+2},\dots\ ,a_{m})=P(n)\]
{\sf by}
\emph{for $0\leq i\leq n-1$
and guess that that $a_{m+1}$ will be close to}
\[T_{n-1}(a_{m-n},a_{m-n+1},\dots\ ,a_{m})=P(n)\]
{\sf Page 79, line 2 replace $k=b=1$ by $k=c=1$}
{\sf Page 79, line 2 replace $b(t)=1-r(t)$ by $b(t)=1+r(t)$}
{\sf Page 79, first displayed formula, replace $r'(t)=-(1-r(t))r(t)$
by}
\[r'(t)=-(1+r(t))r(t).\]
{\sf Page 79, second displayed formula, replace
$\frac{r(t)}{1-r(t)}=\frac{r(0)}{1-r(0)}e^{-t}$ by}
\[\frac{r(t)}{1+r(t)}=\frac{r(0)}{1+r(0)}e^{-t}.\]
{\sf Page 79, end of first para replace} $r(t)\rightarrow 0$
for all $t>0$. {\sf by}
$r(t)\rightarrow 0$ as $t\rightarrow\infty$.
{\sf Page 120, 2nd displayed formula. Replace} $\tau=gl^{-1}t^{-2}.$
{\sf by}
\[\tau=gl^{-1}t^{2}.\]
{\sf Page 120, 4th displayed formula. Replace} $gl^{-1}t^{-2}=A$
{\sf by}
\[gl^{-1}t^{2}=A.\]
{\sf Page 120, 5th displayed formula. Replace} $t^{2}=A\dfrac{g}{l}$
{\sf by}
\[t^{2}=A\dfrac{l}{g}.\]
{\sf Page 120, 6th displayed formula. Replace} $t=C\sqrt{\dfrac{g}{l}}$
{\sf by}
\[t=C\sqrt{\dfrac{l}{g}}.\]
{\sf Page 120, line -7 replace} varies inversely with
{\sf by} is proportional to
\newpage
{\sf Page 121, REPLACE side note as follows}
According to materials scientists, when molten
glass cools it remains a liquid but one whose coefficient
of viscosity increases as the temperature decreases.
Thus we can blow glass at high temperatures,
mold it at lower temperatures and so on.
However, the demonstration of liquid properties
for glass at room temperature lies at the very
edge of modern experimental technique.
{\sf Page 126, add at end of first side note}
The treatment of dimension I have given
follows the traditional pattern in glossing over
certain points. In~[261] Barenblatt takes a more
modern approach and shows that, if we think a
little harder, we can understand a lot more.
{\sf Page 133, bottom line.
Replace} wave front {\sf by} shock front.
{\sf Page 143, first equation after 7.1 should not
read $(x,y,z,t)=(ct',0,0,t)$ but should read}
\[(x,y,z,t) =(ct,0,0,t),\]
{\sf Page 166. In BOTH Figure 8.2(a) and Figure 8.2(b)
replace $\sqrt{2}$ on $x$-axis by $1/\sqrt{2}$.}
{\sf Pages 167 to 171. Wherever $))b/N$ appears replace
with $))b/(2N)$}
{\sf Page 168 lines 7, 8 and 9. Replace
$f(rb/N)b/N$ by $f(rb/N))b/(2N)$}
{\sf Page 171, second displayed formula, replace $h$
by $\dfrac{h}{2}$}
{\sf Page 171, second to displayed formula, replace $\eta$
by $\eta/2$}
{\sf Pages 175 and 176. Replace $\sqrt{\dfrac{g}{l}}$
in all its four appearances by} $\sqrt{\dfrac{l}{g}}$.
{\sf Page 176, 5th line. Replace $b_{0}(g/l)^{1/2}$ by
$b_{0}(l/g)^{1/2}$.}
{\sf Page 177, 3/4 down. Replace} collected paper with
{\sf by} collected papers) with
{\sf Page 177, replace} \emph{velocity?)}: {\sf by} \emph{velocity?}:
{\sf Page 186, top third TWICE replace $K(t)$ by $Kt$.}
{\sf Page 187, 1/3 down. Replace} not much bigger than $l$
{\sf by} not much bigger than $l^{2}$.
{\sf page 264, figure (e) replace $3$ on top path by $2$.
Replace $1$ on almost vertical path by $0$.}
{\sf Page 271, Figure 11.5. Label on edge $XY$
replace $10p+10$ by $p+10$.}
{\sf page 274, last line but one of Exercise 11.2.6. Replace
$10p+10$ by $p+10$.}
\newpage
{\sf Pages 277 to 278. Can we move last line on 277 to become
first line on 278.}
{\sf Page 278 line 19. Extra space to turn} stop?')are
{\sf into} stop?') are
{\sf Page 290 line -5 replace}
\emph{for all integer $r$ with $R\geq 1$}
{\sf by}
\emph{for all integers $r$ with $r\geq 1$.}
{\sf Page 291, last displayed formula.
Replace $2\pi$ by $(2\pi)^{1/2}$
to give:-}
\[n!\approx (2\pi)^{1/2} e^{-n} n^{n+1/2}.\]
{\sf Page 311, 3/4 way down. Replace} Hodge's {\sf by}
Hodges'.
{\sf Page 331, 3/4 way down. Replace} {\tt A} is encoded
by {\tt SA}, {\tt B} by {\tt SB} and so on. {\sf by}
{\tt A} is encoded
by {\tt S(A)}, {\tt B} by {\tt S(B)} and so on.
{\sf Page 364, 1/4 way down. Replace} Feynmann {\sf by} Feynman.
{\sf Page 391, 7/8 way down. Replace} Hodges's {\sf by}
Hodges'.
{\sf Page 392, 3/8 way down. Replace} Hodges {\sf by}
Hodges'.
{\sf Page 403 Replace the first part of the proof by}
\noindent{\bf Proof} \emph{(i)} Note first that,
since $0\leq u\leq 1/2$, we have
\[\frac{u}{1-u}\leq 1.\]
Thus, by the binomial theorem,
\begin{align*}
1&=(u+(1-u))^{N}=\sum_{r=0}^{N}\binom{N}{r}u^{r}(1-u)^{N-r}\\
&\geq \sum_{r=0}^{uN}\binom{N}{r}u^{r}(1-u)^{N-r}
= (1-u)^{N}\sum_{r=0}^{uN}\binom{N}{r}
\left(\frac{u}{1-u}\right)^{r}\\
&\geq (1-u)^{N}\sum_{r=0}^{uN}\binom{N}{r}
\left(\frac{u}{1-u}\right)^{uN}
=u^{uN}(1-u)^{(1-u)N}\sum_{r=0}^{uN}\binom{N}{r}.
\end{align*}
Multiplying both sides of the inequality by $u^{-uN}(1-u)^{-(1-u)N}$
we have the result.
{\sf Page 405, part (iii) of Exercise 16.2.7, 2nd line. Add to}
if you can, I take {\sf so as to get}
if you can, without using a computer,
I take
{\sf Page 419, First line Exercise 17.4 (ii) replace
$\chi(0)<0$ by $\chi(0)>0$}
{\sf Page 419, Second line Exercise 17.4 (iii) replace
$\chi(t)<0$ by $\chi(t)>0$}
{\sf Page 420, line 19 replace} $n$th generation$\}$ {\sf by}
$n+1$st generation$\}$
\newpage
{\sf Page 428, last line of (i) replace $K=L-\tfrac{1}{4}$
by $K=\tfrac{1}{4}-L$.}
{\sf Page 428 first line of (ii). Replace $K=-\tfrac{1}{4}$
by $K=\tfrac{1}{4}$.}
{\sf Page 428 displayed formula in (iii).
Replace $C(L)=\frac{1}{2}(\frac{1}{4}-L)^{1/2}$ to obtain}
\[C(L)=\left(\frac{1}{4}-L\right)^{1/2}.\]
{\sf Page 428 final displayed formula. Replace
$L-\frac{1}{4}$ by $\frac{1}{4}-L$ to obtain}
\[\frac{dy}{dt}\leq \frac{1}{4}-L<0\]
{\sf Page 440 first displayed formula in (ii). Replace
by}
\[\frac{dI}{dS}=\frac{dI}{dt}\frac{dt}{dS}=
\frac{\beta SI-\gamma I}{-\beta SI}=-1+\frac{\gamma}{\beta S}\]
{\sf Page 442, third displayed formula up should read}
\[J=2S_{0}\left(\frac{S_{0}}{\rho}-1\right).\]
{\sf Page 442, second displayed formula up should read}
\[S_{0}\left(\frac{S_{0}}{\rho}-1\right)=
\left(1+\frac{\nu}{\rho}\right)\]
{\sf Page 443 line 10 replace} Claude {\sf by} Claud
{\sf Page 490 replace} Adam's {\sf by} Adams'
\newpage
{\sf Page 517, line 5 ending with $37-8.$ ADD} Batchelor's
paper was~[262].
{\sf Page 528}
\begin{center}
Additional Bibliography
\end{center}
\noindent
[260] J. D. Altringham. \emph{Bats, Biology and Behaviour.}
OUP, Oxford, 1996
\noindent
[261] G. I. Barenblatt. \emph{Scaling, Self-similarity and
Intermediate Asymptotics.}
CUP, Cambridge, 1996
\noindent
[262] G. K. Batchelor. Kolmogoroff's theory of locally
isotropic turbulence.
\emph{Proceedings of the Cambridge Philosophical Society},
43:533-59, 1947.
\noindent
[263] G. A. Gr\"{a}tzer. \emph{Math into \LaTeX.}
Birkh\"{a}user, Boston, 1996.
\noindent
[264] S. H. Lui. An interview with Vladimir Arnol'd.
\emph{Notices of the AMS}, 44(3):432-8, April 1997.
\vspace{2\baselineskip}
{\sf Page 531, add page reference}
historians, killjoy, routed, 104, 153, 190
{\sf Page 531 under} Knuth K. E. {\sf add item}
three Knuth rule, 505
{\sf Page 531, replace}
lungs, as surfaces of nearly infinite area, 218
{\sf by}
lungs, as fractals, 218, 221
{\sf Page 531 under} menagerie, mathematical {\sf add item}
bats, 54
\vspace{2\baselineskip}
\begin{footnotesize}
\noindent
[Printed out \today. These notes are
written in \LaTeX2e and stored in
directory \verb+~twk/FTP+ on moa
in (I hope) read permitted form in a file labeled
\verb+Cor.tex+. It may be accessed via my web home page
\begin{center}
{\tt http://www.pmms.cam.ac.uk/home/emu/twk/.my-home-page.html}.
\end{center}
Also available:-
`Dr K\"{o}rner's Helpful Guide For Mathematicians Seeking A
Cambridge Research Fellowship',
`In Praise of Lectures' (how to listen to a mathematics lecture),
`An Unofficial Guide To Part~III',
`How to Write a Part~III Essay',
`A Supervisor's Primer'.]
\end{footnotesize}
\end{document}