\magnification=1200 \vsize=20cm\hsize=13.2cm \pretolerance=500 \tolerance=1000 \brokenpenalty=5000 \parindent=0mm \parskip=5pt plus 1pt minus 1pt \newcount\domessage \domessage=0 % new fonts definitions \font\seventeenbf=ecbx10 at 17.28pt \font\fourteenbf=ecbx10 at 14.4pt \font\twelvebf=ecbx10 at 12pt \font\eightbf=ecbx8 \font\sixbf=ecbx6 \font\twelvei=cmmi10 at 12pt \font\eighti=cmmi8 \font\sixi=cmmi6 \font\twelverm=ecrm10 at 12pt \font\eightrm=ecrm8 \font\sixrm=ecrm6 \font\eightsy=cmsy8 \font\sixsy=cmsy6 \font\eightit=ecti8 \font\eighttt=ectt8 \font\eightsl=ecsl8 \font\seventeenbsy=cmbsy10 at 17.28pt \font\fourteenbsy=cmbsy10 at 14.4pt \font\twelvebsy=cmbsy10 at 12pt \font\tenbsy=cmbsy10 \font\eightbsy=cmbsy8 \font\sevenbsy=cmbsy7 \font\sixbsy=cmbsy6 \font\fivebsy=cmbsy5 \font\tenmathx=mathx10 \font\eightmathx=mathx8 \font\sevenmathx=mathxm7 \font\fivemathx=mathxm5 \newfam\mathxfam \textfont\mathxfam=\tenmathx \scriptfont\mathxfam=\sevenmathx \scriptscriptfont\mathxfam=\fivemathx \def\mathx{\fam\mathxfam\tenmathx} \font\tenmsa=msam10 \font\eightmsa=msam8 \font\sevenmsa=msam7 \font\fivemsa=msam5 \newfam\msafam \textfont\msafam=\tenmsa \scriptfont\msafam=\sevenmsa \scriptscriptfont\msafam=\fivemsa \def\msa{\fam\msafam\tenmsa} \font\tenmsb=msbm10 \font\eightmsb=msbm8 \font\sevenmsb=msbm7 \font\fivemsb=msbm5 \newfam\msbfam \textfont\msbfam=\tenmsb \scriptfont\msbfam=\sevenmsb \scriptscriptfont\msbfam=\fivemsb \def\Bbb{\fam\msbfam\tenmsb} \font\tenCal=eusm10 \font\eightCal=eusm8 \font\sevenCal=eusm7 \font\fiveCal=eusm5 \newfam\Calfam \textfont\Calfam=\tenCal \scriptfont\Calfam=\sevenCal \scriptscriptfont\Calfam=\fiveCal \def\Cal{\fam\Calfam\tenCal} \font\teneuf=eusm10 \font\teneuf=eufm10 \font\seveneuf=eufm7 \font\fiveeuf=eufm5 \newfam\euffam \textfont\euffam=\teneuf \scriptfont\euffam=\seveneuf \scriptscriptfont\euffam=\fiveeuf \def\euf{\fam\euffam\teneuf} \font\seventeenbfit=cmmib10 at 17.28pt \font\fourteenbfit=cmmib10 at 14.4pt \font\twelvebfit=cmmib10 at 12pt \font\tenbfit=cmmib10 \font\eightbfit=cmmib8 \font\sevenbfit=cmmib7 \font\sixbfit=cmmib6 \font\fivebfit=cmmib5 \newfam\bfitfam \textfont\bfitfam=\tenbfit \scriptfont\bfitfam=\sevenbfit \scriptscriptfont\bfitfam=\fivebfit \def\bfit{\fam\bfitfam\tenbfit} \def\gS{\hbox{\teneuf S}} % changing font sizes \catcode`\@=11 \def\eightpoint{% \textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm \def\rm{\fam\z@\eightrm}% \textfont1=\eighti \scriptfont1=\sixi \scriptscriptfont1=\fivei \def\oldstyle{\fam\@ne\eighti}% \textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy \textfont\itfam=\eightit \def\it{\fam\itfam\eightit}% \textfont\slfam=\eightsl \def\sl{\fam\slfam\eightsl}% \textfont\bffam=\eightbf \scriptfont\bffam=\sixbf \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\eightbf}% \textfont\ttfam=\eighttt \def\tt{\fam\ttfam\eighttt}% \textfont\msbfam=\eightmsb \def\Bbb{\fam\msbfam\eightmsb}% \textfont\Calfam=\eightCal \def\Cal{\fam\Calfam\eightCal}% \abovedisplayskip=9pt plus 2pt minus 6pt \abovedisplayshortskip=0pt plus 2pt \belowdisplayskip=9pt plus 2pt minus 6pt \belowdisplayshortskip=5pt plus 2pt minus 3pt \smallskipamount=2pt plus 1pt minus 1pt \medskipamount=4pt plus 2pt minus 1pt \bigskipamount=9pt plus 3pt minus 3pt \normalbaselineskip=9pt \setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}% \let\bigf@ntpc=\eightrm \let\smallf@ntpc=\sixrm \normalbaselines\rm} \catcode`\@=12 \def\eightpointbf{% \textfont0=\eightbf \scriptfont0=\sixbf \scriptscriptfont0=\fivebf \textfont1=\eightbfit \scriptfont1=\sixbfit \scriptscriptfont1=\fivebfit \textfont2=\eightbsy \scriptfont2=\sixbsy \scriptscriptfont2=\fivebsy \eightbf \baselineskip=10pt} \def\tenpointbf{% \textfont0=\tenbf \scriptfont0=\sevenbf \scriptscriptfont0=\fivebf \textfont1=\tenbfit \scriptfont1=\sevenbfit \scriptscriptfont1=\fivebfit \textfont2=\tenbsy \scriptfont2=\sevenbsy \scriptscriptfont2=\fivebsy \tenbf} \def\twelvepointbf{% \textfont0=\twelvebf \scriptfont0=\eightbf \scriptscriptfont0=\sixbf \textfont1=\twelvebfit \scriptfont1=\eightbfit \scriptscriptfont1=\sixbfit \textfont2=\twelvebsy \scriptfont2=\eightbsy \scriptscriptfont2=\sixbsy \twelvebf \baselineskip=14.4pt} \def\fourteenpointbf{% \textfont0=\fourteenbf \scriptfont0=\tenbf \scriptscriptfont0=\sevenbf \textfont1=\fourteenbfit \scriptfont1=\tenbfit \scriptscriptfont1=\sevenbfit \textfont2=\fourteenbsy \scriptfont2=\tenbsy \scriptscriptfont2=\sevenbsy \fourteenbf \baselineskip=17.28pt} \def\seventeenpointbf{% \textfont0=\seventeenbf \scriptfont0=\twelvebf \scriptscriptfont0=\eightbf \textfont1=\seventeenbfit\scriptfont1=\twelvebfit\scriptscriptfont1=\eightbfit \textfont2=\seventeenbsy \scriptfont2=\twelvebsy \scriptscriptfont2=\eightbsy \seventeenbf \baselineskip=20.736pt} % main item macros \newdimen\srdim \srdim=\hsize \newdimen\irdim \irdim=\hsize \def\NOSECTREF#1{\noindent\hbox to \srdim{\null\dotfill ???(#1)}} \def\SECTREF#1{\noindent\hbox to \srdim{\csname REF\romannumeral#1\endcsname}} \def\INDREF#1{\noindent\hbox to \irdim{\csname IND\romannumeral#1\endcsname}} \newlinechar=`\^^J \def\openauxfile{ \immediate\openin1\jobname.aux \ifeof1 \ifnum\domessage=1 \message{^^JCAUTION\string: you MUST run TeX a second time^^J}\fi \let\sectref=\NOSECTREF \let\indref=\NOSECTREF \else \input \jobname.aux \ifnum\domessage=1 \message{^^JCAUTION\string: if the file has just been modified you may have to run TeX twice^^J}\fi \let\sectref=\SECTREF \let\indref=\INDREF \fi \ifnum\domessage=1 \message{to get correct page numbers displayed in Contents or Index Tables^^J}\fi \immediate\openout1=\jobname.aux \let\END=\end \def\end{\immediate\closeout1\END}} \newcount\notthispage \notthispage=1 \newbox\titlebox \setbox\titlebox\hbox{\hfil} \newbox\sectionbox \setbox\sectionbox\hbox{\hfil} \newbox\chapterbox \setbox\chapterbox\hbox{\hfil} \def\folio{\ifnum\pageno=\notthispage \hfil \else \ifodd\pageno \hfil {\eightpoint \copy\chapterbox\copy\sectionbox% \kern8mm\number\pageno}\else {\eightpoint\number\pageno\kern8mm\copy\titlebox}\hfil \fi\fi} \footline={\hfil} \headline={\folio} \def\verl{{\vrule width 0.4pt height 12pt depth 6pt}} \def\table#1{\baselineskip=0pt\lineskip=0pt \halign{$\displaystyle{}##\hfil{}$\hfil&& $\displaystyle{}##\hfil{}$\hfil\crcr#1\crcr}} \def\blankline{\phantom{}\hfil\vskip0pt} \def\titlerunning#1{\setbox\titlebox\hbox{\eightpoint #1}} \def\chapterrunning#1{\notthispage=\pageno \setbox\chapterbox\hbox{\eightpoint #1}} \def\title#1{\noindent\hfil$\smash{\hbox{\seventeenpointbf #1}}$\hfil \titlerunning{#1}\medskip} \def\titleleft#1{\noindent$\smash{\hbox{\seventeenpointbf #1}}$\hfil \titlerunning{#1}\medskip} \def\supersection#1{% \par\vskip0.5cm\penalty -100 \vbox{\baselineskip=17.28pt\noindent{{\fourteenpointbf #1}}} \vskip3pt \penalty 500 \titlerunning{#1}} \newcount\numbersection \numbersection=-1 \def\sectionrunning#1{\setbox\sectionbox\hbox{\eightpoint #1} \immediate\write1{\string\def \string\REF \romannumeral\numbersection \string{% \noexpand#1 \string\dotfill \space \number\pageno \string}}} \def\section#1{% \par\vskip0.5cm\penalty -100 \vbox{\baselineskip=14.4pt\noindent{{\twelvepointbf #1}}} \vskip2pt \penalty 500 \advance\numbersection by 1 \sectionrunning{#1}} \def\subsection#1{% \par\vskip0.5cm\penalty -100 \vbox{\noindent{{\tenpointbf #1}}} \vskip1pt \penalty 500} \newcount\numberindex \numberindex=0 \def\index#1#2{% \advance\numberindex by 1 \immediate\write1{\string\def \string\IND #1% \romannumeral\numberindex \string{% \noexpand#2 \string\dotfill \space \string\S \number\numbersection, p.\string\ \space\number\pageno \string}}} \def\chapterjump{ \vfill\eject \ifodd\pageno \else {\headline={\hfil}\null\vskip0pt\vfill\eject} \fi } % Usual sets of numbers \def\bC{{\Bbb C}} \def\bD{{\Bbb D}} \def\bH{{\Bbb H}} \def\bN{{\Bbb N}} \def\bP{{\Bbb P}} \def\bQ{{\Bbb Q}} \def\bR{{\Bbb R}} \def\bZ{{\Bbb Z}} % Calligraphic capital letters \def\cA{{\Cal A}} \def\cC{{\Cal C}} \def\cD{{\Cal D}} \def\cE{{\Cal E}} \def\cF{{\Cal F}} \def\cH{{\Cal H}} \def\cK{{\Cal K}} \def\cL{{\Cal L}} \def\cP{{\Cal P}} \def\cR{{\Cal R}} \def\cV{{\Cal V}} \def\tC{\smash{\tilde C}} \def\hexnbr#1{\ifnum#1<10 \number#1\else \ifnum#1=10 A\else\ifnum#1=11 B\else\ifnum#1=12 C\else \ifnum#1=13 D\else\ifnum#1=14 E\else\ifnum#1=15 F\fi\fi\fi\fi\fi\fi\fi} \def\msatype{\hexnbr\msafam} \def\msbtype{\hexnbr\msbfam} \def\mathxtype{\hexnbr\mathxfam} \mathchardef\smallsetminus="2\msbtype72 \let\ssm\smallsetminus \mathchardef\leqslant="3\msatype36 \mathchardef\geqslant="3\msatype3E \mathchardef\complement="0\msatype7B \mathchardef\sqsymb="0\msatype03 \def\overacute{\mathaccent"0\mathxtype79} \def\overobtuse{\mathaccent"0\mathxtype7D} \let\ge=\geqslant \let\le=\leqslant \let\dsp=\displaystyle \def\square{{\hfill \hbox{ \vrule height 1.453ex width 0.093ex depth 0ex \vrule height 1.5ex width 1.3ex depth -1.407ex\kern-0.1ex \vrule height 1.453ex width 0.093ex depth 0ex\kern-1.35ex \vrule height 0.093ex width 1.3ex depth 0ex}}} \def\qed{\phantom{$\quad$}\hfill$\square$} \def\?{\hbox{$\,$}} \catcode`@=11 \def\cmalign#1#2{\null\,\vcenter{\normalbaselines\m@th \ialign{$\displaystyle ##$\hfil&&\kern#1$\displaystyle ##$\hfil\crcr \mathstrut\crcr\noalign{\kern-\baselineskip} #2\crcr\mathstrut\crcr\noalign{\kern-\baselineskip}}}\,} \catcode`@=12 \def\itemv{% \item{\llap{$\square\kern7pt$}\llap{$\raise0.6pt\hbox{×}\kern6pt$}}} \def\itemf{% \item{\llap{$\square\kern7pt$}}} \def\sqind#1{\kern1.5pt\rlap{\raise5pt\hbox{$\scriptstyle#1$}}\kern-1.5pt} \def\HK{{\rm HK}} \def\ang{\mathop{\rm angle}} \def\cotan{\mathop{\rm cotan}} \def\cotanh{\mathop{\rm cotanh}} \def\supess{\mathop{\rm sup\,ess}} \def\infess{\mathop{\rm inf\,ess}} \def\Id{\mathop{\rm Id}} \def\Re{\mathop{\rm Re}} \def\Im{\mathop{\rm Im}} \def\Aut{\mathop{\rm Aut}} \def\Supp{\mathop{\rm Supp}} \def\lg{\mathop{\rm long}} \def\oscil{\mathop{\rm oscil}} \def\vol{\mathop{\rm vol}} \def\aspect{\mathop{\rm aspect}} \def\aire{\mathop{\rm aire}} \def\longueur{\mathop{\rm longueur}} \def\diam{\mathop{\rm diam}} \def\sign{\mathop{\rm sign}} \def\grad{\mathop{\rm grad}} \def\rot{\mathop{\rm rot}} \def\div{\mathop{\rm div}} \def\note#1#2{\footnote{${}^{\hbox{\sevenrm(#1)}}$} {\baselineskip=8pt {\eightpoint #2}\vskip-15pt}} \def\\{\hfil\break} \def\demi{\textstyle{1\over 2}} \def\ovl{\overline} \def\ovr{\overrightarrow} \def\ul#1{$\underline{\smash{\hbox{#1}}}$} \def\build#1^#2_#3{\mathop{#1}\limits^{#2}_{#3}} \def\boxed#1#2{\hbox{% \leavevmode\thinspace\hbox{\vrule\vtop{\vbox{\hrule\kern1pt \hbox{\thinspace{$\displaystyle \raise2.5pt\hbox{\phantom{\vrule width 0.1pt height #1pt depth #1pt}} #2$}\thinspace}} \kern1pt\hrule}\vrule}\thinspace}} % inclusion of PostScript files \special{header=/home/demailly/psinputs/mathdraw/mdrlib.ps} \def\RGBColor#1#2{\special{color push rgb #1}#2\special{color pop}} \long\def\InsertFig#1 #2 #3 #4\EndFig{\par \hbox{\hskip #1mm$\vbox to#2mm{\vfil\special{" #3}}#4$}} \long\def\InsertPSFile#1 #2 #3 #4 #5 #6\EndFig{\par \hbox{\hskip #1mm$\vbox to#2mm{% \vfil\special{psfile=#5 hscale=#3 vscale=#4}}#6$}} \long\def\InsertImage#1 #2 #3 #4 #5 #6 #7 #8\EndFig{\par \hbox{\hskip #1mm$\vbox to#2mm{% \vfil\special{psfile="`img2eps file #7 height #4 mm width #3 mm gamma #5 angle #6}}#8$}} \long\def\LabelTeX#1 #2 #3\ELTX{\rlap{\kern#1mm\raise#2mm\hbox{#3}}} % bibibliography \def\bibitem#1&% {\hangindent=0.66cm\hangafter=1 \noindent\rlap{\hbox{\eightpointbf #1}}\kern0.66cm{\rm #2}{\it #3}{\rm #4.}} \openauxfile \notthispage=\pageno {\twelvepointbf Démonstration de la formule d'aire du disque à partir de la définition de $\pi=P/D$ en Cours Moyen :\\ ou les prémices de la notion de limite et de calcul infinitésimal} \bigskip\bigskip La preuve suivante a été expérimentée (avec succès, i.e.\ avec compréhension effective des élèves~!) dans une classe de Cours Moyen apparentée SLECC, en 2007-2008. Bien sûr, à ce niveau, il n'y a pas de notations aussi formelles que celles décrites ci-dessous, la procédure repose plutôt sur des manipulations avec papier, compas et ciseaux~! \bigskip \InsertFig 11.000 55.000 { gsave 0.5 0.5 scale -60 0 translate /R 120 def /steps 16 def /delta 360 steps div def R 1.1 mul R 1.1 mul translate 0 1 steps 1 sub { /k exch def 0 0 moveto k delta mul dup cos R mul exch sin R mul lineto 0 0 R k delta mul k 1 add delta mul arc 0 0 lineto closepath gsave k 2 mul steps lt { 0.72 0.72 1 setrgbcolor } { 1 0.72 1 setrgbcolor } ifelse fill grestore stroke} for gsave 0.4 setlinewidth [ 5 2 ] 0 setdash 0 0 moveto 180 delta 0.5 mul sub dup cos R mul exch sin R mul lineto stroke grestore /delta delta 2 div def /dx delta sin R mul 2 mul def R 1.6 mul R -0.5 mul translate gsave 0 1 steps 2 div 1 sub { /k exch def 0 0 moveto delta sin R mul delta cos R mul lineto 0 0 R 90 delta sub 90 delta add arc currentpoint /y exch def /x exch def 0 0 lineto closepath gsave 0.72 0.72 1 setrgbcolor fill grestore stroke gsave x dx add y translate 1 -1 scale 0 0 moveto delta sin R mul delta cos R mul lineto 0 0 R 90 delta sub 90 delta add arc 0 0 lineto closepath gsave 1 0.72 1 setrgbcolor fill grestore stroke grestore dx 0 translate} for grestore gsave 0.4 setlinewidth [ 5 2 ] 0 setdash 0 0 moveto 0 R lineto stroke dx steps mul 0.5 mul 0 translate 0 0 moveto 0 R lineto stroke 0 0 R 90 90 delta add arc stroke grestore 0 -30 moveto 0 180 10 vector stroke 0 -30 moveto dx steps mul 0.5 mul 0 10 vector stroke dx steps mul 0.5 mul 35 add 0 translate 0 0 moveto 0 -90 10 vector stroke 0 0 moveto R 90 10 vector stroke grestore } \LabelTeX 7 48 Disque\kern1.4cm $\longrightarrow$\kern1.4cm parallélogramme (ou rectangle)\ELTX \LabelTeX -9 -5 $\displaystyle \pi={P\over D}~~\Rightarrow~~ P=\pi\times D=2\times \pi\times R$\ELTX \LabelTeX 72 0 base$\displaystyle{}\simeq{P\over 2}=\pi\times R$\ELTX \LabelTeX 115 22 $R$\ELTX \EndFig \bigskip À la limite, en augmentant le nombre de secteurs triangulaires, on voit donc que l'aire du disque est donnée par $\pi\times R\times R=\pi R^2$. Bien entendu, ce travail suppose que l'on ait au préalable soigneusement traité l'aire du rectangle, du triangle et du parallélogramme, avec là encore les découpages géométriques classiques pour justifier les formules. Le statut de la formule $P=\pi D=2\pi R$ est différent, dans ce cas il s'agit d'une {\it définition} du nombre~$\pi$\?: c'est le rapport du périmètre au diamètre, qui est indépendant du cercle considéré (on justifiera intuitivement que si le diamètre double ou triple, il en est de même pour le périmètre, ce qui formellement résulte du théorème de Thalès$\,\ldots$). Il est bien sûr souhaitable d'expérimenter en enroulant quelques tours d'une ficelle autour d'un tuyau de diamètre connu, pour trouver une valeur approchée de~$\pi$. \end % Local Variables: % TeX-command-default: "uTeX" % End: