% !TEX TS-program = XeLaTeX
\documentclass[]{article}

% This was created by Chris Staecker
% You are free to do whatever you want with it.

% Uses the Rameau font family, which I downloaded here: 
% https://www.onlinewebfonts.com/search?q=rameau
% Also Antiquarian, from here:
% https://allbestfonts.com/antiquarian/
% Also Orniste tfb, from here:
% https://www.ffonts.net/Orniste-tfb.font.download
% Also Sackers Italian Script, from here:
% https://www.dafontfree.net/sackers-italian-script-std-regular/f16849.htm


% Export to 1000dpi for squareshop print file

\newcommand{\wid}{20cm} % width of central box
\newcommand{\ty}{.01*\wid}
\newcommand{\smalltextscale}{.45}
\newcommand{\textcolwid}{.67*\wid}
\newcommand{\textpad}{5*\ty} %vertical space between chart and text paragraphs
\newcommand{\textcolx}{.1*\wid} %left text columns centers at this x-value
\newcommand{\trigpad}{.15*\wid} %horz space between chart and trig rectangles
\newcommand{\trigwid}{2*\ty} %width of each small trig bar

\newcommand{\paperdim}{23cm}

\usepackage[ paperwidth=\paperdim, paperheight=\paperdim ]{geometry}

%\newcommand{\langs}[2]{\it #1} % use this one for French
\newcommand{\langs}[2]{\it #2} % this one for English


\usepackage{fontspec}
\setmainfont[ItalicFont=Rameau W01 Italic, WordSpace=1.2]{Antiquarian}

\newcommand{\boldy}[1]{{\fontspec{Antiquarian}\Large #1}}

\usepackage{nopageno,fullpage}

\usepackage[dvipsnames]{xcolor}

\usepackage{tikz}

\usepackage{qrcode}


\usepackage{background}

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scale=1,
color=black,
opacity=0.4,
angle=0,
contents={%
  \includegraphics[width=\paperwidth,height=\paperheight]{parchment-paper-light-texture}
  }%
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\newcommand{\col}{MidnightBlue}
\newcommand{\lightcol}{ProcessBlue}

 
\newcommand{\bigbox}[1]{\noindent\raisebox{-\height}[0pt][0pt]{%
  \makebox[\linewidth][l]{#1}}}
  




\tikzset{lightgridline/.style={very thin,draw=\lightcol}}

\begin{document}

\begin{center}
\bigbox{
\begin{tikzpicture}[draw=\col,text=\col,scale=.6,every node/.style={scale=0.6}]

% vert light lines
\foreach \x in {10,...,100} {
\draw[lightgridline] ({(log10(\x)-1)*\wid},0) -- ({(log10(\x)-1)*\wid},\wid);
}

% horz light lines
\foreach \y in {10,...,100} {
\draw[lightgridline] (0,{(log10(\y)-1)*\wid}) -- (\wid,{(log10(\y)-1)*\wid});
}

% light diagonals
\foreach \y in {10,...,100} {
 \draw[lightgridline] (0,{(log10(\y)-1)*\wid}) -- ({(log10(\y)-1)*\wid},0);
 \draw[lightgridline] ({(log10(\y)-1)*\wid},\wid) -- (\wid,{(log10(\y)-1)*\wid});
}


% squares line
\draw[thin,draw=\lightcol] (0,0) -- 
node[thick,below,rotate=45,scale=\smalltextscale,pos=.035]{\it\langs{Carrés}{Squares}} 
node[above=5,left=1,rotate=-45,scale=\smalltextscale,pos=.301]{\bf 4}
node[above=1,right=4,rotate=-45,scale=\smalltextscale,pos=.65]{\bf 20}
node[above=1,right=6,rotate=-45,scale=\smalltextscale,pos=.697]{\bf 25}
node[above=1,right=6,rotate=-45,scale=\smalltextscale,pos=.736]{\bf 30}
node[above=1,right=6,rotate=-45,scale=\smalltextscale,pos=.77]{\bf 35}
node[above=1,right=6,rotate=-45,scale=\smalltextscale,pos=.92]{\bf 70}
node[above=2,right=5,rotate=-45,scale=\smalltextscale,pos=.95]{\bf 80}
(\wid,\wid);
\foreach \x in {2,3,5} {
 \pgfmathsetmacro{\z}{(log10(\x*10)-1)*\wid/2}
 \node[above=2, left=-2,rotate=-45,scale=\smalltextscale] at ({(log10(\x*10)-1)*\wid/2},{(log10(\x*10)-1)*\wid/2}) {\bf \x};
}
\foreach \x in {10,15,40,45,50,60,90} {
 \node[above=3, right=1,rotate=-45,scale=\smalltextscale] at ({(log10(\x*10)-1)*\wid/2},{(log10(\x*10)-1)*\wid/2}) {\bf \x};
}


% CUBES LINE
% first branch
\draw[thin,draw=\lightcol] (0,0) -- 
node[thick,above,rotate=60,scale=\smalltextscale,pos=.05]{\it\langs{Cubes}{Cubes}} 
node[thick,above=3, right=1,rotate=-45,scale=\smalltextscale,pos=.467]{\bf 5}
node[thick,above=3, right=1,rotate=-45,scale=\smalltextscale,pos=.668]{\bf 10}
node[thick,above=3, right=1,rotate=-45,scale=\smalltextscale,pos=.785]{\bf 15}
node[thick,above=3, right=1,rotate=-45,scale=\smalltextscale,pos=.867]{\bf 20}
(\wid/2,\wid) -- (\wid/2, \wid+\ty);
\node[above=-5, right=-1, rotate=90, scale=\smalltextscale] at (\wid/2, \wid+2*\ty) {\it\langs{Cub.}{Cub.}};

% second branch
\draw[thin,draw=\lightcol] ({\wid/2},-\ty) -- (\wid/2,0) -- node[thick,above,rotate=60,scale=\smalltextscale,pos=.374]{\it\langs{Cubes}{Cubes}} 
node[thick,above=3, right=1,rotate=-45,scale=\smalltextscale,pos=.133]{\bf 50}
node[thick,above=3, right=1,rotate=-45,scale=\smalltextscale,pos=.333]{\bf 100}
node[thick,above=3, right=1,rotate=-45,scale=\smalltextscale,pos=.534]{\bf 200}
node[thick,above=2, right=2,rotate=-45,scale=\smalltextscale,pos=.65]{\bf 300}
node[thick,above=-1, right=5,rotate=-45,scale=\smalltextscale,pos=.734]{\bf 400}
node[thick,above=3, right=1,rotate=-45,scale=\smalltextscale,pos=.8]{\bf 500}
node[thick,above=2, right=2,rotate=-45,scale=\smalltextscale,pos=.852]{\bf 600}
node[thick,above=1, right=4,rotate=-45,scale=\smalltextscale,pos=.895]{ \bf 700}
(\wid,\wid);
\node[below=5, rotate=90, scale=\smalltextscale] at ({\wid/2-\ty/sqrt(3)},-\ty) {\it\langs{Cub.}{Cub.}};

% PI R^2 LINE
% first branch
\draw[thin, draw=\lightcol] (0, {(log10(pi*10)-1)*\wid}) -- 
node[above, rotate=45,scale=\smalltextscale,pos=.54]{\it\langs{Surf. cercle}{Area circle}}
node[thick,above=3, right=1,rotate=-45,scale=\smalltextscale,pos=.105]{\bf 4}
node[thick,above=2, right=2,rotate=-45,scale=\smalltextscale,pos=.2]{\bf 5}
node[thick,above=3, right=1,rotate=-45,scale=\smalltextscale,pos=.28]{\bf 6}
node[thick,above=3, right=1,rotate=-45,scale=\smalltextscale,pos=.35]{\bf 7}
node[thick,above=3, right=1,rotate=-45,scale=\smalltextscale,pos=.409]{\bf 8}
node[thick,above=3, right=1,rotate=-45,scale=\smalltextscale,pos=.457]{\bf 9}
node[thick,above=3, right=1,rotate=-45,scale=\smalltextscale,pos=.504]{\bf 10}
node[thick,above=3, right=1,rotate=-45,scale=\smalltextscale,pos=.679]{\bf 15}
node[thick,above=2, right=3,rotate=-45,scale=\smalltextscale,pos=.802]{\bf 20}
node[thick,above=3, right=2,rotate=-45,scale=\smalltextscale,pos=.897]{\bf 25}
({(log10(pi*10)-1)*\wid}, \wid) -- ++(\ty,\ty);
\node[above=-8, right=5, rotate=45, scale=\smalltextscale] at ({(log10(pi*10)-1)*\wid}, \wid+3*\ty) {$\pi r^2$};

% second branch
\draw[thin, draw=\lightcol] ({(log10(pi*10)-1)*\wid-\ty},-\ty) -- ({(log10(pi*10)-1)*\wid},0) --  
node[thick,above=2, right=2,rotate=-45,scale=\smalltextscale,pos=.202]{\bf 50}
node[thick,above=3, right=1,rotate=-45,scale=\smalltextscale,pos=.503]{\bf 100}
node[thick,above=3, right=1,rotate=-45,scale=\smalltextscale,pos=.805]{\bf 200}
node[above, rotate=45,scale=\smalltextscale,pos=.54]{\it\langs{Surf. cercle}{Area circle}}
(\wid,{(log10(pi*10)-1)*\wid});
\node[below, left=-4, rotate=45, scale=\smalltextscale] at ({(log10(pi*10)-1)*\wid-\ty},-\ty) {$\pi r^2$};

% 4/3 PI R^3 LINE
% first branch
\newcommand{\xone}{1/2*(\wid - (log10(4/3*pi*10)-1)*\wid)} %x-value where first branch hits the top
\draw[thin, draw=\lightcol] (0,{(log10(4/3*pi*10)-1)*\wid}) -- 
node[thick,above=1, right=3,rotate=-45,scale=\smalltextscale,pos=.665]{\bf 10}
node[thick,above=3, right=1,rotate=-45,scale=\smalltextscale,pos=.975]{\bf 15}
({\xone}, \wid) -- ++(\ty/2, \ty);
\node[above,right,rotate=60,scale=\smalltextscale] at ({\xone}, {\wid+\ty}) {$\frac{4}{3}\pi r^3$};

% second branch
\draw[thin, draw=\lightcol] ({\xone-\ty/2},-\ty) -- ({\xone}, 0) -- 
node[rotate=62,above,scale=\smalltextscale,pos=.093]{\langs{Vol. de la sphère}{Vol. of sphere}}
node[above=2, right=2,rotate=-45,scale=\smalltextscale,pos=.075]{\bf 20}
node[above=2, right=2,rotate=-45,scale=\smalltextscale,pos=.14]{\bf 25}
node[above=3, right=1,rotate=-45,scale=\smalltextscale,pos=.192]{\bf 30}
node[above=3, right=1,rotate=-45,scale=\smalltextscale,pos=.237]{\bf 35}
node[above=3, right=1,rotate=-45,scale=\smalltextscale,pos=.275]{\bf 40}
node[above=3, right=1,rotate=-45,scale=\smalltextscale,pos=.31]{\bf 45}
node[above=2, right=2,rotate=-45,scale=\smalltextscale,pos=.339]{\bf 50}
node[above=1, left=1,rotate=-45,scale=\smalltextscale,pos=.398]{\bf 60}
node[above=0, left=-1,rotate=-45,scale=\smalltextscale,pos=.442]{\bf 70}
node[above=0, left=0,rotate=-45,scale=\smalltextscale,pos=.482]{\bf 80}
node[above=0, left=0,rotate=-45,scale=\smalltextscale,pos=.515]{\bf 90}
node[above=0, left=0,rotate=-45,scale=\smalltextscale,pos=.547]{\bf 100}
node[above=0, left=2,rotate=62,scale=\smalltextscale,pos=.615]{\langs{Vol. \,sphère}{Vol. \,sphere}}
node[above=0, left=0,rotate=-45,scale=\smalltextscale,pos=.664]{\bf 150}
node[above=0, left=0,rotate=-45,scale=\smalltextscale,pos=.747]{\bf 200}
node[above=0, left=0,rotate=-45,scale=\smalltextscale,pos=.812]{\bf 250}
node[above=-2, left=0,rotate=-45,scale=\smalltextscale,pos=.865]{\bf 300}
node[above=0, left=0,rotate=-45,scale=\smalltextscale,pos=.908]{\bf 350}
node[above=-1, left=0,rotate=-45,scale=\smalltextscale,pos=.947]{\bf 400}
node[above=0, left=0,rotate=-45,scale=\smalltextscale,pos=.981]{\bf 450}
({\xone + \wid/2}, \wid) -- ++(\ty/2, \ty);
\node[below,left, rotate=60, scale=\smalltextscale] at ({\xone-\ty/2}, -\ty) {$\frac{4}{3}\pi r^3$};

% third branch
\draw[thin,draw=\lightcol] ({\xone+\wid/2-\ty/2},{-\ty}) -- ({\xone+\wid/2},0) -- 
node[above=-2,left=0,rotate=-45,scale=\smalltextscale,pos=.02]{\bf 500}
node[above=-2,left=0,rotate=-45,scale=\smalltextscale,pos=.106]{\bf 600}
node[above=0,left=3,rotate=-45,scale=\smalltextscale,pos=.178]{\bf 700}
node[above=-2,left=0,rotate=-45,scale=\smalltextscale,pos=.24]{\bf 800}
node[above=0,left=2,rotate=-45,scale=\smalltextscale,pos=.295]{\bf 900}
node[above=-2,left=0,rotate=-45,scale=\smalltextscale,pos=.344]{\bf 1000}
node[above=-2,left=0,rotate=-45,scale=\smalltextscale,pos=.667]{\bf 2000}
node[above=-2,left=1,rotate=-45,scale=\smalltextscale,pos=.855]{\bf 3000}
node[above=-2,left=-1,rotate=-45,scale=\smalltextscale,pos=.99]{\bf 4000}
node[above=0, left=2,rotate=62,scale=\smalltextscale,pos=.432]{\langs{Vol. \,sphere}{Vol. \,sphère}}
(\wid, {(log10(4/3*pi*10)-1)*\wid)});
\node[below,left, rotate=60, scale=\smalltextscale] at ({\xone+\wid/2-\ty/2}, -\ty) {$\frac{4}{3}\pi r^3$};

% horz heavy lines
\foreach \y in {2,...,20} {
 \draw[ thick] (-\ty,{(log10(\y*5)-1)*\wid}) --  (\wid+\ty,{(log10(\y*5)-1)*\wid});
}

% y-axis labels
\foreach \y in {2,...,10} {
  \node[anchor=east] at (-1*\ty, {(log10(\y*10)-1)*\wid}) {\y};
  \pgfmathsetmacro\z{int(10*\y)}
  \node[anchor=west] at (\wid+\ty, {(log10(\y*10)-1)*\wid}) {\z};
}

% vert heavy lines
\foreach \x in {2,...,20} {
 \draw[ thick] ({(log10(\x*5)-1)*\wid},-\ty) -- ({(log10(\x*5)-1)*\wid},\wid+\ty);
}

% x-axis labels
\foreach \x in {1,...,10} {
  \node[] at ({(log10(\x*10)-1)*\wid},-1.5*\ty) {\x};
}
\foreach \x in {2,...,9} {
  \pgfmathsetmacro\z{int(10*\x)}
  \node[align=left] at ({(log10(\x*10)-1)*\wid},\wid+1.5*\ty) {\z};
}

% heavy diagonals
\foreach \y in {2,...,20} {
 \draw[ thick] (0,{(log10(\y*5)-1)*\wid}) -- ({(log10(\y*5)-1)*\wid},0);
 \draw[ thick] ({(log10(\y*5)-1)*\wid},\wid) -- (\wid,{(log10(\y*5)-1)*\wid});
}

% 2 pi r line
\draw[dashdotted] (0,{(log10(2*pi)*\wid}) -- (\wid,{(log10(2*pi)*\wid)});


% left side labels

%makes label with v-center at given y-coordinate- converts to log scale, so use the linear value from .1 to 1
\newcommand{\leftlabel}[4]{
\node[scale=\smalltextscale,anchor=east] at (-2.8*\ty,{(log10(#2)+1)*\wid}) {\langs{#3}{#4}};
\draw[thin] ({-2.5*\ty},{(log10(#2)+1)*\wid}) to[out=0,in=180] (0,{(log10(#1)+1)*\wid});
}

\leftlabel{.1055}{.1055}{Pieds car.\ en déc.\ car}{Sq pieds in sq decim.}
\leftlabel{.127}{.127}{$\frac{1}{10}$ Boisseaux en liters}{$\frac{1}{10}$ Boisseau in liters}
\leftlabel{.152}{.152}{Setiers en hectolitres}{Setier in 100 liters}
\leftlabel{.160934}{.160934}{Milles anglais en kil.}{Miles in km}
\leftlabel{.1852}{.1852}{Mil.\ géog.\ ou mar.\ en kil.}{Nautical miles in km}
\leftlabel{.1949}{.19}{Toises en mètres}{Toises in meters}
\leftlabel{.19836}{.194}{$\frac{1}{100}$ Po.\ cub.\ en déc.\ cub.}{$\frac{1}{100}$ cubic pouces in liters}
\leftlabel{.2256}{.2256}{Lignes en millimèt.}{Ligne in mm}
\leftlabel{.254}{.254}{Po.\ angl.\ en centim.}{Inches in cm}
\leftlabel{.2707}{.2707}{Pouces en centim.}{Pouces in cm}
\leftlabel{.3048}{.28}{Pi.\ angl.\ en decim.}{Feet in decim.}
\leftlabel{.3059}{.31}{$\frac1{10}$ Onces en gram.}{$\frac1{10}$ ounces in g}
\leftlabel{.31415}{.318}{\boldy{Surf.\ cercle}\, $\pi r^2$}{\boldy{Area of circle}\, $\pi r^2$}
\leftlabel{.3248}{.3248}{Pieds en décimètres}{Pieds in decim.}
\leftlabel{.3419}{.34}{$\frac{1}{10}$ Arp.\ de Paris en ares}{$\frac1{10}$ arpent de Paris in ares}
\leftlabel{.3428}{.35}{Pieds cub.\ en déc.\ cub.}{Cubic pieds in decim.}
\leftlabel{.37986}{.37}{Toises car.\ en mèt.\ car}{Sq. toises in sq.\ meters}
\leftlabel{.3824}{.38}{Gros en grammes}{Gros in grams}
\leftlabel{.3898}{.392}{Li.\ de poste en kil.}{Lieue de poste in km}
\leftlabel{.4448}{.4448}{Li de 25 au d.\ en kil.}{Lieue de 25 deg.\ in km}
\leftlabel{.4/3*pi}{.4/3*pi}{\boldy{Vol.\ de la sph.}\, $\frac43 \pi r^3$}{\boldy{Vol.\ of sphere}\, $\frac43 \pi r^3$}
\leftlabel{.4895}{.4895}{$\frac1{10}$ Livres en kilogr.}{$\frac1{10}$ Livres in kg}
\leftlabel{.5107}{.5107}{$\frac1{10}$Arpent des eaux et forets in square m}{$\frac1{10}$ Arp.\ d.\ eaux et for.\ en a.}
\leftlabel{.5311}{.5311}{Grains in centigrams}{Grains en centigr.}
\leftlabel{.5556}{.5556}{Lieue de mar in km}{Li.\ mar do 20 au d.\ en k.}
\leftlabel{.2*pi}{.2*pi}{\boldy{Circonfér.}\ $2\pi r$}{\boldy{Circumpherence} $2\pi r$}
\leftlabel{.7327}{.72}{Pouces car.\ en cent.\ car}{Sq.\ pouces in sq.\ cm}
\leftlabel{.740347}{.74}{Toises cub.\ en m.\ cub}{Cub. toise in cub. meters}
\leftlabel{.7415}{.76}{Mil.\ all.\ de 115 au d.\ en k.}{Bavarian mile in km}

%RIGHT SIDE LABELS
\newcommand{\rightlabel}[4]{
\node[scale=\smalltextscale,anchor=west,align=left] at ({\wid+2.8*\ty},{(log10(#2)+1)*\wid}) {\langs{#3}{#4}};
\draw[thin] ({\wid+2.5*\ty},{(log10(#2)+1)*\wid}) to[out=180,in=0] (\wid,{(log10(#1)+1)*\wid});
}

% line labels
\rightlabel{.4188}{.4188}{\boldy{Vol.\ sph} $\frac43\pi r^3$}{\boldy{Vol.\ sph} $\frac43\pi r^3$}
\rightlabel{.1*pi}{.318}{\boldy{Surf.\ cer.\ }$\pi r^2$}{\boldy{Area circle} $\pi r^2$}

% g & such
% probably "intensite de la pesanteur"
\rightlabel{.981}{.97}{$g$ \boldy{Int. de la pes.}}{$g$ \boldy{Strength of gravity}}
\rightlabel{.4905}{.48}{$\frac12 g$}{$\frac12 g$}
\rightlabel{.4429}{.4429}{$\sqrt{2g}$}{$\sqrt{2g}$}
\rightlabel{.3132}{.308}{$\sqrt g$}{$\sqrt g$}
\rightlabel{.1962}{.194}{$\frac1{10} 2g$}{$\frac1{10} 2g$}


% polygons: each value is the reciprocal of the area of the unit regular polygon
\rightlabel{.4/sqrt(3)}{.4/sqrt(3)}{\boldy{A. triangle}}{\boldy{A. triangle}}
\rightlabel{.1}{.1}{\boldy{A. carré}}{\boldy{A. square}}
\rightlabel{.5812}{.5812}{\boldy{A. pentagone}}{\boldy{A. pentagon}}
\rightlabel{.3849}{.3849}{\boldy{10 A. hexagone}}{\boldy{10 A. hexagon}}
\rightlabel{.2071}{.2071}{\boldy{10 A. octogone}}{\boldy{10 A. octogon}}
\rightlabel{.1618}{.1618}{\boldy{10 A. ennéagone}}{\boldy{10 A. nonagon}}
\rightlabel{.1299}{.1299}{\boldy{10 A. decagone}}{\boldy{10 A. decagon}}
\rightlabel{.8931}{.88}{\boldy{100 A. dodecagone}}{\boldy{100 A. dodecagon}}

% densities- these numbers from "description et usage", p41
\rightlabel{.1}{.103}{Eau distillée}{Distilled water}
\rightlabel{.105}{.105}{Argent fondu}{Molten silver}
\rightlabel{.1135}{.1135}{Plomb fondu}{Molten lead}
\rightlabel{.136}{.136}{$\frac1{10}$ Mercure à $0^\circ$}{$\frac1{10}$ Mercury at $0^\circ$}
\rightlabel{.193}{.187}{$\frac1{10}$ Or fondu}{$\frac1{10}$ Molten gold}
\rightlabel{.2206}{.2206}{$\frac1{10}$ Platine laminé}{$\frac1{10}$ Rolled platinum}
\rightlabel{.6860}{.6860}{Zinc fondu}{Molten zinc}
\rightlabel{.721}{.721}{Fonte}{Cast iron}
\rightlabel{.729}{.74}{Etain fondu}{Molten pewter}
\rightlabel{.779}{.779}{Fer forgé}{Wrought iron}
\rightlabel{.854}{.82}{Cuivre jaune \\\it passé à la filière}{Die cast brass}
\rightlabel{.890}{.86}{Cuivre rouge}{Copper}

% TOP TEXT
\node[scale=1.2] at (\wid/2, \wid+.17*\wid) {\langs{\fontspec{Orniste tfb} Abaque {\tiny ou} \normalsize Compteur Universel,}{\fontspec{Orniste tfb} Abaque {\tiny or} \normalsize Universal Calculator,}};
\node[scale=1.7] at (\wid/2, \wid+.13*\wid){\langs{%
\fontspec{SackersItalianScriptStd} donnant à vue, à moins de $\frac1{200}$ près, les résultats de tous les calculs d'Arithmétique, de Géometrie et de Mécanique pratique, \&c}%
{\fontspec{SackersItalianScriptStd}giving on sight, accurate within $\frac1{200}$, the results of all calculations of Arithmetic, Geometry, and practical Mechanics, etc.}};
\node[scale=.8] at (\wid/2, \wid+.095*\wid){\langs{%
\boldy{par Léon Lalanne, ancien élève de l'Ecole Polytechnique, ingénier des Ponts et Chaussées.}}%
{\boldy{by Léon Lalanne, former student of the Ecole Polytechnique, engineer of the Ponts et Chaussées.}}};
\node[scale=2.4] at (\wid/2,\wid+.065*\wid){\langs{%
\fontspec{SackersItalianScriptStd}Cet Abaque a été approuvé par l'Académie des Sciences le 11 Septembre 1843}
{\fontspec{SackersItalianScriptStd}This Abaque was approved by the Academy of Sciences on September 11, 1843}};

\node[anchor=east,scale=\smalltextscale] at (0,{\wid+2*\ty}){\langs{%
\boldy{POIDS {\small ET} MESURES.}}%
{\boldy{WEIGHTS {\small AND} MEASURES.}}};

\node[anchor=west,scale=\smalltextscale,align=center] at (\wid+\ty,{\wid+2*\ty}) {\langs{%
\boldy{DENSITÉS} \\ \boldy{ET POLYGONES.}}{%
\boldy{DENSITIES} \\ \boldy{AND POLYGONS.}}};

% BOTTOM TITLE TEXT
\node[scale=.8] at ({\wid/2},{-.8*\textpad}) {\langs{%
\boldy{INSTRUCTION ABRÉGÉE POUR L'USAGE DE L'ABAQUE.}}{%
\boldy{BRIEF INSTRUCTIONS FOR USE OF THE ABAQUE}}};

% bottom text, left column
\node[anchor=north,align=left,scale=.55,text width=\textcolwid] at ({\textcolx},{-\textpad}) {\langs{%
\quad \boldy{Lecture des nombres sur l'Abaque}. \it Le nombre correspondant à un point,
\it soit sur les bords du cadre, soit sur une des droites inclinée de l'intérieur
\it de la figure, s'obtiendra facilement en considérant les chiffres 1, 2, 3, 4....10,
\it 20, 30.... 100 placés sur ces bords, comme représentant à volonté des unités
\it entières ou décimales d un ordre quelconque.

\quad\it Ainsi le 5\textsuperscript{e} point de division entre 2 et 3 peut reprèsenter à volonté les 
\it nombres 2{,}5, 25, 250 \&c et 0{,}25, 0{,}025 \&c

\quad \it Mais sur les lignes inclinées transversalement qui portent les désignations 
de \boldy{carrés} \it et de \boldy{surface du cercle} \it on ne devra lire que les nombres qui y sont 
\it inscrits et leurs multiples ou sous-multiples par 100, 10\, 000, 1\,000\,000, \&c; sur
\it celles qui sont relatives au \boldy{volume de la sphère} \it et aux \boldy{cubes},\it on ne lira que 
\it les nombres inscrits et leurs multiples ou sous-multiples par 1\,000, 1\,000\,000, \&c......

\quad Principe général de l'Abaque.\it Le produit de deux nombres se trouve absolument
\it comme dans la table attribuée vulgairement à Pythagore, par la lecture du nombre
\it de la ligne inclinée dans ce sens \begin{tikzpicture}[scale=.2]\draw (0,1) -- (1,0);\end{tikzpicture} qui est à la rencontre des deux droites 
\it l'une verticale, l'autre horizontale, correspondant aux deux facteurs.

\quad \it Ainsi le produit de 2 par 3 se trouve sur la droite inclinée portant le chiffre 6; 
\it celui de 13 par 2{,}5 tombant entre les lgnes 3{,}2 et 3{,}3, on prendra 325 pour la valeur
\it absolue du produit, qui est réellement 32{,}5 en plaçant convenablement la virgule.
}{%
\quad \boldy{Reading numbers on the Abaque}. The number corresponding to a point, 
\it either on the edges of the frame or on one of the diagonal lines inside
\it the figure, can be easily obtained by considering the digits 1, 2, 3, 4....10,
\it 20, 30.... 100 placed on these edges, representing whole or decimal units \
\it of any order as desired.

\quad \it Thus, the 5th division mark between 2 and 3 can represent the 
\it numbers 2.5, 25, 250, etc., and 0.25, 0.025, etc.

\quad \it However, on the diagonal lines labeled \boldy{squares} 
\it and \boldy{circle area}, 
\it only the numbers written on them and their multiples or quotients
\it by 100, 10,000, 1,000,000, etc., should be read. On the lines related 
\it to the volume of the sphere and cubes, only the written numbers and their 
\it quotients by 1,000, 1,000,000, etc., should be read.

\quad \boldy{General principle of the Abaque}. \it The product of two numbers is 
\it determined exactly as in the table commonly attributed to Pythagoras, by 
\it reading the number on the diagonal line in this direction \begin{tikzpicture}[scale=.2]\draw (0,1) -- (1,0);\end{tikzpicture} where
\it the vertical and horizontal lines meet, corresponding to the two factors.

\quad \it Thus, the product of 2 by 3 is found on the diagonal line with 
\it the digit 6. For the product of 13 by 2.5 falling between the lines 3.2 and 
\it 3.3, the digits of the product will be 325, which represents the answer 
\it 32.5 when placing the decimal point appropriately.%
}};


% bottom text, middle column
\node[anchor=north,align=left,scale=.55,text width=\textcolwid] at ({\wid/2},{-\textpad}) {\langs{%
\quad Réciproquement on fera la division de 32,5 par 13 en partant du point 
\it de rencontre de la droite inclinée 325 avec la verticale 13 et en suivant une
\it horizontale jusqu'à la division 2,5 du bord vertical du cadre.

\quad \it Rien n'est donc plus facile que d'obtenir le résultat de la multiplication et de la division
\it d'un nombre quelconque de quantités et notamment le quatrième terme d'une proportion.

\quad \boldy{Puissances et racines}. \it Les \boldy{carrés} \it et les \boldy{cubes} \it se trouvent sur les lignes transversales
\it qui portent ces désignations, en partant des nombres comptés sur te bas du cadre; et
\it réciproquement, les \boldy{racines carrées} \it et \boldy{cubiques} \it s'obtiennent en partant des lignes des
\it carrés et des cubes et en descendant sur les lignes du bas du cadre.

\quad\it Pour les puissances $\frac32$ qui se présentent dans diverses questions {\it d}'\boldy{hydraulique}, \it il faut
\it partir du bord vertical du cadre, et faire la lecture sur la ligne des cubes la plus proche.

\quad\it Exemples: $\sqrt 2 = 1{,}41$; $\sqrt[3]{1318} = 11{,}0$; $\overline{4{,}5}^{\frac32} = \sqrt{(4{,}5)^3} = 9{,}54$.

\quad \it Pour obtenir une puissance du 4\textsuperscript{e}, du 5\textsuperscript{e}, ..... du $n$\textsuperscript{e} degré, ce qui est utile dans les \boldy{règles}
\boldy{d'intéret} \it composé, il suffit de tracer sur l'Abaque des transversales inclinées à 1 de base 
\it pour 3,4 .....$n-1$ de hauteur; la premiére partant du point 1, et les autres se succédant
\it mutuellement comme les lignes des cubes.

\quad \boldy{Circonférence, cercle, sphère}. \it On se sert des transversales inclinées qui portent ces
\it dénominations pour obtenir la longueur d une circonférence, la superficie d'un cercle
\it et le volume d'une sphère, le rayon étant compté sur le bord inférieur du cadre.
%
}{%
\it \quad Conversely we would divide 32.5 by 13 starting from the intersection point 
\it of the diagonal line 325 with the vertical line 13 and following a 
\it horizontal line to the quotient 2.5 on the vertical edge of the frame.

\it \quad Nothing could be easier than obtaining the result of the multiplication and division
\it of any number of quantities, in particular the fourth term of a proportion.

\quad \boldy{Powers and roots.} \it The \boldy{squares} \it and \boldy{cubes} are found on the diagonal 
\it lines labeled as such, starting from the numbers counted at the bottom of the frame. 
\it Conversely, square roots and cube roots are obtained by starting from the lines 
\it of squares and cubes and descending to the bottom lines of the frame.
\it For powers of 3/2 that arise in various questions of \boldy{hydraulics}, one should 
\it start from the vertical edge of the frame and read on the nearest cube line.

\quad \it Examples: $\sqrt 2 = 1{,}41$; $\sqrt[3]{1318} = 11{,}0$; $\overline{4{,}5}^{\frac32} = \sqrt{(4{,}5)^3} = 9{,}54$.

\it To obtain a power of the 4th, 5th, ..., or $n$th degree, which is useful in \boldy{compound interest} \it calculations, simply draw diagonal lines on l'Abaque starting at 1 
\it with slopes of 3, 4, ..., $n-1$; each following the others like the lines of cubes.

\quad \boldy{Circumference, circle, sphere.} \it The diagonal lines labeled as such are 
\it used to obtain the length of a circumference, the area of a circle and the volume 
\it of a sphere, with the radius being measured on the bottom edge of the frame.
}};

% bottom text, right column
\node[anchor=north,align=left,scale=.55,text width=\textcolwid] at ({\wid-\textcolx},{-\textpad}) {\langs{%
\it Ainsi le rayon étant 5, la circonférence 31{,}4, le cercle est 78{,}6 et la sphère 527. Les questions inverses se résolvent aussi facilement.

\quad \boldy{Conversions des poids et mesures}. \it Les hauteurs comprises sur le bord à gauche 
\it du cadre entre le point de départ 1 t les petits traits correspondant aux conversions
\it des poids et mesures, serviront de multiplicateurs pour changer des mesures anciennes 
\it en nouvelles; et elles serviront de diviseurs pour le problème inverse.
\quad \boldy{Poids des volumes de diverses substances}.\it De même, aux noms des diverses
\it  substances placés à côté du bord à droite, correspondent des multiplicateurs
\it ou des diviseurs, suivant la nature de la question à résoudre.

\quad \boldy{Pesanteur}. \it Les questions relates à la chûte des corps dans le vide, au pendule,
\it à l'écoulement des liquides, se résoudront facilement à l'aide des nombres correspondant
\it à $g$, à $\frac12g$, à $\sqrt{2g}$, $\sqrt g$, à $2g$, lesquels ont été marqués sur le bord à droite du cadre.

\quad \boldy{Polygones réguliers}. \it On obtiendra l'aire d'un de ces Polygones en divisant le carré
\it de son côté par un nombre correspondant marqué sur le bord à droite du cadre
\it avec la lettre \boldy{A}.

\bigbreak
\bigbreak
\bigbreak
\bigbreak

\quad \boldy{N.B.} \it Pour plus amples détails, voir l'\boldy{instruction} imprimée qui se trouve
\it chez les mêmes dèpositaires, ainsi que les autres modèles d'Abaque.
}{%
\it Thus, given a radius of 5, the circumference is 31.4, the area of a circle is 78.6, and  
\it the volume of a sphere is 527. The inverse questions can be solved just as easily.

\quad \boldy{Conversions of weights and measures.} \it The heights marked on the left edge 
\it of the frame between the starting point 1 and the small marks corresponding to the 
\it conversions of weights and measures serve as multipliers to convert old style 
\it measurements into metric, and they serve as divisors for the inverse problem.

\quad \boldy{Weight of volumes of various substances.} \it Similarly, next to the names of  
\it various substances placed along the right edge, there are corresponding multipliers 
\it or divisors depending on the nature of the question to be solved.

\quad \boldy{Gravity.} \it Questions related to the falling of bodies in a vacuum, the pendulum, 
\it and the flow of liquids can be easily solved using the numbers corresponding 
\it to $g$, $\frac12g$, $\sqrt{2g}$, $\sqrt g$, and $2g$, which have been marked on the right edge of the frame.

\quad \boldy{Regular polygons.} \it The area of a regular polygon can be obtained by dividing the  
\it square of its side by the corresponding number marked on the right edge of the frame
\it with the letter \boldy{A}.

\bigbreak

\quad \boldy{N.B.} \it For more detailed information, please refer to the printed \boldy{instruction manual} available from the same distributors, as well as other models of the Abaque.%
}};

% SINES & COSINES, LEFT SIDE
%value light lines
\foreach \y in {10,...,100} {
\draw[lightgridline] ({-\trigpad-2.5*\trigwid},{(log10(\y)-1)*\wid}) -- ++({.5*\trigwid},0);
}

% left light scale lines 
\foreach \t in {95.833,95.916,96,96.08333,...,100.0,100.16666,100.3333,...,110.0,110.25,110.5,...,120.0,120.5,121.0,...,140.0,141.0,142.0,...,160.0,165.0,170.0,175.0} {
 \def\y{\fpeval{(1+ln(abs(cosd(\t)))/ln(10))}*\wid}
 \draw[lightgridline] ({-\trigpad-3.5*\trigwid},\y) -- ++({\trigwid},0);
}

%right light scale lines
\foreach \t in {90.5833333,90.6666666,...,95.7} {
 \def\y{\fpeval{(2+ln(abs(cosd(\t)))/ln(10))}*\wid}
 \draw[lightgridline] ({-\trigpad-2*\trigwid},\y) -- ++({\trigwid},0);
}

% frame for scales
\draw (-\trigpad,0) rectangle ({-\trigpad - 4.5*\trigwid},\wid);
\foreach \x in {0,-1,-2,-2.5,-3.5} {
 \draw ({-\trigpad+\x*\trigwid},0) -- ++(0,\wid);
}

\node[anchor=east,rotate=90,scale=.7] at ({-\trigpad-.5*\trigwid},-\ty) {\langs{\it Cosinus}{\it Cosine}};
\node[anchor=east,rotate=90,scale=.7] at ({-\trigpad-1.5*\trigwid},-\ty) {\langs{\it Sinus}{\it Sine}};
\node[anchor=east,rotate=90,scale=.5] at ({-\trigpad-2.25*\trigwid},-\ty) {\langs{\boldy{Nombres}}{\boldy{Values}}};
\node[anchor=east,rotate=90,scale=.7] at ({-\trigpad-3*\trigwid},-\ty) {\langs{\it Sinus}{\it Sine}};
\node[anchor=east,rotate=90,scale=.7] at ({-\trigpad-4*\trigwid},-\ty) {\langs{\it Cosinus}{\it Cosine}};

% value heavy lines
\foreach \y in {2,...,20} {
\draw[] ({-\trigpad-2.5*\trigwid},{(log10(\y*5)-1)*\wid}) -- ++({.5*\trigwid},0);
}

%left heavy little scale lines
\foreach \t in {96,96.5,97,97.5,98,...,100,101,102,...,140,145,150,...,160,170} {
 \def\y{\fpeval{(1+ln(abs(cosd(\t)))/ln(10))}*\wid}
 \draw ({-\trigpad-3.5*\trigwid},\y) -- ++({\trigwid},0);
}

%right heavy little scale lines
\foreach \t in {91,91.5,...,95.5} {
 \def\y{\fpeval{(2+ln(abs(cosd(\t)))/ln(10))}*\wid}
 \draw[] ({-\trigpad-2*\trigwid},\y) -- ++({\trigwid},0);
}

% biggest left scale lines with numbers
\foreach \t in {96,97,98,99,100,102,104,106,108,110,115,120,125,130,135,140,150,160,180} {
 \def\y{\fpeval{(1+ln(abs(cosd(\t)))/ln(10))}*\wid}
 \pgfmathsetmacro{\z}{int(180-\t)}
 \pgfmathsetmacro{\w}{int(90-\z)}
 \pgfmathsetmacro{\v}{int(180-\w)}
 \draw ({-\trigpad-4.5*\trigwid},\y) -- ++({2*\trigwid},0);
 \node[above=-2] at ({-\trigpad-4*\trigwid}, \y) {\t};
 \node[below=-2] at ({-\trigpad-4*\trigwid}, \y) {\z};
 \node[above=-2] at ({-\trigpad-3*\trigwid}, \y) {\v};
 \node[below=-2] at ({-\trigpad-3*\trigwid}, \y) {\w};
}

% biggest right scale lines with numbers
\foreach \t in {91,92,93,94,95} {
 \def\y{\fpeval{(2+ln(abs(cosd(\t)))/ln(10))}*\wid}
 \pgfmathsetmacro{\z}{int(180-\t)}
 \pgfmathsetmacro{\w}{int(90-\z)}
 \pgfmathsetmacro{\v}{int(180-\w)}
 \draw ({-\trigpad},\y) -- ++({-2*\trigwid},0);
 \node[above=-2] at ({-\trigpad-.5*\trigwid}, \y) {\t};
 \node[below=-2] at ({-\trigpad-.5*\trigwid}, \y) {\z};
 \node[above=-2] at ({-\trigpad-1.5*\trigwid}, \y) {\v};
 \node[below=-2] at ({-\trigpad-1.5*\trigwid}, \y) {\w};
}

% TANS & COTANS, RIGHT SIDE
%value light lines
\foreach \y in {10,...,100} {
\draw[lightgridline] ({\wid+\trigpad+2*\trigwid},{(log10(\y)-1)*\wid}) -- ++({.5*\trigwid},0);
}

% left light scale lines 
\foreach \t in {90.6,90.7,...,91,91.08333,91.1666666,...,95.7} {
 \def\y{\fpeval{(2+ln(abs(cotd(\t)))/ln(10))}*\wid}
 \draw[lightgridline] ({\wid+\trigpad+\trigwid},\y) -- ++({\trigwid},0);
}

%right light scale lines
\foreach \t in {95.75, 95.833333,...,100,100.166666,100.33333,...,110,110.25,110.5,...,120,120.5,121,...,135} {

 \def\y{\fpeval{(1+ln(abs(cotd(\t)))/ln(10))}*\wid}
 \draw[lightgridline] ({\wid+\trigpad+2.5*\trigwid},\y) -- ++({\trigwid},0);
 \def\z{(1+log10(-cot(\t)))}
% \pgfmathparse{\z}\message{\t,\pgfmathresult}
}

% frame for scales
\draw (\wid+\trigpad,0) rectangle ({\wid+\trigpad + 4.5*\trigwid},\wid);
\foreach \x in {0,1,2,2.5,3.5} {
 \draw ({\wid+\trigpad+\x*\trigwid},0) -- ++(0,\wid);
}

\node[anchor=east,rotate=90,scale=.7] at ({\wid+\trigpad+.5*\trigwid},-\ty) {\langs{\it Cotangentes}{\it Cotangent}};
\node[anchor=east,rotate=90,scale=.7] at ({\wid+\trigpad+1.5*\trigwid},-\ty) {\langs{\it Tangentes}{\it Tangent}};
\node[anchor=east,rotate=90,scale=.5] at ({\wid+\trigpad+2.25*\trigwid},-\ty) {\langs{\boldy{Nombres}}{\boldy{Values}}};
\node[anchor=east,rotate=90,scale=.7] at ({\wid+\trigpad+3*\trigwid},-\ty) {\langs{\it Tangentes}{\it Tangent}};
\node[anchor=east,rotate=90,scale=.7] at ({\wid+\trigpad+4*\trigwid},-\ty) {\langs{\it Cotangentes}{\it Cotangent}};

% value heavy lines
\foreach \y in {2,...,20} {
\draw[] ({\wid+\trigpad+2*\trigwid},{(log10(\y*5)-1)*\wid}) -- ++({.5*\trigwid},0);
}

%left heavy little scale lines
\foreach \t in {91,91.5,...,95.6} {
 \def\y{\fpeval{(2+ln(abs(cotd(\t)))/ln(10))}*\wid}
 \draw ({\wid+\trigpad+\trigwid},\y) -- ++({\trigwid},0);
}

%right heavy little scale lines
\foreach \t in {96,96.5,...,100,101,102,...,135} {
 \def\y{\fpeval{(1+ln(abs(cotd(\t)))/ln(10))}*\wid}
 \draw[] ({\wid+\trigpad+2.5*\trigwid},\y) -- ++({\trigwid},0);
}

% biggest left scale lines with numbers
\foreach \t in {91,...,95} {
 \def\y{\fpeval{(2+ln(abs(cotd(\t)))/ln(10))}*\wid}
 \pgfmathsetmacro{\z}{int(180-\t)}
 \pgfmathsetmacro{\w}{int(90-\z)}
 \pgfmathsetmacro{\v}{int(180-\w)}
 \draw ({\wid+\trigpad+0*\trigwid},\y) -- ++({2*\trigwid},0);
 \node[above=-2] at ({\wid+\trigpad+.5*\trigwid}, \y) {\t};
 \node[below=-2] at ({\wid+\trigpad+.5*\trigwid}, \y) {\z};
 \node[above=-2] at ({\wid+\trigpad+1.5*\trigwid}, \y) {\v};
 \node[below=-2] at ({\wid+\trigpad+1.5*\trigwid}, \y) {\w};
}

% biggest right scale lines with numbers
\foreach \t in {96,...,100,102,104,...,110,115,120,125,130,135} {
 \def\y{\fpeval{(1+ln(abs(cotd(\t)))/ln(10))}*\wid}

 \pgfmathsetmacro{\z}{int(180-\t)}
 \pgfmathsetmacro{\w}{int(90-\z)}
 \pgfmathsetmacro{\v}{int(180-\w)}
  \draw ({\wid+\trigpad+2.5*\trigwid},\y) -- ++({2*\trigwid},0);
 \node[above=-2] at ({\wid+\trigpad+4*\trigwid}, \y) {\t};
 \node[below=-2] at ({\wid+\trigpad+4*\trigwid}, \y) {\z};
 \node[above=-2] at ({\wid+\trigpad+3*\trigwid}, \y) {\v};
 \node[below=-2] at ({\wid+\trigpad+3*\trigwid}, \y) {\w};
}


% Chris Staecker note at bottom
\node[anchor=west,scale=.7] at ({-\trigpad-4.5*\trigwid},-7*\textpad) {\fontspec{Baskerville}Typeset \& translated by Chris Staecker. More info: \quad \qrcode[height=.75cm]{https://cstaecker.fairfield.edu/\string~cstaecker/machines/lalanne.html}};

\end{tikzpicture}


}
\end{center}











\end{document}