# Align two equations, second split over two lines. Enclosed by leading left brace

I am attempting to reproduce an existing typeset equation. Several hours into this. I have tried every combination of align, aligned, cases, split, empheq, multline, eqnarray, that I can think of.

The equation that I am attempting to reproduce is illustrated here:

Edit: removed original code.

2nd Edit: Satisfied with result - change code and result image.

3rd Edit: Slight adjustment to adjustwidth - equation now well positioned horizontally, line breaks and page breaks back to where I would like. Incorporate suggestion @egreg.

The reason for the \usepackage[textwidth=10cm, textheight=20cm]{geometry} is that gives me line and page breaks very close to the format of the original facsimile, hence makes transcription easier.

The original article(fourth in a series of five): Nouvelles annales de mathématiques 3e série, tome 3(1884), p. 545-561


\documentclass[12pt, leqno]{article}
\usepackage{empheq}
\usepackage{changepage}
\usepackage[textwidth=9.5cm, textheight=20cm]{geometry}
\usepackage[french]{babel}
\usepackage[math-style=french]{unicode-math}
\setmathfont{Latin Modern Math}
\pagenumbering{gobble}
\DeclareMathOperator{\tr}{tr.}
\begin{document}
gentes quelconques $(λ_0,θ_0$ et $(λ,θ)$ en divisant cette
aire en triangles infinitésimaux par les tangentes menées
aux points intermédiaires entre $M_0$ et $M$ la surface de
chacun de ces petits triangles est égale à
$\frac{\text{\textsc{i} }}{2}t(t+dt)\sin dθ,$
ou, en négligeant les infiniment petits d’ordre supérieur, à
$\frac{t^2dθ}{2}\text{\quad ou\quad }\frac{\text{\textsc{i} }}{2}\left(\frac{dλ}{dθ}\sin θ\right)^2dθ;$
il viendra, par suite, pour l’expression de l’aire cherchée
$\tag{8}σ=\frac{\text{\textsc{i} }}{2}\int_{θ_0}^{θ}\left(\frac{dλ}{dθ}\sin θ\right)^2dθ.$

Si l’on veut l’aire Σ comprise entre la courbe, l’axe
$Ox$ et les ordonnées $M_0P_0$ et $MP$, on a

\small
\begin{empheq}[left=\mkern-15mu\empheqlbrace]{align}\tag{9}
\begin{alignedat}{2}
Σ&=σ+\mathrlap{\tr MTP-\tr M₀T₀P₀} \\
& =\frac{\;\text{\textsc{i}}}{2}\biggl[
& ∫_{θ₀}^θ&\biggl(\frac{dλ}{dθ}\sin θ\biggr)^{\mkern-5mu 2}dθ \\
& & {}+{}&\biggl(\frac{dλ}{dθ}\sin θ\biggr)^{\mkern-5mu 2}\sin θ\cos θ
-\biggl(\frac{dλ₀}{dθ₀}\sin θ\biggr)^{\mkern-5mu 2}\sin θ\cos θ \biggr].
\end{alignedat}
\end{empheq}

\vspace{2em}
\centerline{VIII. — \textsc{Applications}}

\vspace{1em}
48. \textit{Trouver une courbe dont la tangente ait une
longueur constante l.} — Cette courbe est celle que
décrit un point lié par un fil inextensible à un point qui
se meut sur une droite indéfinie; de là le nom de \textit{tractrice.}

La formule (5) (n° 44) donne immédiatement pour
l’équation différentielle de la courbe
$l=\frac{dλ}{dθ}\sin θ\text{\quad ou \quad}dλ=\frac{l\,dθ}{\sin V}.$
\end{document}



I obtain:

For context: the facsimile of the page in question.

• Should the left-hand curly brace span just the last two rows, or all three? Please advise.
– Mico
May 15, 2021 at 18:42
• @Mico Yes. Thanks, and sorry, the article facsimile is unclear. I am editing my original post to correct and perhaps make my intention more clear. May 16, 2021 at 1:23
• Just figured out how to shove the minipage to the left. I simply enclosed the minipage in an {\hspace{2cm} \begin{minipage} ... \end{minipage} }. almost perfection! Will edit code and image in original question. May 16, 2021 at 2:57
• I'd avoid the inconsistencies of the original: if you look closely, “tr.” is followed by a thin space in the first instance, but not in the second one. You get the thin space everywhere if you do \DeclareMathOperator{\tr}{tr.} and use \tr instead of \mathrm{tr.}. May 16, 2021 at 14:16
• @egreg Noted, Thanks. May 16, 2021 at 15:22

Here is a possibility, with empheq and alignedat:

\documentclass[11pt, leqno]{article}
\usepackage{empheq}
\usepackage[textwidth=10cm, textheight=20cm]{geometry}
\usepackage[french]{babel}
\usepackage[math-style=french]{unicode-math}
\setmathfont{Latin Modern Math}
\pagenumbering{gobble}

\begin{document}

\begin{empheq}[left=\empheqlbrace]{equation}\tag{9}
\begin{alignedat}{2}
Σ&=σ+\mathrlap{tr.MTP-tr.M_0T_0P_0} \\
& =\frac{\;\text{\textsc{i}}}{2}
\bigl[\int_{θ_0}^θ\biggl(\frac{dλ}{dθ}\sin θ\biggr)^{\mkern-5mu 2}dθ
& & +\biggl(\frac{dλ}{dθ}\sin θ\biggr)^{\mkern-5mu 2}\sin θ\cos θ \\
& & & -\biggl(\frac{dλ_0}{dθ_0}\sin θ\biggr)^{\mkern-5mu 2}\sin θ\cos θ \biggr]
\end{alignedat}
\end{empheq}

\end{document}


Edit: For a layout closer to the original, one can use a smaller font (for the equation) and nest the equation in an adjustwidth environment, from changepage:

\documentclass[11pt, leqno]{article}
\usepackage{empheq}
\usepackage{changepage}
\usepackage[textwidth=10cm, textheight=20cm]{geometry}
\usepackage[french]{babel}
\usepackage[math-style=french]{unicode-math}
\setmathfont{Latin Modern Math}
\pagenumbering{gobble}

\begin{document}

Si l’on veut l’aire $\Sigma$ comprise entre la courbe, l’axe $Ox$ et les ordonnées $M_0P_0$ et $MP$, on a

\small
\begin{empheq}[left=\mkern-15mu\empheqlbrace]{align}\tag{9}
\begin{alignedat}{2}
Σ&=σ+\mathrlap{\mathrm{tr}.MTP-\mathrm{tr}.M₀T₀P₀} \\
& =\frac{\;\text{\textsc{i}}}{2}\biggl[
& ∫_{θ₀}^θ&\biggl(\frac{dλ}{dθ}\sin θ\biggr)^{\mkern-5mu 2}dθ \\
& & {}+{}&\biggl(\frac{dλ}{dθ}\sin θ\biggr)^{\mkern-5mu 2}\sin θ\cos θ
-\biggl(\frac{dλ₀}{dθ₀}\sin θ\biggr)^{\mkern-5mu 2}\sin θ\cos θ \biggr].
\end{alignedat}
\end{empheq}

\end{document}


• Brackets are definitely needed after the 1/2 and at the end of that line. See scan in the question. May 16, 2021 at 0:26
• @Bernard Thank you. I have edited my original question to incorporate your answer and hopefully correct some misunderstanding I provoked. May 16, 2021 at 2:45
• @JeffHiggins: If I understand correctly, you'd like to habe the equation number in the left margin? May 16, 2021 at 8:31
• @Bernard Yes. I have edited my original question to include an image of the page in question. Thanks for your help. May 16, 2021 at 12:46
• @JeffHiggins: Please take a look at my edit, if it is what you want. May 16, 2021 at 13:05

Something like this?

Note the use of oldstyle numerals in the equation number, the \frac{1}{2} term and 0-subscripts in row 1, as well as the extra kerning in the exponents associated with the instances of \Bigl(...\Bigr)^{2}.

\documentclass[11pt,leqno]{article}
\usepackage{mathtools} % for 'dcases' env.
\DeclareMathOperator{\tr}{tr.} % trace operator, right?
\usepackage[a4paper,textwidth=11cm, textheight=20cm]{geometry}
\usepackage[french]{babel}
\usepackage{fontspec}
\setmainfont{Latin Modern Roman}[Numbers=OldStyle]
\usepackage[math-style=french,]{unicode-math}
\setmathfont{Latin Modern Math}
\pagenumbering{gobble}

\begin{document}

\begin{align*}
&\mkern6mu Σ =σ+\tr MTP-\tr M_0T_{\!0}P_{\!0} \\
\tag{9}
&\begin{dcases}
\begin{aligned}
&=\frac{\text{1}}{\text{2}} \biggl[\, \int_{θ_0}^θ\Bigl(\frac{dλ}{dθ}\sin θ\Bigr)^{\!2}dθ
\\

• You increase \textwidth :-). With original width of text equation is longer and consequently equation number is pushed to own line. So, I move one term from second line to the first. May 15, 2021 at 18:47