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

Using answer from @Bernard:

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 ouquad }frac{text{textsc{i} }}{2}left(frac{dλ}{dθ}sin θright)^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 θright)^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

begin {adjustwidth}{-2.5em}{-4em}
small
 begin{empheq}[left=mkern-15muempheqlbrace]{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 θbiggr)^{mkern-5mu 2}dθ 
     & & {}+{}&biggl(frac{dλ}{dθ}sin θbiggr)^{mkern-5mu 2}sin θcos θ
     -biggl(frac{dλ₀}{dθ₀}sin θbiggr)^{mkern-5mu 2}sin θcos θ biggr].
    end{alignedat}
    end{empheq}
end{adjustwidth}

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 θtext{quad ou quad}dλ=frac{l,dθ}{sin V}.]
end{document}

I obtain:

For context: the facsimile of the page in question.

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}^θbiggl(frac{dλ}{dθ}sin θbiggr)^{mkern-5mu 2}dθ
 & & +biggl(frac{dλ}{dθ}sin θbiggr)^{mkern-5mu 2}sin θcos θ 
 & & & -biggl(frac{dλ_0}{dθ_0}sin θbiggr)^{mkern-5mu 2}sin θcos θ biggr]
end{alignedat}
end{empheq}

end{document}

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

begin {adjustwidth}{-1.5em}{}
small
 begin{empheq}[left=mkern-15muempheqlbrace]{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 θbiggr)^{mkern-5mu 2}dθ 
     & & {}+{}&biggl(frac{dλ}{dθ}sin θbiggr)^{mkern-5mu 2}sin θcos θ
     -biggl(frac{dλ₀}{dθ₀}sin θbiggr)^{mkern-5mu 2}sin θcos θ biggr].
    end{alignedat}
    end{empheq}
end{adjustwidth}

end{document} 

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}^θBigl(frac{dλ}{dθ}sin θBigr)^{!2}dθ
 
&qquad +Bigl(frac{dλ}{dθ}sin θBigr)^{!2}sin θcos θ
-Bigl(frac{dλ_0}{dθ_0}sin θBigr)^{!2}sin θcos θ biggr],.
end{aligned}
end{dcases}
end{align*}

end{document}