I would like to insert quotations to first pages devoted to \part{}
of a document at the bottom right as shown in the picture below.
Could you help me? Thanks!

The following method should work independent of the document class.
The output page of the part page is manipulated via package atbegshi to add the part note at the lower right corner of the text area.
First macro \partnote ensures, that the current page will be the part page by calling \cleardoublepage (flushing of floats, emitting an even numbered fill page depending on the document class settings). Then the node is placed via \AtBeginShipoutNext on the next output page.
\AtBeginShipoutUpperLeft starts a picture environment at the upper left corner of the page. The \put commands move to the lower right corner of the text area using e-TeX's \dimexpr for calculations. Package picture allows the specification of dimen or length values in picture's commands (e.g. `\put).
The following example also shows frames to visualize the page layout via option showframe of package geometry. (Option pass means that the current layout settings are not changed by package geometry.)
The full example file:
\documentclass[a5paper, landscape]{article}
% Options a5paper and landscape to get a smaller image for TeX.SX
\usepackage[pass, showframe]{geometry}% only for the frame
\usepackage{atbegshi}
\usepackage{picture}
\newcommand*{\partnotefont}{\itshape}
\newcommand{\partnote}[1]{%
\cleardoublepage
\AtBeginShipoutNext{%
\AtBeginShipoutUpperLeft{%
\put(\dimexpr\ifodd\value{page}\oddsidemargin\else\evensidemargin\fi
+ 1in + \textwidth\relax,
-\dimexpr\topmargin + 1in + \headheight + \headsep + \textheight){%
\makebox(0,0)[rb]{%
\partnotefont
\begin{tabular}{@{}c@{}}#1\end{tabular}%
}%
}%
}%
}
}
\begin{document}
\partnote{%
Necessity of the method in the search for truth\\
Ren\'e Descartes (1596--1650)
}
\part{Tintin in America}
\end{document}
TeX allows the depth of the lowest box on a page to stick into the bottom margin. The amount is limited by \maxdepth. The following refinement measures the depth of the bottom line of the box and allows it to stick into the bottom margin:
\documentclass[landscape,a5paper]{article}
\usepackage[pass,showframe]{geometry}% only for the frame
\usepackage{atbegshi}
\usepackage{picture}
\usepackage{varwidth}
\newcommand*{\partnodefont}{\itshape}
\newcommand{\partnote}[1]{%
\cleardoublepage
\AtBeginShipoutNext{%
\AtBeginShipoutUpperLeft{%
\put(\dimexpr\ifodd\value{page}\oddsidemargin\else\evensidemargin\fi
+ 1in + \textwidth\relax,
-\dimexpr\topmargin + 1in + \headheight + \headsep + \textheight){%
\makebox(0,0)[rb]{%
\partnodefont
\sbox0{\begin{varwidth}[b]{\linewidth}\centering#1\end{varwidth}}%
\ifdim\dp0>0pt
\dp0=\dimexpr\dp0 - \ifdim\dp0>\maxdepth \maxdepth\else\dp0 \fi
\fi
\usebox{0}%
}%
}%
}%
}
}
\begin{document}
\partnote{%
Necessity of the method in the search for truth\\
Ren\'e Descartes (1596--1650)
}
\part{Tintin in America}
\end{document}
The base line of the note's lower line is now at the bottom of the text area.
\documentclass{...}and ending with\end{document}. – R. Schumacher Apr 12 '15 at 01:26