  The subject of these Notes is the new proof, proposed in [F. H{\'e}lein,
In{\'e}galit{\'e} isop{\'e}rim{\'e}trique et calibrations, Annales de
l'Institut Fourier 44, 4 (1994), 1211-1218] of the classical isoperimetric
inequality in the plane. This proof is far from being the first one, but its
interest is that it uses essentially integration by parts and Stoke's formula
in a simple manner, like in a calibration. Here we expound again this proof and
discuss in which sense our proof works like a calibration, or it can be
understood in the framework of the theory of null Lagrangians, which goes back
to Weierstrass, Mayer, Hilbert, Weyl, Carath{\'e}odory, and Lepage.
