  Glasses are ubiquitous in daily life and technology. However the microscopic
mechanisms generating this state of matter remain subject to debate: Glasses
are considered either as merely hyper-viscous liquids or as resulting from a
genuine thermodynamic phase transition towards a rigid state. We show that
third- and fifth-order susceptibilities provide a definite answer to this
longstanding controversy. Performing the corresponding high-precision nonlinear
dielectric experiments for supercooled glycerol and propylene carbonate, we
find strong support for theories based upon thermodynamic amorphous order.
Moreover, when lowering temperature, we find that the growing transient domains
are compact - that is their fractal dimension d_f = 3. The glass transition may
thus represent a class of critical phenomena different from canonical
second-order phase transitions for which d_f < 3.
