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.The glassy state of matter, despite is omnipresence in nature and technology (1), continues to be one of the most puzzling riddles in condensed-matter physics (1, 2): For all practical purposes, glasses are rigid like crystals but they lack any long-range order. Some theories describe glasses as kinetically constrained liquids (3), becoming so viscous below the glass transition that they seem effectively rigid. By contrast, other theories (4, 5) are built on the existence of an underlying thermodynamic phase transition to a state where the molecules are frozen in well defined, yet disordered positions. This so-called "amorphous order" cannot be revealed by canonical static correlation functions, but rather by new kinds of correlations [i.e. point-to-set correlations or other measures of local order (6, 7)] that have now been detected in recent numerical simulations (7)(8)(9). In these theories, thermodynamic correlations lock together the fluctuations and response of the molecules, which collectively rearrange over some length-scale , ultimately leading to rigidity. In this thermodynamic scenario, is proportional to a power of ln(τ α /τ 0 ) where τ α is the structural relaxation time and τ 0 is the microscopic time-scale, generally smaller than 1 arXiv:1606.04079v1 [cond-mat.dis-nn] 13 Jun 2016 1ps (4, 5). Because equilibrium measurements require a time larger than τ α , they cannot be performed in the range where is very large since this would require exponentially long times. This limitation is essentially why the true nature of glasses is still a matter of intense debate.Here, we propose a pioneering strategy to unveil the existence of a thermodynamic length that grows upon cooling. Instead of only varying the temperature T , we also vary the non-linear order k of the response of supercooled liquids. This is motivated by a general, although rarely considered (10), property of critical points: At a second order critical temperature T c , the linear susceptiblity χ 1 associated to the order parameter is not the only diverging response. As a function of temperature, all the higher order responses χ 2m+1 with m ≥ 1 diverge even faster than χ ...