And just as knowledge of latent schematism may be necessary to formulate axioms, knowledge of axioms may be necessary to acquire knowledge of latent schematisms. Unlike the density of a solid, which can be calculated relatively directly by measuring the weight of a quantity of the solid and dividing by the volume, measurement of the density of a gas must be made indirectly, for the weight cannot be directly measured on a scale. He heated the phial so that some of the alcohol evaporated, filling the bladder, then removed the bladder and measured how much of the alcohol had evaporated from the phial. Using the principle of conservation of mass, an axiom of nature, he reasoned that the mass lost from the phial was equal to the mass that filled the bladder, and hence he was able to calculate the density of the vapor. Another simpler example of this is the way in which knowledge of the laws of optics made possible the development of the microscope, which in turn revealed the hidden inner schematism of objects, which led to knowledge of further laws.
Prediction is impossible without both a hypothesized law and a hypothesized initial state.
And since our hypotheses are tested by their ability to make successful predictions, our axioms and our hypothesized schematisms are always tested together. There can be no knowledge of laws without knowledge of latent schematisms, and no knowledge of latent schematism without knowledge of laws. Permission granted to distribute in any medium, commercial or non-commercial, provided all copyright notices remain intact.