Until the nineteenth century, engineering science was founded on a view of dimensional homogeneity that required the following: • Parameters must not be multiplied or divided. • Dimensions must not be assigned to numbers. • Equations must be dimensionless. This view made it impossible to create equations such as the laws of modern engineering science. Modern engineering science is founded on Fourier’s radically different nineteenth century view of dimensional homogeneity. His view allows the following, and makes it possible to create the laws of modern engineering science: • Parameters may be multiplied or divided. • Dimensions may be assigned to numbers. • Equations may or may not be dimensionless. Fourier did not prove the validity of his radically different view of dimensional homogeneity. He merely stated that his view of dimensional homogeneity “is the equivalent of the fundamental lemmas which the Greeks have left us without proof”. Presumably, his colleagues accepted his unproven view because he solved problems they were unable to solve. A critical appraisal of Fourier’s unproven view of dimensional homogeneity results in the following conclusions: • Parameters cannot rationally be multiplied or divided. Only the numerical values of parameters can rationally be multiplied or divided. • Dimensions cannot rationally be assigned to numbers. If dimensions could be assigned to numbers, any equation could be regarded as dimensionally homogeneous. • Equations are inherently dimensionless and dimensionally homogeneous because symbols in parametric equations can rationally represent only numerical value. The changes required by the appraisal conclusions result in a much simpler engineering science because parameters such as material modulus and heat transfer coefficient are abandoned. They are abandoned because problems are readily solved without them, and because when dealing with nonlinear behavior (as in the inelastic region and in various forms of convection heat transfer), they are extraneous variables that greatly complicate solutions. Examples in the text demonstrate how to solve problems without using parameters such as material modulus or heat transfer coefficient.

dimensional homogeneity, Fourier, heat transfer coefficient, inelastic region, laws of engineering, modulus, parameter symbols