DOI: https://doi.org/10.21203/rs.3.rs-59813/v1

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 view of dimensional homogeneity. His view *allows* the following, and makes it possible to create equations such as 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 view of dimensional homogeneity. He merely stated* *that his view of dimensional homogeneity is equivalent to *unspecified *axioms left behind by the ancient Greeks. Presumably, his colleagues accepted his *unproven* view because it enabled him to solve 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 because symbols in parametric equations can rationally represent *only* numerical value.

The appraisal and the changes in modern engineering science required by the appraisal conclusions are presented in the text.

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