inch to the fourth

In US design the second moment of area in in⁴ is what sits in the denominator of every deflection formula. A simply supported beam under a uniform load deflects $\delta = 5wL^4/(384EI)$, and with $E$ for steel at 29 000 ksi, lengths in inches and $w$ in kip per inch, the answer comes out in inches with no conversion needed. That internal consistency is why the customary system's structural corner works despite the mess elsewhere: kips, inches, ksi and in⁴ are a coherent set.

The fourth-power dependence on depth is where the design intuition lives. A 2×10 joist laid on edge is roughly 3.4 times stiffer than a 2×8 in the same span, from depth alone, so a floor that bounces is almost always solved by going deeper rather than wider. The same logic drives the shape of every rolled section: material near the neutral axis contributes almost nothing to $I$, so an efficient bending member is thin in the middle and heavy at the extremes, constrained only by the need to keep the thin web from buckling.