Among the most crucial assumptions in the solution of any engineering problem, whether it be a problem in engineering science or in engineering design, is the assumption of how any particular mode of failure will occur. Indeed, it is the analyst's or designer's preconceived ideas about failure that drive the analysis or design (which beyond conceptualization necessarily proceeds by some form of analysis), and virtually every theoretical calculation or experimental measurement on an analytical or scale model of a real or projected system is most significant in its relationship to how the system is imagined to fail in theory or practice.

As an illustration of these ideas, a fundamental problem that Galileo (1638) identified as central to determining the strength of materials will be explicated in this chapter. Galileo's famous problem of the cantilever beam may be taken as a paradigm of all engineering design problems, and his flawed analysis may serve as a paradigm of logical error in the solution of such problems. The unassailable genius of Galileo and the undeniable greatness of his many contributions to engineering mechanics make his error all the more suitable as the basis for a paradigm intended to emphasize the ease into which we all can fall into error.

Galileo's Problem of the Cantilever Beam

The several examples of notable failures of Renaissance engineering with which Galileo opens his treatise served for him as counterexamples to the prevailing hypothesis that geometry alone was sufficient to analyze and design structures and machines.