Mathematics rarely evolves in a vacuum. Major algorithms are almost always born out of practical necessities, and the Gram-Schmidt orthogonalization process is no exception. Long before it became a standard undergraduate linear algebra exercise, the core philosophy of turning a set of messy, dependent vectors into a clean, orthogonal basis was driven by a single, persistent problem: how to find the best possible approximation of data and functions through the method of least squares. In the late eighteenth and early nineteenth centuries, giants like Pierre-Simon Laplace grappled with errors in astronomical observations and data fitting. When attempting to minimize sum-of-squares errors, implicit forms of orthogonalization naturally emerged. Although Laplace did not frame his work as an explicit vector-space algorithm—indeed, the modern concept of a vector space was decades away—his approach inherently relied on building orthogonal components to simplify calculations. These early pionee...