Assembling natural numbers in such nice patterns often has interesting consequences, and so it is in this case. Each of the aforementioned matrices is endowed with positive definiteness of a very high order: for every positive real number r the matrices with entries a\-, b\-, c\-, and d\are positive semidefinite. This special property is called infinite divisibility and is the subject of this paper. Positive semidefinite matrices arise in diverse contexts: calculus (Hessians at minima of functions), statistics (correlation matrices), vibrating systems (stiffness matrices), quantum mechanics (density matrices), harmonic analysis (positive definite functions), to name just a few. Many of the test matrices used by numerical analysts are positive definite. One of the interests of this paper might be the variety of examples that are provided in it. The general theorems and methods presented in the context of these examples are, in fact, powerful techniques that could be used elsewhere. In this introductory section we begin with the basic definitions and notions related to positive semidefinite matrices.