One of the most frequently used means of reasoning and interpreting is Pythagorean analogia, that is, four-place analogies of the form A : B :: C : D (A is to B as C is to D). Charles Sanders Peirce thought argument from analogy was the most complete form of reasoning since it is composed of the three major classes of argument: abduction, deduction, and induction. Such four-place analogies have often been proposed as counter-examples to the main theorem of Peirce’s logic of relations, specifically that monadic, dyadic, and triadic relations are separately necessary and jointly sufficient for a relationally complete logic. Central to this theorem is the claim that there are genuine triadic relations, that is, relations which cannot be analyzed into combinations of relations of lesser adicity. In this essay I will argue that Pythagorean analogia not only fail to be to counterexamples to Peirce’s thesis that there are genuine triadic relations, but instead turn out to exemplify that claim beautifully. Twentieth-Century mathematical logician Marshall Stone, a key scholar in mathematical logic, averred ‘a cardinal principle of modern mathematical research maybe be stated as a maxim: “One must always topologize”.’ Peirce was decades in advance of Stone’s insight because he maintained that the logic of relations should be represented topologically. This essay will first survey the fundamentals of Peirce’s logic of relations adequate to explicate and then defend Peirce’s Composability-of-Relations Theorem by means of a diagrammatical logic involving one-dimensional, topological networks. This diagrammatical logic, called Peircean Relational Graph Theory, is a dual of standard mathematical graph theory. A second topological model for Peirce’s logic of relations consisting of two-dimensional surfaces with and without boundaries will be sketched. This diagrammatic logic is called Peircean Relational Surface Theory. Both of these topological logics as well as algebraical logic will each be employed to show that Pythagorean analogia must be composed of genuine triadic relations. Pythagorean analogia will reveal that Peircean Relational Surface diagrams have diagrammatical virtues lacking in Peircean Relational Graph diagrams. (Note: The mathematics necessary to follow this essay are contained therein.)