Charles Sanders Peirce who is one of the most original and productive logicians of the nineteenth century is the founder of logic of relations. Two volumes of his Collected Papers which was gathered after his death consists of his writings on logic. In this work we tried to explicate his first paper on the logic of relations which was written in 1870 and which takes place in volume 3 of his Collected Papers. Peirce, first of all, introduces De Morgan's notation and states that this notation can be used outside the logic of absolute terms. Then after emphasizing that the relation of inclusion is a transitive relation he gives definitions and properties of equality, addition and multiplication operations. Here Peirce distinguishes between normal multiplication and the multiplication of functions. He also defines exponentiation, subtraction, division, taking root and logarithm. He says that there are three kinds of terms in the logic of relations which are individual terms, infinitesimal relatives and elementary relatives. Afterwards he gives the general formulas about the logic of relations. Eventually he classifies elementary relatives in different ways. Since Peirce had created many new terms in his essay we gave these terms and their Turkish equivalents in a vocabulary at the end of this work.