This landmark work by Richard Dedekind includes his method for construction irrational numbers from rational numbers, thus demonstrating the continuity of the real number line using algebraic methods. The method Dedekind outlines here is the standard method taught to students today. From this point, Dedekind goes on to reconstruct the regular arithmetic operations. In the second section, Dedekind goes further in establishing a rigorous basis for number systems up to and including Cantor's transfinite numbers.