The living forms represented in this paper are sets of parts that spontaneously increase in organization. Their organizations are measured by an information-theoretic function derived from the work of Boltzmann and Shannon. We briefly review its derivation in the context of the troubled role of mathematics in biology, and then define the function. We illustrate its nature by measuring the 22 different organizations of a set of eight things; and we facilitate its use by defining the parameters that determine an (...) amount of organization. The measure is then applied to show that the organization of limb pairs on free-living arthropods, based on data given by Cisne, confirms a pattern of increasing organization in their evolution from the Cambrian era to the present. Further applications measure the changes in organizations of ideal (theoretical) life forms, and contrasting changes in inanimate systems. Our main results represent the reproduction of unicellular organisms, and the formation of hierarchies, as processes of increasing organization. (shrink)