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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. Biology and mathematicsIn the preface to the first edition of Mathematical Principles of Natural Philosophy Newton (1946, p. xviii) expressed the wish to apply these principles, which were admirably successful for the understanding of mechanics, to other phenomena: I wish we could derive the rest of the phenomena of Nature by the same kind of reasoning from mechanical principles, for I am induced by many reasons to suspect that they may all depend upon certain forces by which the particles of bodies, by some causes hitherto unknown, are either mutually impelled towards one another, and cohere in regular figures, or are repelled and recede from one another. These forces being unknown, philosophers have hitherto attempted the search of nature in vain; but I hope the principles here laid down will afford some light either to this or to some truer method of philosophy.Newton's wish was realized for phenomena of Heat when Ludwig Boltzmann, in the 1870's, showed that the second law of thermodynamics could be understood by applying probability theory and the laws of mechanics to the motions of atoms of a gas. These motions involved collisions of the atoms with each other, as well as

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