By Robert Rosen
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All measurements in mechanics ultimately reduce to measurements of length, mass, and time, and consequently every physical quantity arising in mechanics can be expressed in a unique way from a sequence of measurements of mass, length, and time alone. For instance, velocity is defined to be length per (unit) time; acceleration is velocity per (unit) time, or length per time per time; force is mass times acceleration or mass times length per time per time, etc. The basic entities, mass, length, time, are called the fundamental mechanical dimensions, and the dimension of every mechanical quantity is some monomial of the form maJllt C, where THE VASCULAR SYSTEM 47 a, b, c are rational integers.
5) Here K is a constant of proportionality and the bracketed expression is, of course, the volume of the path A DC. 4). 4). 6) do not agree. How weIl do the simple arguments above compare with what is observed in areal vascular system? 5) the more realistic cost functional? The literature contains a number of empirical laws on branching in the vasculature, ascribed to Wilhelm Roux, and which are cited by d' Arcy Thompson3 as folIows: (1) If an artery bifurcates into two equal branches, these branches come off at equal angles to the main stern.
Hancock, Theory 0/ Maxima and Minima, reprinted Dover, New York, 1960. A modem treatment of the CaIculus of Variations may be found in L. M. Gelfand and S. V. Fomin, Calculus 0/ Variations, Prentice-Hall, New York, 1963. A book with a similar title, stressing applications to physics and engineering, is that of R. Weinstock, McGraw-HilI, New York,1952. 40 OPTIMALITY PRINCIPLES IN BIOLOGY 8. This form for the functional J(y) emerged very early from a study of the classical problems of the calculus of variations.
