vj THE MEANING OF SYMMETRY 357 



tutis: evitando, quam maxime fieri potest, incommoditates et 

 prolixitates." The principle of least action grew up, and grew 

 quickly, out of cruder, narrower notions of "least time" or "least 

 space or distance." Nowadays it is developing into a principle of 

 "least action in space-time," which shall still govern and predict 

 the motions of the universe. The infinite perfection of Nature is 

 expressed and reflected in these concepts, and Aristotle's great 

 aphorism that "Nature does nothing in vain" lies at the bottom 

 of them all. 



In all cases where the principle of maxima and minima comes 

 into play, as it conspicuously does in films at rest under surface- 

 tension, the configurations so produced are characterised by obvious 

 and remarkable symmetry'^. Such symmetry is highly characteristic 

 of organic forms, and is rarely absent in living things — save in such 

 few cases as Amoeba, where the rest and equilibrium on which 

 symmetry depends are likewise lacking. And if we ask what 

 physical equilibrium has to do with formal symmetry and structural 

 regularity, the reason is not far to seek, nor can it be better put 

 than in these words of Mach'sf: "In every symmetrical system 

 every deformation that tends to destroy the symmetry is com- 

 plemented by an equal and opposite deformation that tends to 

 restore it. In each deformation, positive and negative work is done. 

 One condition, therefore, though not an absolutely sufficient one, 

 that a niaximum or minimum of work corresponds to the form of 

 equiUbrium, is thus supj)ned by symmetry. Regularity is successive 

 symmetry; there is no reason, therefore, to be astonished that the 

 forms of equiUbrium are often symmetrical and regular." 



A crystal is the perfection of symmetry and of regularity; 

 symmetry is displayed in its external form, and regularity revealed 

 in its internal lattices. Complex and obscure as the attractions, 

 rotations, vibrations and what not within the crystal may be, we 

 rest assured that the configuration, repeated again and again, of 



* On the mathematical side, cf. Jacob Nteiner, Einfache Beweise der isoperi- 

 metrischen Hauptsatze, Abh. k. Akad. Wisa. Berlin, xxiii, pp. 116-135, 1836 (1838). 

 On the biological side, see {int. al.) F. M. Jaeger, Lectures on the Principle of Symmetry, 

 and its application to the natural sciences, Amsterdam, 1917; also F. T. Lewis, 

 Symmetry. . .in evolution, Amer. Nat. lvii, pp. 5-41, 1923. 



f Science of Mechanics, 1902, p. 395; see also Mach's article Ueber die physika- 

 lische Bedeutung der Gesetze der Symmetrie, Lotos, xxi, pp. 139-147, 1871. 



