356 THE FORMS OF CELLS [ch. 



deeply and intrinsically there enter into this whole class of problems 

 the method of maxima and minima discovered by Fermat, the "loi 

 universelle de repos" of Maupertuis, "dont tous les cas d'equilibre 

 dans la statique ordinaire ne sont que des cas particuhers", and the 

 lineae curvae maximi miniynive proprietatibus gaudentes of Euler, by 

 which principles these old natural philosophers explained correctly 

 a multitude of phenomena, and drew the lines whereon the founda- 

 tions of great part of modern physics are well and truly laid. For 

 that physical laws deal with minima is very generally true, and is 

 highly characteristic of them. The hanging chain so hangs that the 

 height of its centre of gravity is a minimum ; a ray of hght takes 

 the path, however devious, by which the time of its journey is a 

 minimum ; two chemical substances in reaction so behave that their 

 thermodynamic potential tends to a minimum, and so on. The 

 natural philosophers of the eighteenth century were engrossed in 

 minimal problems ; and the differential equations which solve them 

 nowadays are among the most useful and most characteristic equa- 

 tions in mathematical physics. 



"Voici," said Maupertuis, "dans un assez petit volume a quoi je 

 reduis mes ouvrages mathematiques ! " And when Lagrange, fol- 

 lowing Euler 's lead*, conceived the principle of least action, he 

 regarded it not as a metaphysical principle but as "un resultat 

 simple et general des lois de la mecaniquef." The principle of 

 least action J explains nothing, it tells us nothing of causation, 

 yet it illuminates a host of things. Like Maxwell's equations and 

 other such flashes of genius it clarifies our knowledge, adds weight 

 to our observations, brings order into our stock-in-trade of facts. 

 It embodies and extends that "law of simplicity" which Borelli 

 was the first to lay down: "Lex perpetua Naturae est ut agat 

 minimo labore, mediis et modis simplicissimis, facillimis, certis et 



* Euler, Traite des Isoperimetres, Lausanne, 1744. 



t Lagrange, Mecanique Analytique (2), ii, p. 188; ed. in 4to, 1788. 



{ This profound conception, not less metaphysical in the outset than physical, 

 began in the seventeenth century with Fermat, who shewed (in 1629) that a ray 

 of light followed the quickest path available, or, as Leibniz put it, via omnium 

 facillima; it was over this principle that Voltaire quarrelled with Euler and 

 Maupertuis. The mathematician will think also of Hamilton's restatement of the 

 principle, and of its extension to the theory of probabilities by Boltzmann and 

 Willard Gibbs. Cf. (int. al.) A. Mayer, Geschichte des Prinzips der kleinsten Action, 

 1877. 



