498 THE FORMS OF TISSUES [ch. 



The resultant of all four is equal to the resultant of W at a, and 

 3Pf at p; it lies, therefore, on the straight line ap, and at the 

 point 0, such that ao = '6of. Therefore, in the triangle pod, as 

 in the other three similar triangles in the figure, cos pod = 1/3, 

 and cos aod = — 1/3. Our tables tell us that the angle pod, whose 

 trigonometrical value is the very simple one of cos pod ^ 1/3, has; 

 in degrees and minutes to the nearest second, the seemingly less 

 simple value of 70° 31' 43"; and its supplement, the angle aod, 

 has the corresponding value of 109° 28' 16". 



This latter angle, then, of 109° 28' 16", or very nearly 109 degrees 

 and a half, is the angle at which, in this and throughout every other 

 three-dimensional system of liquid films, the edges of the partition- 

 walls meet one another. It is the fundamental angle in simple 

 homogeneous partitioning of three-dimensional space. It is an 

 angle of statical equilibrium, an angle of close-packing, an angle 

 of repose. In the simplest of carbon-compounds, the molecule of 

 marsh-gas (CH4), we may be sure that this angle governs the 

 arrangement of the H-atoms; it determines the relation of the 

 carbon-atoms one to another in a diamond — simplest of crystal- 

 lattices; it defines the intersections of the bubbles in a froth, and 

 of the cells in the honeycomb of the bee. 



It is sometimes called the "tetrahedral angle"; it might be better 

 called (for a reason we shall see presently) "Maraldi's angle." The 

 whole story is less a physical than a mathematical one; for the 

 phenomena do not depend on surface tension nor on any other 

 physical force, but on such relations between surface and volume 

 as are involved in the properties of space. If we take four little 

 elastic balloons, half fill them with air, smear them with glycerine 

 to lessen friction, place them in a bottle and exhaust the air therein, 

 they wull expand, adjust themselves together, and group themselves 

 in a tetrahedral configuration, whose partition walls, edges and 

 centre of symmetry are just those of our experiment of the soap-films. 



This characteristic angle, though it leads in ordinary angular measurement 

 to an endless decimal of a second, is nevertheless a very simple and perfectly 

 definite magnitude. It is a strange property of Number that it fails to express 

 certain simple and definite magnitudes, such as tt, or y/2, or Vs, or this four- 

 fold angle made by four lines meeting symmetrically in three-dimensional space. 

 It is not these magnitudes that are peculiar, it is Number itself that is so ! In 

 all of these cases we have to import a new symbol; and in this case, when we 



