viij OF HEXAGONAL SYMMETRY 499 



draw it not from arithmetic but from trigonometry, and define our angle as 

 cos — ^ , nothing can or need be more precise or siinpler. We may put the 

 same thing a little differently, and say that Number itself fails us, now and 

 then, to express what we want, although we have all the ten digits and their 

 apparently endless permutations at our command. In such a deadlock, we 

 have only to bring one new symbol, one new quantity, into use; and at once 

 a wide new field is open to us. 



Out of these two angles — the Maraldi angle of 109° etc., and the 

 plane angle of 120° — we may construct a great variety of figures, 



Fig. 179. Diagram of hexagonal cells. After Bonanni. 



plane and solid, which become still more complex and varied when 

 we consider associations of unequal as well as of co-equal cells, and 

 thereby admit curved as well as plane intercellular partitions. Let 

 us consider some examples of these, beginning with such as w^e need 

 only consider in reference to a plane. 



Let us imagine a system of equal cylinders, or equal spheres, in 

 contact with one another in a plane, and represented in section by 

 the equal and contiguous circles of Fig. 179. I borrow my figure 

 from an old Italian naturalist, Bonanni (a contemporary of Borelli, 

 of Ray and Willoughby, and of Martin Lister), who dealt with this 

 matter in a book chiefly devoted to molluscan. shells*. 



* A. P. P. Bonanni, Kicreatione delV occhlo e delta mente, nelV Osservatione delle 

 Chiocciole, Roma, 1681. 



