Viii INTRODUCTION. 



The use of formulas for expressing succinctly such struc- 

 tures as the teeth of bivalves or gastropods would be a great 

 convenience if it were not that nearly every investigator in- 

 vents a new system for himself. Without uniform currency, 

 terminology loses its value as an intellectual tool. For this 

 reason, and because such formulas appeal to some but not to 

 all naturalists, little use has been made of them in this work. 



In the Introduction to Vol. XXIV, p. viii, the tooth-formula 

 used by Westerlund was noticed, and a more elaborate one 

 was proposed. As the latter has never been used practically, 

 it need not be considered further, being at best rather clumsy. 



Westerlund 's formula gives merely the number of teeth on 

 each margin of the aperture, parietal, columellar, and palatal 

 including basal, thus: 2-1-3. This is easily understood and 

 often convenient for succinct statements in keys or tables. 



More elaborate formulas for brief statement of the aper- 

 tural denticulation of Vertigo (but applicable also to other 

 Pupillida3) have been published by Dr. V. Sterki (Proc. U. S. 

 Nat. Mus., xi, 1889, pi. 42, fig. 5) and by Mr. C. M. Steenberg 

 (Danmarks Fauna, Landsnegle, 1911, p. 156, fig. 127). 

 Though these formulas have been used but little outside of 

 the publications mentioned, the lists of symbols are given 

 below as a matter of record. 



Both the Sterki and the Steenberg formulas express the 

 structures fully. The chief objection to them is that they are 

 rather difficult for one not a specialist on Pupillidae to inter- 

 pret. Steenberg 's symbols do not distinguish between pri- 

 mary and secondary teeth, and are based upon the Danish 

 language ; Sterki does not recognize the lamella angidaris as 

 a primary tooth. In writing his formulae, Dr. Sterki uses dots 

 in place of figures for the secondary teeth. 



It appears to me that an ''international" formula uniting 

 the best features of both could be constructed by designating 

 the six primary teeth by the initials of their Latin names, 

 and the secondary denticles by dots; the parietal, columellar 

 and baso-palatal parts of the formula to be separated by 

 hyphens. In all formulae the enumeration proceeds anti- 

 clockwise. 



