ARITHMETICAL INSTRUMENTS. 29 



degree, to which we have above referred, is specially deserving of 

 mention in this place, because it admits of mechanical applications 

 to the theory of wheel-work, and also because it can be repre- 

 sented by a simple geometrical illustration. 



We may mention, for a similar reason, another important 

 research connected with the theory of numbers, viz., the calcula- 

 tion of Tables giving for each number the least number by which 

 it is divisible ; or, if it is a prime number, indicating that it is so. 

 Such tables (which considerably abbreviate certain computations) 

 have been constructed for the first nine millions ; the tables of the 

 fourth, fifth, and sixth millions exist, however, in manuscript only, 

 and have never been published. The first attempt to form a Table 

 of Primes was made by Eratosthenes, and the partly mechanical 

 method adopted by him (and called after its inventor, " the sieve 

 of Eratosthenes") has been adopted in principle, though with 

 appropriate modifications, by his successors. 



There is in general so little appearance in those laws of nature 

 with which we are acquainted of any adherence to integral or 

 whole numbers, that we may be allowed to call attention to two 

 important classes of phenomena which form an exception to this 

 remark. We refer to the laws of chemical combination, and to 

 the laws of crystallography. 



If we imagine chemical substances existing in the ideal condi- 

 tion of perfect gases, the law of chemical combination may be 

 expressed in its most abstract form by saying that if two perfect 

 gases combine chemically, and form a compound which is also 

 supposed to exist in a perfectly gaseous condition, the volumes of 

 the two gases before combination and of the gas resulting from 

 their combination are to one another as three whole numbers. 



The law of integral numbers to which the faces of a crystal are 

 subject is sufficiently illustrated by the models in the section of 

 mineralogy ; and it would be out of place to discuss it here. 



It may, however, be proper to remark that the whole numbers 

 which present themselves in the formulae whether of chemistry or 



