Mr. W. H. Wiilenn on Uaitation. 375 



vity " is distinguished by its unalterability under the influence 

 of heat, and general constancy under all conditions. Could a 

 more constant cause be imagined than the above ? and could a 

 more simple one be desired ? or could any other means of satis- 

 fying all the conditions of the problem bo conceived of? If 

 simplicity be a mechanical recommendation, the simplicity of 

 the above conditions will recommend themselves. We say 

 simplicity ; for surely we have the ne plus ultra of simplicity, 

 when no postulates at all are required, but the total sum of 

 effects may be said simply to evolve themselves out of pure 

 dynamics. 



L. On Unitation. — YII. Practical Remarks thereon, together 

 with Examples. By W. H. Walenn, Mem. Phys. Soc.*' 

 [Continued from vol, i. p, 549.] 

 24. ~|~N considering the nature of a unitate and of the ope- 

 -^ ration of unitation, the method of thought is new to 

 mathematical minds, and does not readily fall in with the 

 routine of ordinary mathematical work ; for this reason it is 

 necessary to treat the matter somewhat in detail. The whole 

 subject is fundamental^ and is characterized by constant refer- 

 ence to first principles of the simplest kind. It is its very 

 simplicity that creates a difficulty in the facile apprehension 

 of a unitate as a function, and of unitation as an operation. 

 From what has been written it will be perceived, upon exami- 

 nation, that it must by no means be confounded with the 

 remainder to a division*. The denomination of the remainder 

 to a division is determined by the denomination of the last 

 figure of the quotient ; but a unitate, in its strictest sense, is 

 an integer ; for it is one of a series of integral values (each 

 member of which is made up of units), the series being 

 determined by the base of the system of unitates, and extend- 

 ing from 1 or unity to the numerical value of the base itself. 



Thus, in dividing a given number by 20, to find the remainder 

 to that division, if the number has a decimal portion, the re- 

 mainder will be less than unity f. For instance, the division 



— ^ — gives for remainder '003 ; but the unitate of the num- 

 ber 40*123 to the base 20 is 3, since it must be one of the 

 numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 

 18, 19, 20. The name unitation is given to the operation to 

 note this distinction in the most marked manner possible. 

 This peculiarity is pointed out in the Philosophical Magazine 



* Phil. Mag. [V.] vol. ii. p. 348. t Ibid. p. 345. 



