86 



of each of the combined operations severally and the third operation, 

 in the particular order thus specified, provided all the operations and 

 products are performable on the same operand. 



The above general conceptions and laws of combined operations 

 hold for any operations whatsoever with their appropriate operand 

 objects ; but the nature of the operations and operands requires 

 especial study. In mathematics, objects are only considered with 

 respect to their three most general properties : first, as contem- 

 platable in discontinuous succession, whence number and Arithmetic ; 

 secondly, as contemplatable in continuous succession, whence ex- 

 tension and Geometry ; and thirdly, as contemplatable in a con- 

 tinuous succession bearing a relation to another continuous succession, 

 whence motion in time and Mechanics. The problem of mathe- 

 matics is, first, to discover the laws of these successions as respects 

 results (that is, statically), by means of considerations drawn from 

 contemplating operations (that is, dynamical) ; secondly, to investi- 

 gate the relations of these laws, giving rise to statical algebra ; 

 thirdly, to reduce all dynamical to statical laws, as in dynamical 

 algebra ; and fourthly, to make the expression of all the results de- 

 pendent on the most simple, viz. those of common arithmetic. The 

 purpose of the problem is to prepare the mind for the further investi- 

 gation of nature, and to increase practical power immediately. 



In Arithmetic we conceive objects spread out in a scale, and by 

 aggregating those contained between any one and the beginning of 

 the scale, form statical groups, whose distinctive character is derived 

 from the scale. The operation by which any group is formed from 

 .the first object is termed an integer, the especial laws of which are 

 .next investigated. All objects being interchangeable in respect to 

 discontinuous succession, an aggregate is not changed by altering the 

 disposition of its parts. This leads to the first two laws of commu- 

 tation and association in addition. The possibility of arranging 

 objects at once in two horizontal directions, and a third vertical 

 direction, leads to the laws of commutation and association in multi- 

 plication. Combining these with the two former, we have the law 

 of commutative distribution. From the laws of association in multi- 

 plication is immediately deduced the law of repetition or indices. 



Having obtained these laws, we proceed to study their relations in 

 the algebra of integers, first, statically, in order to reduce all results 



