Mr. A. J. Ellis on the Systematization of Mathematics. 225 
on the same object produce different resultant objects, the operation 
of transforming one of these resultant objects into the other, is re- 
garded as the quotient of the two former operations. Two opera- 
tions are termed reciprocal when their product is unity. Hence the 
quotient of two operations is the product of the one and of the reci- 
procal of the other. When two objects are combined in any manner 
so as to produce a third, and the two first are formable from any 
fourth by two known operations, the single operation by which the 
third object can be also formed from the fourth, is termed the same 
combination of the two first operations. From this we gain the con- 
ception of null or zero, as the operation of annihilating any object in 
respect to any place. The product of a combination of two opera- 
tions and a third operation, is the same combination of the products 
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 /aw 
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 
