218 ON THE INSUFFICIENCY OF TAYLOR's THEOREM, 



chine might conveniently replace — and, 2dly, of algebra, elementary and 

 transcendent, which prepares and reduces complicated relations for the ap- 

 plication of arithmetic, or, in other words, brings them within reach of the 

 machine. 



3. The applied branch of numerical logic — nearly as ill named as that which 

 is pure — has for its object to reduce all the relations of time and space, and 

 such moral relations as admit the notion of definite ao^o^reofation to relations of 

 number; or, more correctly, to reduce all modes of exact aggregation to the re- 

 lations of that particular arrangement of number which constitutes the branch 

 preceding. 



4. As I cannot persuade myself to use a term so ill adapted to its purpose as 

 ''the calculus," I must employ, in its place, the circuitous definition of "the 

 known and connected algebraic arrangements," and shall then observe that, 

 throughout this paper, any .expression composed of the known and connected 

 algebraic arrangements is termed a function of numerical logic, as any ex- 

 pression which relates to modal aggregation, unreduced to relations of num- 

 ber, is termed a modal function. 



5. The two great instruments which numerical logic employs in reducing 

 the unknown to the known are equalities and developments. In the present 

 mode of considering the subject, nearly all developments are derived from 

 those which are equalities, by the simple process of omitting the remainder; 

 so that, denoting a development by the sign ^, either of the propositions 



12 3 m 



U=:U+U-f-U + U+ &C. . U + R 



or 



(1) 



00 11 32 33 mm 



u = uv + uv+uv + uv. . .UV-f-R 



(2) 



may be said to involve the proposition 



12 3 m 



u 7^ u, u, u, u . . . u (3) 



6. A variety of instances, more general than these, might be given, but our 

 present object is limited to the manner in which the third proposition arises 



12 m 



from the second, and to that form of the "classifying functions" v, v, v . . . v, 

 which gives rise to the differential and the integral calculus. 



To conceive the first of these neatly, we remark, that many developments 

 are not only definite, or such as can be found by fixed rules, but also prime, or 



