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continuous law, so that a z+1 : a z finally approaches towards a limit, 

 the limit towards which the series approaches as its form approaches 

 neutrality is a z — a z +i divided by a z —a z +i. And this limit is 

 always ■§-, If there be in the series a cycle of laws involving an even 

 number of terms, so that 



a 2nz — #2nz+lj tt2nz+] — #2raz+2> • • • &2nz-\-2n — \ — Q>2nz+2n 



approach in ratio to k , k v . ..k 2n -i> then the two series a — a x -\-a 2 — ... 

 and a l — a., + « 3 — ..., which have unity for their sum, have the 

 ratio of k + k 2 + . . . + k 2n -2 to k L + k 3 + . . . +k 2n -i- But if the 

 cycle have an odd number of terms, each of these series is g, just as 

 if the law hud been continuous. The demonstration is founded upon 

 the following theorem : — If P + Pj+ • • • and Q + Q 1 + . . . be di- 

 verging series, whether of increasing or decreasing terms, their two 

 infinite sums are in the final ratio of P z to Q z . Applications of this 

 theorem are given to the determination of a large number of terms 

 of l ra -f-2 w -{- ... when n is — I or greater, and to the determination 

 of the usual approximation to 1 . 2 . 3 . . . n when n is great. 



2. " On Infinity, and on the Sign of Equality." 



The author professes himself satisfied of the subjective reality of 

 the notions of infinitely great and infinitely small. His paper, as far 

 as it deals with various objections by various modes of answer, is 

 not capable of abstract ; but four points, on which he especially 

 relies, may be stated as follows :— 



1 . The concepts of the mind are divisible into imageable and un- 

 imageable : the first can be pictured and placed before the mind's 

 eye ; the second cannot. The mathematician, dealing in great part 

 with imaged concepts, is apt to repel the unimageable, as if it 

 could not be a legitimate object of mathematical reasoning. But all 

 that is necessary to reasoning is knowledge of the connexion of sub- 

 jects and predicates. Infinite quantity is unimageable in its relation 

 to finite quantity, but not therefore inconceivable, nor destitute of 

 known attributes. A million of cubic miles is as destitute of image 

 as infinite space ; nevertheless it is a conception the attributes of 

 which give known propositions. 



2. Number, or enumeration as distinguished from multitude, is a 

 concept from which no notion of infinity can be gained ; but much 

 perplexity has arisen from the attempt to make it a teacher of this 

 subject. Abstract number has more than one affection which is de- 

 rived from the concrete in such manner that the two abstractions, 

 number and its affection, cannot have their function explained ex- 

 cepting by return to the concrete. Such affections are the divisible 

 unit, on which the doctrine of fractions is founded, and the opposi- 

 tion of positive and negative. The representation of infinite and of 

 infinitesimal number is a third affection of the numerical, which 

 cannot be explained on purely numerical notions. 



3. The infinite is not a land of terminus to the finite, but another 

 status of magnitude, such that no finite, however great, is anything 

 but an infinitesimal of the infinite. And the same may be said of 

 each order of infinity with reference to the one below it. 



