EXAMINATION OF ANTINOMIES. 531 
that is conceivable in thought, lies between two extremes, which, as contradictory of each 
other, cannot both be true, but of which, as mutual contradictories, one must.” The most 
important of these supposed contradictions deserve a passing notice. 
“‘]. Finite cannot comprehend, contain the Infinite—Yet an inch or minute, say, are 
finites, and are divisible ad infinitum, that is, their terminated division is incogitable. 
“2. Infinite cannot be terminated or begun.—Yet eternity ab ante ends now; and 
eternity a post begins now. So apply to Space. 
“3, There cannot be two infinite maxima—Yet eternity ab ante and a post are two in- 
finite maxima of time. 
“4, Infinite maximum if cut into two, the halves cannot each be infinite, for nothing 
can be greater than infinite, and thus they could not be parts; nor finite, for thus two 
finite halves would make an infinite whole. 
“5, What contains infinite quantities, cannot be passed through,—come to an end. An 
inch, a minute, a degree contains these ; ergo, etc. Take a minute; this contains an infi- 
nitude of protended quantities, which must follow one after another; but an infinite series 
of successive protensions can, ex termino, never be ended ; ergo, etc. 
“6, An infinite maximum cannot but be all inclusive. Time ab ante and a post infinite 
and exclusive of each other; ergo. 
“7, An infinite number of quantities must make up either an infinite or a finite whole. 
I. The former.—But an inch, a minute, a degree, contain each an infinite number of 
quantities; therefore, an inch, a minute, a degree, are each infinite wholes; which is 
absurd. IJ. The latter—An infinite number of quantities would thus make up a finite 
quantity ; which is equally absurd. 
“8. If we take a finite quantity (as an inch, a minute, a degree), it would appear 
equally that there are, and that there are not, an equal number of quantities between 
these and a greatest, and between these and a least. . . . 
“13. A quantity, say a foot, has an infinity of parts. Any part of this quantity, say an 
inch, has also an infinity. But one infinity is not larger than another; therefore an inch 
is equal to a foot.’* 
262. The ambiguity that runs through all these propositions, is the same that has 
already been noticed. In each proposition, the term infinite is used with two or more 
meanings, and the different properties of different relative infinites, are contrasted with 
the supposed properties of a supposed absolute infinite, Hamilton himself points out this 
* This fallacy resembles the algebraical demonstration already given, that 18. If one infinity is not larger 
than another, then * — =, and 1—n. In reality, 0 and oc may each have an infinite number of values, and 
any reasoning that is based either upon the infinitely great or the infinitely small, may lead us into error, unless 
we keep all the conditions in view,—those which are limiting, as well as those which are infinite. 
