There is no doubt that every whole can be subdivided into part. without limit, or, in common language, can be divided into an infinite number of parts. An old fallacy, mentioned in Mottos, receives its explanation from the preceding. If we make a= 1 —r, the equation carried ad infinitum becomes 1=(1 —r)+ (1—r) r + (1 , &c., cal inf.
By giving different values to r, we have therefore an infinite num• ber of ways of subdividing unity into an infinite number of parts. 11 then we take a problem in which au antecedent is followed by a con. Sequent; and if dividing the antecedent into an infinite number of parts, we consider separately the parts of the consequent which belong to those of the antecedent, we shall of course divide the conse quent into an infinite number of parts. It would be a gross, fallacy tc infer that the consequent must be infinitely great, because it is pro duced in a never-ending succession of parts, since that never-ending succession was produced by dividing the avowedly finite antecedent Into an infinite number of parts. No one could fail to detect the
following :—" Let m be divided into an infinite number of parts, a, c, &c. ; let each of these parts be doubled ; then the result is made ur of 2e, 25, 2c, &c., ad infinitum; conaequeutly 2a + +2es- , &c., being made up of an infinite number of quantities, is infinite.' Nevertheless this fallacy was not only produced in an ingenious form as a sophism [MOTION), but has even reappeared in modern times as e serious argument. The sophism is known by the name of " Acluillce and the Tortoise." The swiftest of men rune after the slowest of beasts without (says the sophism) the possibility of ever overtaking it. Foi if, when they set out, Achilles be at A and the tortoise at T, then by the time Achilles has run over AT, how fast soever he may run, tin tortoise will have gone over some length, say TB; while the hero goes over TB, his dinner (for dinner he may have out of it, in spite of tlo sophism) goes over no, and so on ad infinitum. How then, asks tin