I II D I T C, &e. 31 objector, is it possible that Achilles can ever come up with tin tortoise, since it is unquestionable (and this is perfectly correct), tha let him go as far as he may, he must always come up to where tin tortoise was before lie can reach the point at which he is ; so that i requires an infinite number of parts of time (but here the sophism quietly introduces an infinite time) to catch the tortoise 1 The answe is, that Achilles will certainly overtake the tortoise nt a finite distance from A, say at any contrivance which subdivides A Si into at infinite number of parts, does the mane with the time in whirl Achilles runs over A 11; and there is no more reason to say that time is therefore infinitely great, than to say that As is made infinite]: great by the subdivision. This would be a sufficient answer, sine, it would throw upon the sophist the anus of showing that infinite number of parts of time makes an infinite time; but a nor complete answer consists in positive proof that it is not so, a follows : Let be called a, and let Achilles movo m times as fast as th tortoise ; then T B is necessarily the nail part of A T, B C of T D, C D of n c, &c. Hence, if 1 be the time in which Achilles moves over A T, this time, added to his times of going over T B, B C, C D, &c., or t, tχm, &c., make up / 1 1 1 1 t I1 + + + + &c., ad inf. ) Now if ni be greater than 1 (for unless Achilles move faster than the tortoise, it is admitted he can never catch it), the series above named is 1 1 ), so that the whole time is tm,-;.-(nu 1), and ns
the whole length A 11 is 1), the same answer as would be produced by common methods. The sophism divides this length into the infinite number of parts a a a &c.
and taking the times due to each, t t t &c., sit' assumes the sum of the latter to be infinite.
In the work of a celebrated political economist there is the follow iug argument, to show that a tax ou wages must fall ou the labourers; for if it did not so fall, wages would rise, whence the price of goods would rise, which would again cause a rise of wages, and this again a rise in goods, and so on ad infinitunt, which is iuferred to be absurd. This is of course precisely a repetition of the preceding ease ; and granting all the premises, the conclusion by no means follows. For that conclusion is that the rise would go on without limit, which need not be the case.
The best way of remembering the summation of a geometrical series is by a verbal rule, as follows :The sum is the difference between the first term in and the first term out, divided by the differ ence between the common multiplier and unity. Thus the sum of 30 + 90+ 270 +810 is 2430 30 divided by 3 1, or 1200.