On the Fundamental Principles of Mathematics. 335 



ber may be. An aggregate such as S' would then be relatively- 

 infinite in comparison with Q. 



The like must, a fortiori, be true, if, in any or all of the pro- 

 cesses of successive multiplication, the multiplier were more than 

 2 j so that the multiplicand would be more than doubled. 



If, by an inverse process, there were taken from Q, its half, and 

 then from the remainder its half, &c. &c., a sufficient number of 

 times, we should, in the end, obtain a quantity so small, in com- 

 parison with Q,, that no multiplier of it could be found sufficiently 

 large to reproduce as much as &. 



For in this case, if the number of individual inverse processes 

 were equal to that of the direct processes in the former case, and 

 L be the last remainder; then, beginning with L, we must, in 

 eitect, repeat the process of continued doubling as often as before, 

 in order to reproduce as much as Q, ; or Q will itself be relatively 

 infinite in comparison with L; or L will be "an infinitesimal' 5 in 

 comparison with Q,. 



The like must be true, a fortiori, if at any step in the process, 

 more than half were removed. 



As, moreover, Q, is relatively infinite in comparison with L, 

 and S' again relatively infinite in comparison with Q; so again, 

 by continued doubling, beginning with S', might another aggre- 

 gate be obtained, which would be relatively infinite in comparison 

 with S', &c. &c. On this it is unnecessary to dwell; as one 

 mode of exhibiting the differential calculus, owes its peculiarity 

 to the employment of quantities such as these.* 



It is important however to observe, that this description of in- 

 finity is the only one which can be predicated of number, velocity, 

 mere mechanical force, &c. &c. 



For no number can be so great, that a sum of units might not 







exist 



* It may not be amiss, here, to notice an argument against the consistency of the 

 results of mathematics which may be thus exemplified. An inch may be divided, 

 and the remainder subdivided, <fcc., by the process already explained, and thus the in- 

 finitesimal of an inch obtained ; and the aggregate of all such infinitesimals into which 

 the whole inch might be divided, would be equal to the inch itself. Now if the inch 

 ^ere passed over by a moving body, the pass e over each infinitesimal would oc- 

 C1 *py some portion of time. But the number of such portions of time would be in- 

 finite; since the number of the intinit imals of the inch is infinite. Hence (says the 

 bjector) it must require an eternity for a body to move i r an inch — which is absurd. 



The conclusion is indeed absurd, but that conclusion follows nor rom the premises. 

 For as the inch is relatively infmii i. a, infinite in rison with its inrii I, 



*& the restricted sense of relative infinity; but still finite and capabi m< 

 nient by a compar n with other standards ; so the time requisite to pas- over an 

 inch v ii a uniform velocity, will h rel rly infinite, i. e., in comparison with the 

 time in which the infinh d of an inch would tbtfte be passed o r; while t also 



might be finite and capable of accurate measurement by a comparison with another 

 standard. This portion of time then could only be called a rclat eternity; if that 

 ^ere not an abuse of the term. If the whole inch were trav< fed in a antral his rela- 

 tive eternity would endure but for a minute ; while it would still be true that the 

 ijifinitesimal of an inch would be traversed in an infinitesimal of that minute. 





