184 liEV. H. J. CLARKE. 



in whicL annihilation alternates witli reproduction. The 

 truth is, it was not perceived that repetition, however rapid, 

 is generically distinct from transition, and, indeed, from 

 continuity or extension of any kind, although, in so far as 

 they severally yield magnitudes or values, ratios observable 

 within the limits of one genus may admit of comparison with 

 those of another, and thus furnish material for equations. 

 And so it came to pass that, on the supposition of a race being 

 proposed such as that in which Achilles is fancifully depicted 

 vainly striving to overtake a tortoise, the subtle philosopher, 

 although, we may presume, he would not have been prepared 

 to stake anything upon the success he seemed to promise the 

 slow-paced competitor, was able to satisfy himself that, in the 

 dispute as to what the issue must be, he had at any rate the 

 best of the argument. 



Now, no metaphysical incongruity, it is true, forbids the 

 use of arithmetic in the calculation of times, velocities, dis- 

 tances, dimensions, and so forth ; but whatever value a 

 unit may represent, its repetition is only accidental, and 

 no arithmetical process can change its nature. It is utterly 

 inconceivable that by repetition a point should produce a 

 line, or a line a surface, or a sui-face a solid. To look for 

 such transformations of genus would be less reasonable 

 than to expect to see a pile of twelve penny-pieces meta- 

 morphosed into that silver coin which is called the shilling. 

 A unit of any conceivable value, if finite, of course admits 

 of hypothetical mtdtiplwatioii, but no ivvolution affecting it 

 can take place, except that of its numerical coefficient. If 

 a i*epresent the number of times a rectilinear unit is to be 

 taken, then a to the power of 2 will denote the number of 

 corresponding square units required to form a square of which 

 a may be taken to indicate a side. Similarly, a to the power 

 of 3 will signify the number of cubic units contained in the 

 cube which may now be imagined as standing upon the square. 

 Thus it will appear that, if we should be called upon to assign 

 a geometrical significance to a to the power of 4, we might 

 say that it suggests a har formed by repeating the con- 

 structed cube (now adopted as a unit), as many times as there 

 are numerical units in a. The association of arithmetical 

 relations with those of extension is plainly accidental. It can 

 only be effected through the medium of a concept which is not 

 logically inherent in that of the latter, namely, the con- 

 cept of the unit, and innumerable are the cases in which the cal- 

 culations it involves can never attain to more than approximate 

 exactness. I must, therefore, confess myself at a loss to 



