Harding. — Certain Decimal and Metrical Fallacies. 97 



genuine relations, for the suggestion that there is anything at 

 all "natural" in decimal fractions, especially as contrasted 

 with vulgar fractions, is a complete inversion of fact. Take 

 that convenient school-book illustration the apple ; halve and 

 quarter it. Which form most correctly aud naturally ex- 

 presses the proportion the sections bear to the whole ? The 

 "half" or "quarter" represents the actual fact accurately 

 expressed by the vulgar fraction, and by that alone. To 

 describe the fruit as divided into sections of 0-5 or 0-25 is 

 to introduce an artificial convention remote from the fact, 

 warrantable only when, for purposes of calculation, we are 

 considering the pieces in association with others actually 

 divided into fifths, tenths, or twenty-fifths. The " Autocrat " 

 has told how, after defending his landlady's pie, " I took more 

 of it than was good for me — as much as 85°, I should think — 

 and had an indigestion in consequence." This does not 

 strike an appreciative reader as intended for the "natural" 

 mode of expression, but rather as a piece of playful pedantry 

 on the author's part. Vulgar fractions are so universally and 

 unmistakablv natural that no convention or notation, how- 

 ever ingenious, can dislodge them. Therefore the German 

 typefounder of to-day, being a man of common-sense, cata- 

 logues his smaller wares by the "half-kilo." (his nearest 

 approach to the discarded pficnd), the American coins his 

 half- and quarter-dollar, and so to the end of the chapter will 

 weights and measures necessarily be popularly divided, 

 thought of, and spoken of in halves and quarters, whatever 

 official system may be in vogue, or by whatever written 

 symbol these values are expressed. 



The radix of 2 is one of the most important in practical 

 use, and in many cases it is the only practicable one. It 

 governs not only our own, but the other traditional systems of 

 weights ; it is the obvious corollary of any system of balanc- 

 ing. In book-folding it is in all ways the most convenient, 

 and its sole rival, the sextuple fold, is almost extinct. The 

 names "twelves" and "twenty-fours" are still in use to 

 indicate certain shapes and sizes, but as a matter of practice 

 such are nearly always now printed and folded as eights- 

 or sixteens. A " decimo " sheet would be an obvious im- 

 possibility. Two of the multiples of 2 — eight and sixteen — 

 have been suggested as notational substitutes for ten ; but as 

 an arithmetical radix either of them would be little, if any, 

 better, apart from the consideration that one is inconveniently 

 small and the other just as inconveniently large. The latter 

 would further involve a cumbrous notation and an extensive 

 nomenclature, as an American would-be reformer, who boldly 

 selected sixteen as his radix, discovered many years ago. De 

 Morgan, in his " Budget of Paradoxes," has made merry over 



7— Trans. 



