452 report — 1878. 



The meaning of the new name, change-ratio, may be given by the following 

 definition. 



Definition. — In respect to changes of any variable, the change-ratio is the 

 ratio of the new value to the old, for any change from one value of that variable to 

 another. 



Further, in Everett's treatise, the relation already referred to among the units 

 of length, time, and acceleration, is stated in some other ways, which I will next 

 cite: — 



length" 



" The dimensions of acceleration are ... ^ 2 • 



(limey 



unit of length „ 

 " The dimensions of the unit of acceleration are 7 — -, » time') 2 ' 



Instead of either of these, I would substitute this — 



_ .„.«-,. change-ratio of unit of length 



Change-ratio 01 unit of acceleration = s-r -■ — 7. — rr — ,.,. ,, ■ 



6 (change-ratio 01 unit 01 time)-* 



This expression states clearly and correctly all the truth which is meant to be con- 

 veyed by the previous statements. 



In order now to be enabled to speak, in language brief and free from 

 ambiguity, of any numerical expression whatever, whether whole or fractional, 

 greater or less than unity, or unity itself, I shall use the word numeric, which I 

 recommend for general use, to comprise all the meanings which at present are con- 

 veyed in common use, but with much of troublesome ambiguity, by words or 

 phrases such as number , fraction, number or fraction, number and fraction, number 

 or proper fraction or improper fraction. I recommend that, as soon as possible, 

 the word number should be restricted to its only proper signification, which is often 

 at present designated in an objectionable way by the two words " whole number" 

 but which is often also expressed, and really properly so, by the single word 

 number* 



Now we have no right to speak of dividing one quantity by another of a dis- 

 similar kind, except, merely for brevity, in the case of dividing the numeric 

 expressing the one quantity by the numeric expressing the other quantity, after we 

 have fixed upon units of the two things. Thus we have no right to speak of unit 

 of length divided by unit of time, nor to employ, unless perhaps for brevity, and 

 under an implied protest, such a notation as 



Unit of length 

 Unit of time 



Further, we have no right to speak of " second power of unit of time," nor of 

 "square of unit of time." The warns power seems admirably well suited (whether 

 by deliberate design entirely, or partly by good chances) for its uses in reference to 

 what are called powers, whether integral or fractional, of any numerics, as for 

 instance 



X 2 , X 3 , X*, X 2 - 13 . &c, &c. ; 



* Thus, for instance, in the public regulations for Post Office Savings Banks, 

 issued by authority of the Postmaster-General (' British Postal Guide '), the intima- 

 tion is made that " At these banks deposits of one shilling, or any number of shillings, 

 will be received," this being, however, subject to some restrictions, which need not 

 be mentioned here, merely assigning limits to the amounts that will be accepted 

 from any one person. Now the words here quoted would convey a false statement 

 of what the Post Office authorities really mean to announce, if the word "number " 

 were allowed to mean a fractional numerical expression. The announcement is 

 obviously framed on the presumption that the word "number " in it can only mean 

 legally what in the present paper is referred to as its only proper signification, that, 

 namely, which is commonly designated as "a whole number " or "an integer." If 

 an intending depositor, understanding the word " number " in the extended sense in 

 which it is very often, and also quite authoritatively used, would offer a deposit of 

 7i*., his offer would be refused, as it would amount to Is. id., which is not contem- 

 plated in the regulations as an amount to be accepted as a deposit. 



More examples to the same effect might be cited from usages in practical 

 business affairs ; and also from usages of scientific writers in arithmetic and in other 

 branches of mathematics, but the one here given may suffice. 



