AND ON LOGIC IN GENERAL. 



205 



thus they announced a maxim which bewilders those who have not the key ; Quantitas non 

 suscipii tnagis et minus. 



They meant that a thing which has one amount of quantum is not a thing of which we 

 have more perception of its having quantitas than another thing of another amount of quantum. 

 Nevertheless, even in the hands of the old logicians, quantity came to have the meaning which 

 we now* give it, that which exists where it is possible to conceive of answer to the question 

 How much ? or How many ? — measured solely by the quantum of the answer. Accordingly, 

 to equate two quantities, is to assert them to be the same quantities : a furlong of the wall 

 of China is the same quantity of length as 220 yards of the pier at Brighton. The res divi- 

 sibilis is a frequent use of the word in common life. Thus a person may buy a quantity of 

 timber, and may sell that quantity again, meaning the very same planks, not as many cubic 

 feet out of what he had before. To equate quantities, even if the word be thus restricted, can 

 only mean to identify them in a very loose and inaccurate acceptation. If a person should be 

 tried for stealing the timber just spoken of, his counsel could not properly ask the prosecutor, 

 even when speaking concretely of quantity, ' Now, Sir, on your oath, can you equate the 

 quantity found on the prisoner's premises to the quantity you bought at the yard ?' 



The singular theory of predication above described seems to depend, not merely on the 

 adoption of the res divisibilis by persons unpractised in mathematical thought, though this 

 must have had some share ; but also on a tendency, fostered by an invented quantification, to 

 confusion between two attributes of the relation of whole and part; one, the notion of contain- 

 ing and contained, the other, the notion of more and less. Containing and contained is con- 

 vertible with whole and part, if terminal ambiguity be conceded to both, or denied to both ; 

 but more and less, though contained in either on the same terms, is not convertible with either. 

 The mathematicians, to whom more and less frequently constitutes the whole matter of thought 

 for which they introduce the relation of whole and part, do very often use the name of the 

 whole relation for the name of the component which they are considering : thus a problem may 

 occur in which a bottle of sherry in London in 1857 may be (quantitatively only) part of a cask 

 of claret at Bordeaux in 1800. The mathematicians thus speak of the compound where they 

 mean only the component : the logicians whom I am now describing insist that the compound 

 shall mean no more than the component. 



* The mathematician becomes so accustomed to the abstract 

 quantitas that he often forgets — and some have even denied — 

 the quantitative science of the res divisibilis itself. This 

 science, to which the curious in words may refuse the name of 

 arithmetic in its addition or subtraction, until concrete number 

 is expressly introduced, becomes arithmetic in its multiplica- 

 tion and even takes from abstract arithmetic: for the multiplier 

 must be an abstract number. In its division, either magnitude 

 divided by magnitude gives an abstract quotient, or magnitude 

 divided by abstract number gives a magnitude as quotient. So 

 that, granting application of magnitude to magnitude, addition 

 and subtraction may be treated independently of any notion of 

 number, and number itself may be learnt in the quotient of 

 division, made antecedent to multiplication. When we come 

 to fractions in abstract arithmetic, which we cannot do without 



making an abstract reS divisibilis of the unit, we see a cognate 

 distinction. Addition, subtraction, and division with integer 

 quotient can be fully conceived and reduced to rule so soon 

 as the common denominator is obtained, or the modes of divid- 

 ing the unit all reduced to one : multiplication remains incon- 

 ceivable until the idea of part of a time is introduced. 



Euclid's system of proportion is a specimen of the arith- 

 metic of the res divisibilis. When it is abandoned, and what 

 is called an arithmetical definition is introduced, elementary 

 writers generally leap the difficulty of expressing magnitudes 

 numerically, and start from the expressed magnitudes, taken 

 merely as numbers. But the way of bridging this chasm in- 

 volves some matters which are of great importance. The con- 

 nexion of number and magnitude should not be left to mother 

 wit. 



