AND ON THE LOGIC OF RELATIONS. 



337 



of a calculator who has enough of calculi, life, and patience. And number is defined by 

 the more or less of counting which has taken place in its formation ; further counting onwards 

 is the process required in addition; counting backwards is the process required in sub- 

 traction: and to these all other processes can be reduced. The last unit, or item* of 

 numeration, tells the result of all that has been done. Suppose any one to contend that 

 arithmetic should never transcend pure counting, and he would be a faithful imitator of 

 argument about logic, as not infrequently expressed, and always implicitly maintained. The 

 arithmetician I have supposed should argue from the fundamental character of the counting 

 process : he should leave practice and progress out of sight, should refuse to allow the possibility 

 of abstractions which might end in the differential calculus, and should contend for the pure 

 form of arithmetical thought. Every merchant's clerk would laugh at his book of arithmetic, and 

 would be joined by every speculator on that theory of numbers at which he could never arrive. 

 But our arithmetician should stand firm upon the fact that men naturally count on their 

 fingers. And though those who count on the fingers do not want him, and those who can 

 do better will not have him, he can retire within himself, satisfied that he is the true philo- 

 sopher of arithmetic, and the sole depository of the science. And, all unreasonable as he 

 is, he would be more reasonable than the logician. For it is the truth that all arithmetical 

 result can be obtained by counters : it is not the truth that all inference can be obtained 

 by ordinary syllogism, in which the terms of the conclusion must be terms of the premises. 

 If any one will by such syllogism prove that because every man is an animal, therefore every 

 head of a man is a head of an animal, I shall be ready to — set him another question. 



When the logician contends that a syllogism which is not onymatic can be reduced 

 to one which is, he always proceeds by a statement of the combination of relations, for his 



use Halley's words when inviting to the calculation of the 

 logarithms of all prime numbers under 100,000 to 25 or 30 

 places of figures, "should prompt him to undertake" to 

 verify this assertion, he ought to find the following as the solu- 

 tion of the celebrated equation «* — 2a? — 5 = 0. I will not say, 

 with Halley, " I can assure him that the facility of this method 

 will invite him thereto." 



* = 3-09455 14815 42326 59148 23865 40579 30296 38573 06105 

 62823 91803 04128 52904 53121 89983 48366 71462 67281 

 77715 77578 60839 52118 90629 63459 84514 03984 20812 

 82370 08437 22349 91 



This result, which will place the power of Homer's me- 

 thod in its proper light of evidence, was calculated in 1850 by 

 my pupil Mr John Power Hicks, since of Lincoln College, 

 Oxford, and has not been published till now. A hundred 

 places had previously been calculated by another pupil, Mr 

 William Harris Johnston (Mathematician, Vol. in. p. 289) 

 whose solution was unknown to Mr Hicks. Neither solution 

 was merely numerical exercise; both were performed upon a 

 knowledge of, and by incitement of, the tardiness of mathema. 

 ticians, as well abroad as at home, in recognising the true 

 place of Horner's discovery in fundamental arithmetical ope- 

 ration. 



• In my last paper I criticised the phraseology of logicians 

 when they say that the difl'erence between one and another 

 individual of the same species is numerical. An able de- 

 fender referred me to the Greek original of the phrase: in 



Porphyry, &c., things which, being different, do not differ 

 eUei, differ dpid/xm. My thorough conviction that the Greeks 

 never altered the vernacular in scientific terms led me to an 

 examination of the word dpi6/i6^, the results of which appear 

 in the Transactions of the Philological Society for 1859. 

 The original meaning of apidfiSs, never lost, though soon 

 associated with the secondary sense of total, is the item of 

 enumeration, the unit of a collection, which standing alone 

 would be //oi/as. Thus Aristotle, (Metaphysics, book xi. or xii.) 

 speaking of the primary meaning, afiirms that /xotidt and 

 dpidfioi do not differ in quantity. When an dpSpLot was 

 spoken of as large, the departure from the original meaning is 

 precisely that which takes place in our own language when a 

 sale is said to be made at a high figure, meaning much money 

 to count. The word sum gives occasion to similar remarks. 

 Summa and sum meant number indicated by the highest unit 

 of counting: neither had reference to addition more than to 

 subtraction, number to be subtracted being also sum and 

 summa deducibilis. The totum was summa totalis, and sum 

 total still remains in use, sounding like tautology : but the 

 fact is that totalis, when it dropped off, left its meaning fixed 

 in summa. The school-word for arithmetic, summing, is not 

 a derivation from the leading rule, addition, but means, or 

 meant, numbering generally. Logicians would speak more to 

 the purpose, in English, if they substituted monadical for 

 numerical: nothing can make a numerical difference to an 

 English ear except a difference of numerical quantity. 



43—2 



