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GENERAL INFORMATION ON SUBJECTS OF MATHEMATICS, NATURAL PHILOSOPHY AND HISTORY, CHRONOLOGY, GEOGRAPHY, STATISTICS, &c.
I.-ON ANCIENT AND MODERN USAGE IN
THE year 1850, to which the present number of our work belongs, is a year one designation of which depends upon the settlement of a controversy. About the end of last century, there was a fierce contest upon its proper termination: some said that the nineteenth century began with January 1, 1800, others with January 1, 1801. The former would say that 1850 is the first year of the second half of the century, the latter that it is the last year of the first half. In taking up this subject, we do not look at the century question (which is of no consequence whatever, and is easily settled in favour of the second-named interpretation) so much as at the mode in which it arose, and at the genus of which it is a species. We make it a peg on which to hang a short dissertation on the distinction of ancient and modern reckoning, which the lawyer generally ignores, the scholar often disregards, and the mathematician almost always denies the existence of upon demonstration.
Those who are accustomed to settle the meaning of ancient phrases by self-examination, will find some strange conclusions arrived at by us; but nothing, we believe, which may not be justified by even a moderate examination of old writers.
Language and counting both came before the logical discussion of either. It is not allowable to argue that something is or was, because it ought to be or ought to have been. That two negatives make an affirmative, ought to be: if no man have done nothing, the man who has done nothing does not exist, and every man has done something. But in Greek, and in uneducated English, it is unquestionable that "no man has done nothing" is only an emphatic way of saying that no man has done anything; and it would be absurd to reason that it could not have been so, because it should not.
The manner in which any common reckoning of time is made would, we might suppose, be a matter admitting of neither dispute nor ambiguity, and of little, if any, change. It is the object of this paper to point out that such is not the case—that language is to this day but ill adapted to express precise meaning-that serious and not sufficiently marked changes have taken place in the modes of reckoning-and that the confusion which these changes have made continues.
At the same time, there is nothing on which we are so positive, each for himself, as upon what is and what is not, right in the matter of reckoning time. Every one has, or thinks he has, a permanent meaning attached to the phrases in common use; which meaning no small number think these phrases must of necessity bear: others, aware of the very different senses which the phrases have borne, are content to admit that their meanings are conventional, but are prepared to contend for the existence of a well-settled and universal convention.
Suppose that at ten o'clock on Monday morning, a person engages to do something in four days. There are four distinct meanings, each of which will bear argument or citation of authority, and each of which may have been in the understanding of the speaker or of the hearer.
First, not counting Monday, on which the engagement is made, Tuesday, Wednesday, Thursday, Friday, may be claimed as appertaining to the four days; in such manner that the pledge cannot be considered as broken, until some moment of Saturday has arrived without its performance. This is one extreme case, and is the debtor's version.
Secondly, counting Monday, the day of the engagement, the four days may be reckoned as Monday, Tuesday, Wednesday, Thursday, and the claimant may consider that at any moment of Thursday proof of performance is due. This is the other extreme case, and is the creditor's version.
Thirdly, different tendencies towards a mixed mode of interpretation may lead to the result that Friday is the day on which the performance may be claimed: and for this many will pronounce, when they consider the question, from mere indecision between the preceding two cases.
Fourthly, those who consider a day as capable of beginning at any moment will say that from ten o'clock on Monday morning to ten o'clock on Friday morning it is four complete days; and that therefore proof of performance may be claimed on Friday, but not before ten o'clock in the morning.
If we had to make a meaning for the phrase, we might well fix on the third, which perhaps would be most generally agreed on in our day as the proper interpretation. But we are to con
sider the meanings which have prevailed: this is a very different thing, and will require a little discussion of the stages of the process of counting.
The earliest process of arithmetic is that of counting units, the unit being considered both as the commencement and the ultimate subdivision of the process: in such manner that between ten and eleven, for example, there is nothing imaginable; nothing more between ten and eleven feet than between ten and eleven horses. As to the latter instance, we should still agree with the ancients: we should refuse to admit of any number of horses to ride upon between ten and eleven, being wholly unused to see such chargers as Baron Munchausen's; though we might admit ten and a half horses as a possible sale of dog's meat. But with regard to feet we should be inclined to assert that there are lengths between ten feet and eleven feet, and that he restricts modern language, to say the least, who asserts that there is no number of feet between ten and eleven, though it may be proper to say that there is no intermediate number of complete feet, no whole number of feet. But we are familiar* with fractions: down to the beginning of the sixteenth century, the Romans, and all the Europeans who used Latin, were so strange to the idea of fractions of numerable units, that the books of arithmetic hardly contain a notion of them. editions of Boethius (the most common text-book of the learned) published in the fifteenth century, though perfectly free from all allusion to fractions, actually use lengths, by which fractions of a unit can be shown, to indicate the integers of their processes. And this usage is precisely due to the absence of the notion of fractions. A modern teacher uses dots, drawings of pebbles, or horses, or men, by which to inculcate purely monadic counting. He avoids length or area, or other simple magnitude, in the first instance, because he does not want to suggest the fraction before its time, which he thinks he should certainly do, if he employed a unit capable of division into parts like itself. But the teacher of the fifteenth century felt
*It is not perhaps allowable to say, as yet, that power of applying notions of arithmetic, except in routine ciphering, is very common. A man of high scientific station, now deceased, who was long a member of the House of Commons, used to say that there never were, at any one time of his continuance in it, more than three men in the House who had a tolerable notion of fractions. The following will show that a palpable absurdity will pass before the eyes of generations of men of letters without notice. In Boswell's Life of Johnson (chapter viii. of the edition with chapters) there is given a conversation between Dr. Adams and Johnson, in which the latter asserts that he could finish his Dictionary in three years. "ADAMS. But the French Academy, which consists of forty members, took forty years to compile their Dictionary.-JOHNSON. Sir, thus it is. This is the proportion. Let me see: forty times forty is sixteen hundred. As three to sixteen hundred, so is the proportion of an Englishman to a Frenchman. No one of the numerous editors of Boswell has made a note upon this, though many things as slight have been commented upon it was certainly not Johnson's mistake, for he was a clear-headed arithmetician. How many of our readers will stare, and wonder what we are talking about, and what the mistake is ?
no such fear and there cannot be imagined such a proof of it, as the use of divisible magnitude to signify number in a book so thoroughly monadic as the arithmetic of Boethius. The earliest editions of the arithmetical books of Euclid do the same thing.
In our day some notion of fractions is learnt so early and has become so familiar, that the monad,* or indivisible unit, has almost disappeared. There are few elements of computation which we are not accustomed to separate into subordinate elements, which are actually parts of themselves: not merely conventionally, as in the case of the twelve copper pence which are held equivalent to the silver shilling (this could be done in the old system), but actually, as in the case of the twelve inches which are the very parts of the foot. We have accordingly forgotten the old maxim that unity has no parts, which was so well fixed in the minds of our forefathers that they likened unity in arithmetic to the point in geometry. It was in 1585 that Stevinus (one of the most original minds of his day; no less man was wanted) dared to say that it is 0 in arithmetic which answers to the point in geometry, and to pray that the Author of nature would have pity upon the unfortunate eyes of those who could not see it to be so.
We are arriving at the other extreme, or shall do so, if books of arithmetic do not soon begin to inculcate the distinction of monad and magnitude. A man had a letter to send, for which the postage was one penny. Not having a penny stamp by him, he cut a twopenny one in half, and affixed one of the halves to his letter. The post-office clerks, who are monadists, considered this as an unpaid letter, and charged it accordingly; on which the fractionist, considering himself unjustly used, presented more than one memorial to the higher authorities.
It was not impossible, in the old reckoning, to imagine that the monad of the reckoning was only a part of its divisible space, the other part being nonexistent: for example, that time should be reckoned by days, of twelve hours each, the intermediate nights being blotted out. The founder of our æra, Dionysius Exiguus, does this: in the last of the rules presently alluded to, he makes his day to be twelve hours, multiplies his number of days by twelve to get the hours, and adds three hours each year to get the intercalary day once in four years: istæ tres horæ faciunt in IIII annis diem.
It may be convenient to revive this old term, in the sense stated. Any magnitude whatever may be considered as a monad, or unit for repetition incapable of division.
The question will naturally arise, might not Dionysius by possibility be making use of an hour twice as long as the usual one? Fortunately he explains himself on this point, indirectly. He pronounces against a curious notion extant in his time, that the additional day of leap-year was a commemoration of the long day on which Joshua caused the sun to stand still: here day means term of daylight, but dies is still the phrase used. Dionysius had a very odd notion of the meaning of bissextile, He thinks that