ARITHMETIC. 



257 



in arithmetic to the mere lalxwr of writing ; but 

 nothing would thus be gained in practice, since, ad- 

 vancing with excessive slowness, it would soon re- 

 quire a multiplicity of words, and a fatiguing compli- 

 cation of characters. The binary scale appears best 

 adapted to the descending progression ; for the 

 fractions produced by a continued bisection, are, from 

 the equal competition of buyer and seller, naturally 

 introduced into commercial transactions, and employ- 

 ed almost exclusively among the Eastern nations. 

 This subdivision is likewise used with convenience in 

 Europe, for ascertaining the smaller weights. The 

 next step in the progress to numeration, was probably 

 to assume the double pair, or four, as the root of the 

 scale. In counting over small articles, it is custom- 

 ary, for the sake of expedition, to take a couple in 

 each hand ; and therefore, the throw, or, in older 

 language, the warp, becomes, in this way, the 

 measure of tale. The ancient Mexicans appear to 

 have reckoned by fours, and to have afterwards ad- 

 vanced, in their numeration, by combining the pro- 

 ducts of four with those of ten. Nor is it altogether 

 improbable that Pythagoras might have alluded to 

 such a system of computation, in celebrating the mys- 

 tical properties of his famous tetractys, or quaternion. 

 But nature has furnished the great and universal 

 standard of computation, in the fingers of the hand. 

 A 11 nations, accordingly, have reckoned by Jives ; 

 and some barbarous tribes have scarcely advanced 

 farther. A ristotle, who was aware of the principle, 

 lias noticed the existence of such a people in Africa. 

 After the fingers of the one hand had been counted 

 over, it was a second, and perhaps a distant step, to 

 proceed to those of.the other. The primitive words 

 expressing numbers probably exceeded not five. 

 To denote six, seven, eight, and nine, the North 

 American Indians repeat the five, with the succes- 

 sive addition of one, two, three, and four. The same 

 composition is apparent in the various dialects spoken 

 by the inhabitants of the islands which are so widely 

 scattered over the Southern Ocean. Could we safely 

 trace the descent and affinity of the abbreviated 

 terms denoting the numbers from five to ten, it seems 

 highly probable that we should discover a similar 

 process to have taken place in the formation of the 

 most refined languages. The alphabet must in 

 yeneral have been framed before any regular system 

 for notation of numerals was invented. In forming 

 such a system, the obvious method was to imitate as 

 nearly as possible the progress by which the mind 

 ascends the scale of numeration ; but the simplicity 

 mid uniformity of this procedure were in the sequel 

 frequently disturbed, by adopting such alphabetic 

 characters as happened to resemble the compound 

 symbols, or by employing, for the sake of abbrevia- 

 tion, the initial letters of words significant of the 

 numbers themselves. The Roman numerals, having 

 undergone little subsequent change, may be con- 

 sidered as the most ancient specimens of notation. 

 To denote one, a simple vertical stroke was assumed 

 | ; and the repetition of this expressed two, three, &c. 

 Two decussating strokes X marked the next step in 

 the scale of numeration, or ten ; and that symbol was 

 repeated to signify twenty, thirty, &c. Three strokes, 

 or an open square [3. were employed to denote a 

 It nndred, or the thirdstage of numeration ; and four 

 interwoven strokes |^J, sometimes incurved ("*), or 

 even divided clr>, expressed a thousand. Such are 

 all the characters absolutely required inn very limited 

 system of numeration. The necessary repetition of 

 them, however, as often occasionally as nine times, 

 was soon found to be tedious and perplexing. Reduced 

 or curtailed marks were, therefore, employed to ex- 

 press the intermediate multiple's of five ; and this 

 improvement must havv> *aken place at a very early 



period. Thus, fine itself was denoted by the upper 

 half \/, anjj sometimes the under half /\, of the 

 character X f r ten > L r the half of , the mark 

 for a hundred, came to represent fifty ; and the in- 

 curved symbol p% or do, for a thousand, was split 

 into lo, to express Jive hundred. These important 

 contractions having been adopted, another convenient 

 abbreviation was introduced. To avoid the frequent 

 repetition of a mark, it was prefixed to the principal 

 character, and denoted the defect by counting back- 

 wards. Thus, instead of four succeeding strokes 

 1 1 1 1, it seemed preferable to write ;. ' ; for eight and 

 nine, the symbols were ||X an( l IX 5 a "d ninety was 

 expressed by XEZ This mode of reckoning by the 

 defect was peculiar to the Romans, and has evidently 

 affected the composition of their numerical terms. 

 Instead of octodecim and novendecim, it is held more 

 elegant, in the Latin language, to use undevigintt 

 and duodeviginti. But the alphabetic characters now 

 lent their aid to numeration. The uniform broad 

 strokes were dismissed, and those letters which most 

 resembled the several combinations were adopted in 

 their place. The simple stroke | for one, and the 

 marks V, X> an d L_ for Jive, ten, and fifty, were re- 

 spectively supplied by the letters I, V, X and L. The 

 symbol for a hundred, was aptly denoted by C, 

 which had originally a square shape, and happened, 

 besides, to be the initial letter of the very word 

 centum. The letter D was very generally assumed 

 as a near approximation to the symbol j;) for five 

 hundred ; and M not only represented the angular 

 character for a thousand, but was likewise, though 

 perhaps accidentally, the first letter of the word mille. 

 The last improvement attempted in the Roman sys- 

 tem of numerals, was devised for the purpose of ex- 

 pressing the numbers beyond a thousand. This in- 

 novation belongs evidently to an advanced period of 

 society, and appears never to have been very gener- 

 ally embraced. The method of proceeding, however, 

 was perfectly analogical. Taking the complex sym- 

 bol clo for a thousand, the intermediate stroke was 

 retained, while the C on each side of it was succes- 

 sively repeated, to mark the ascending progression 

 by tens. Thus c c I o o and c c c J o o o were made to 

 signify, respectively, 10,000 and 100,000. The 

 halves, again, of these compounded characters, or 

 I D o and I o r> o, were employed to denote 5,000 and 

 50,000. The oldest form of notation among the 

 Greeks, and the system of numerals retained by the 

 Romans, were utterly incapable of any material 

 improvement. They might serve laboriously to 

 register a number that was not very large ; but they 

 could not afford the slightest aid in performing an 

 arithmetical computation. By what ingenuity, for 

 instance, could even such small numbers as 48 and 

 34 be multiplied together, if expressed by the com- 

 plicated symbols XLVIII and XXXIV, where both 

 the units and the tens are equally involved ? But the 

 Romans were late in acquiring any taste for refine- 

 ment, and remained, during the whole course of their 

 history, profoundly ignorant of science. In the few 

 simple calculations which they had occasion to make, 

 the Romans were obliged to have recourse to a sort 

 of mechanical process, employing pebbles or counters. 

 Boys were taught that humble art at school, and 

 carried with them, as implements of computation, a 

 locnlus, or box filled with- pebbles, and a board on 

 which these were placed in rows, called Abacvs, (q. 

 v.) It is curious to observe, that the term calculation 

 itself claims no higher descent than from calculus, a 

 pebble. The labour of counting and arranging those 

 pebbles was afterwards sensibly abridged, by drawing 

 across the board a horizontal line, above which each 

 single pebble had the power of five. In the progress 

 of luxury, tali, or dies made of ivory, were usj-u in- 

 2 KT. 



