NOTATION. 



267 



and so on ; so that each character signifies ten times 

 more, according as it stands a place farther to the 

 left. If 7 stands in the place of the units, it signifies 

 7 units ; if it stands in the place of the ten-thousands, 

 it signifies 7 times ten thousand. To determine the 

 place of a character, zeros are made to the right of 

 it : thus, to express seventy thousand, we write 

 7C,000, which shows that 7 stands in the fifth place, 

 which is that of the ten-thousands. The system pro- 

 ceeds still farther : if we have to express seventy 

 thousand one hundred and thirty-four, we do not 

 write 70,000, and 100, and 30, and 4, but 70,134. 

 Place the above numbers one under the other, and the 

 reason is obvious: 



70000 



100 



30 



4 



It appears that it is unnecessary to retain three of 

 the zeros to the right of 70,000, the two zeros to the 

 right of 100, and the zero to the right of 30, because, 

 if we leave them out, thus, 



70 



and write the remaining figures all in one line, thus, 

 70,134, each character will have the same place as 

 it had when each number was written out full, and 

 therefore will have its due value. In order to give 

 a number its proper position, a zero is written wher- 

 ever no number is to be expressed in one of the other 

 places, as in the above case, the zero to the right of 

 7, without which the 7 would have denoted seven 

 thousands, not seventy thousands. This is the 

 system of notation of whole numbers, and the mode 

 of expressing fractions differs only in this, that they 

 are numbered from left to right. 



(See Decimal Fractions.) The decimal system, so 

 simple and perfect, is one of the most striking of 

 human inventions, and few tilings have had a greater 

 influence upon the progress of science and civiliza- 

 tion. Little do we dream of the enormous difficulty 

 of calculations with the Roman and Greek notation, 

 and the obstacles which it must have thrown in the 

 way of every art, science, or occupation requiring 

 arithmetical operations. The commerce or the navi- 

 gation of the present day could hardly have been 

 carried on under such systems, and the general habit 

 of keeping regular accounts, which so greatly facili- 

 tates the business of life, and tends so much to pre- 

 serve the good order and peace of society, could 

 hardly have grown up. The origin of this invention, 

 as of many which have exerted the greatest influence 

 on society, is veiled in obscurity. (For a few histori- 

 cal remarks on the introduction of ciphers, see the 

 article Ciphers) As in our decimal system every 

 possible number is expressed by ten characters, so 

 we can imagine a dyadic or binarian system, of which 

 two is the fundamental number, and which, to ex- 

 press analogously every possible number, would 

 require but two characters (say 1 and 0), one being 

 represented by 1, two by 10, three by 11, four by 

 100, five by 101, six by 1 10, seven by 111, eight by 

 1000, nine by 1001, and so on. Leibnitz first deve- 



loped such a system in his Dyadics. In the seme 

 way we can compose systems of three, four, or any 

 given number of characters; and as 12 is a more 

 convenient number for division and subdivision, we 

 knew a German professor of mathematics, who, hav- 

 ing invented for himself two more characters for 1 1 

 and 12, and given them names, made a dodecadic 

 system, by which he performed the most rapid calcu- 

 lations: 12 was the basis of his system, as 10 is of 

 ours, and each character signified 12 times more 

 than its neighbour to the right, and 12 times less 

 than its neighbour to the left : thus, suppose he used 

 for 10 the character X, and for eleven T, then his 

 series would be thus : 



so that 100 expresses, in this system, what we should 

 represent by 144. We have dwelt on this point, be- 

 cause it seems to us that the practice of making ex- 

 periments of this sort would serve to show pupils, 

 more clearly, the nature and character of our system 

 of numeration, the great and sole advantage of which 

 lies in the system, not in the number which serves as 

 its basis. Several tribes have not advanced beyond 

 the rude system of adding successively one to one, 

 and giving a new and arbitrary name to each new 

 number ; but for this reason they cannot count far, 

 and, after going beyond a certain number, they gen- 

 erally say many. The Romans, though they had the 

 decimal numeral system, nevertheless had not the 

 decimal notation. In their notation, they took five 

 units together, and wrote them thus, V ; then they 

 brought two such fives together, from which origi- 

 nated ][, or X ; five tens they wrote L ; one hundred 

 was written C, the initial of centum; five hundreds 

 were expressed by the sign D ; two five hundreds, or 

 one thousand, by M. (See M.) A cipher written to 

 the left of another of higher value, in this system, 

 takes so much from its amount, as, XL is 40, and XC 

 is 90; accordingly 1847 is written MDCCCXLVII. 

 Here no sign has a value according to its position 

 (except, in some degree, X ; but it is not the value 

 of X which is changed, but merely its character : it 

 becomes negative by standing to the left of L), and 

 pronouncing such a number is making an addition. 

 It is utterly impossible for us to imagine the difficulty 

 of their calculations, because, if we take a case of 

 simple addition, as, 



M 



XLVIII 



DV 



CCIX 



XXXVIII 



XCII 



it would be comparatively difficult for us to perform 

 it, although the numbers form themselves in our mind, 

 not according to these signs, but according to our 

 decimal representation, and thus we can perform the 

 operation much easier than they could. They, there- 

 fore, were obliged to have recourse to their abacus 

 (q. v.). We may add, that 500 was represented by 

 ID, as well as by D, and that for every 5 added, this 



