at ARITHMETIC. 



To enable our readers rightly to estimate the advantage which we 

 PUSM* in our notation, we will here describe that of the Greeks, which 

 I* only equalled by that of the Chinese in its near approach to the 



ARITHMETIC. 



doming the principle of ' local value,' ami sub-tituting in its pla. 

 a system of symbols as, without .U-|>arting from the principle of Greek 

 notation, will not confuse the reader l>y the adoption of new digits. 

 ><* the actual mgiw ued by the Greek*, >ee NUMERATION; Ni MEBAI.S. 

 I/et the first nine numlm* be repreaented a usual, but let ten (instead 

 . i in. in which 1 hai local value) be repreaented by 1', twenty by 2', Ac. 

 I.*t 1* be one hundred, 2* two hundred, and so on ; 1'" one thousand, 

 y two thousand, and au on. Let M stand for ten thousand, and let 

 II affixed to a number make its value ten thousand times as great ; 

 thus, 4'2M U 420,000 in our notation. We have hen- improved upon 

 the system of the Greeks, unavoidably, in order not to OOBfnM the 

 reader, since 2000, 200, 20, and 2, would not among them present to 

 the eye that analogy which exists between 2'", 2', 2', and 2, being in 

 fact denoted by 



ft, a, K, and 0. 



We now write some high numbers in our own decimal scale, accom- 

 |nied >>y our imitation of the Greek. 



46379268 4'"6"37M.9"'2'6'8 



07.0 . 6"'7M.3' 



72007106 7"'2"M.7'"r6 



In the first number, vhert tArrt it iw cipher, the Greek looks BO like our 

 own that we might be led to imagine there was no essential ditl./i . M. .-. 

 \\ . might say, that as it would be natural, and was in fact usual, to 

 write the higher numbers first, the mere occurrence of a fourth column 

 would suggest the idea of thousands, so that a notion, which we must 

 call one of lofal ralar, would be inevitably formed. And perhaps it 

 was so : indeed, it is surprising that neither Archimedes, Apollonius, 

 or Diophantus, ever detected and improved the idea. But when we 

 come to look at the second and third numlier, we see immediately that 

 the continual derangement of the columns would prevent this notion 

 from acquiring consistence. The symbol of ramify is wanting ; and we 

 cannot see how great an impediment that defect presented, because we 

 learn 20, 30, Ac., as soon as we learn tin-niii, thirty, &c. And though 

 perhaps 2', 3', &c., might have suggested such a contrivance, yet there 

 was no analogy between * (20) and X (30) and (2) and y (3). 



The ingenuity both of Archimedes and Apollonius was employed in 

 the extension of the preceding system, without alteration of its prin- 

 ciple. That of the Utter we shall imitate. Calling 10,000 M,, let ten 

 thousand times ten thousand be called M,, ten thousand times that 

 number M v and so on, and let any one of these placed immediately 

 after a number mean that the preceding U to be taken ten thousand 

 times if followed by M,, ten thousand times ten thousand if by M,, and 

 o on. The following number, 



1768,9360,0142,0193, 

 would then be represented by 



l"7'6 / 8M r 9'"3"6'M,.l"4'2M l l"a'3, 



on which we may make the same remaks as before. The method of 

 Archimede* (which preceded this) differed from it only in nuking ten 

 million the ra<li.r of the system. We now see why our arithmetic was 

 called rlijtrriuij, cipher coming from an Arabic word signifying raranl. 

 I Hi.- .inJi thought as occurred to Archimedes in the bath [AncHiMEDKS, 

 Hn.. IMV.] might have been fourteen centuries gained to the science. 



We look in vain for anything like local value in the system of the 

 Kgyptians, or any other nation of antiquity who are known with 

 nrtaimly to have very ancient records. That of the Jews was similar 

 to the one just described, so far as it went, and the use of some letters 

 common to both (P. 406) in tin- numeral system, but not so in the 

 alflatfti of the two, proves that the notation of both hail a common 

 onree. [NUMERALS.] 



To the same article we must also refer for the Roman system, which 

 !<! itself throughout Kuro|>e during the first twelve centuries. 

 It is much more rude than the Greek, and is a sufficient proof of the 

 well-known inaptitude of the former people for scientific invention. 



The Chinese had several systems of numeration, nil containing com- 



I -Mi.l-.l-. and somewhat resembling that of tin- (ireeks in 



principle, but with thin important difference, that the symbol for 30, 



for example, lias direct analogy with that for :t. being made l.y the 



jiixtipoirition of a symbol for tea ; so that the improvement upon the 



- wale which we have been obliged to make in onler to explain 



it. n-iiffeni our imitation of the Greek a better resemblance of the 



-. Hut they have no written method of expressing local value ; 



though their Hrhtran-pan f ABACI'S] U a practical use of the principle. 



v we proceed to the history of our own scale, we must extend 

 our remark, that the ' decimal notation' and ' system of local value' art 

 distinct thing*. When we agree that 10 shall stand for ten, we merely 

 express that a number in the second column from the right shall stand 

 for tn timrt ss much ss the same in the first column. But we are at 

 liberty to suppose that a number in the second column shall mean nine. 

 . T any other number of time* what it does in the first. Thus, if 



* a i/uiiiarti scale, in which 10 stands for 5, we reject tl. 



'.. 7, 8, and 9, anri our numerical i 



1 2 



i ! 



I scale runs thus : 

 3 4 10 11 12 13 14 20 21 22 *.-. 

 I I f JJ I t I B I I 8 



! | I I 



I ! 



Thus 20 is in, because 2 in the second column counts five times 2. 

 But if we choose a higher scale than the decimal, we shall 1. 

 invent instead of rejecting symbols ; if, for instance, we take a ili*l, 

 Harf scale in which 10 means twelve, we are left without symbols f..r 

 ten and eleven. Let t and e stand for these ; then our scale of n 

 beginning from ten, is as follows : 



e 10 11 12 13 14 15 16 17 18 19 11 



I 



* I 



Ir 20 fte. 



I i! 







8 I 



o 



k I 



But the scale which best exemplifies the principle is the binary, in 

 which 10 stands for 2, and in which there are consequently no nymb..l~ 

 except 1 and 0. The system of numbers in this scale (from one to ; n 

 is as follows : 



1 10 11 100 101 110 111 1000 1001 1010. 



A Jesuit at Pekin (P. 392) communicated to Leibnitz the following 

 Chinese symbol, called by them the Cova, or liiirnti'iH. and attributed 

 to F.ihi, the founder of the empire. It is suspended in their temple.-, 

 and considered as a mystery : 



If the long line be interpreted to mean one, and the broken line 

 natltinij, these symbols, each being read from the bottom to t 

 give a system of binary arithmetic from to 7 (both inch 

 Leibnitz asserts that there is it larger Cora, which go. - up to t;:;. lint 

 u U" additional information has been obtained upon the subject. 

 which, for anything certainly known to the contrary, may be a hoax, 

 we can only say that there is some presumption that the Chinese l..n^ 

 ago possessed the complete principle of the ' local value.' 



We trace our own knowledge of the decimal system direct t.. the 

 Hindoos, who themselves ascribe it to the divinity. As to the manner 

 of its introduction, there are some differences of opinion on that 

 One and the old account is, that Gerbert, after Pop.' S\ hitter II., 

 found it in Spain among the Moors (P. 415) in the latter part of the 

 10th century. But upon this there are strong reasons for hesitating 

 J.SYI.VI.STKH" II., Uiou. l)iv.] Another and more probable account is, 

 that Leonard of Pisa [BoXACCi, Bioo. Div. ; AI.GKBRA] introduced it, 

 in 1202, in a work entitled 'Liber Abbaci.' Ac. : and some have sup- 

 posed that the ' Alonsine (or Alphonuine) Tables,' Iwing const \ 

 principally by Moors at the court of Alonso, numt have been tl. 

 in which the system appears (P. 413). It is certain that thin 

 had been before the 12th century, and most probably as earlv 

 !'th. in the hands of the Persians and Arabs, who ascribe it to the 

 II indi w, and call it by a name which signifies ' Hindoo science.' It is also 

 , citaiu that the Hindoos themselves have long used it [see VIJA CANMTA 

 anil I. ii lu ATI. names of Hindoo works], and that it ia easy to tl. 

 manner in which our numeral symbols have been derived from those of 

 the Sanscrit. In this latter language there are distu . 

 tent, lie., up to what we should callAiHirfrerf* of tlinii^iuilf / M////..JM eg 

 inillinnt. But whether we are to look to a Hindoo for the invention is 

 a question on which no surmise can be made till some probable 

 of the origin of Hindoo literature can be given. 



The stejis by which the new notation made its way through Kuiope 

 are not capable of being very clearly traced. Montfaucon (I*. 417) 

 found them in an Italian manuscript which was finished in !.".! . 

 many manuscript* of the works of authors a century older < ..Main 

 them, but it i well known that it was usual to substitute t: 

 figures for the old in recopying. In the library of ('orpin < 

 College, Cambridge (P. 418), is a <.. i eclipses from 1300 to 



1848, to which they are subjoined. IH..M-H .lates on inscriptioi 

 been given by Wallis and other* as old as 1330; but, upon cjeamn 

 reason has been found to suspect that 6 has been mistaken for 3. 

 There does not seem to be evidence of any general use of the Ai.il.ic 

 numerals before the invention of printing, and even the works of 

 Caxton do not contain them, except in a w.l. ut. M.-nhants con- 

 tinued their accounts in Koman figures up to the K'.th . . t.; m . in, 

 the whole, we think that the general use of th. . numerals in 

 works did not much precede, if at all, the diffusion of algebra. 



There has been much argument in favour of the assertion th 

 principle of local value in of older date in Europe than is commonly 

 supposed, and was known even to the Romans. We suspend our 

 opinion on this unfinished discussion, and refer the reader to the 

 writings of M. Cliasles. 



