NUMERAL CHARACTERS. 



NUMERATION. 



i "i 



(which, according to HeUbronner, U alluded to in a written work by 

 Herodian only), U supposed to be u old aa the time of Solon. The 

 Egyptian hieroglyphic y*tem U on the same principle, but without 

 abbreviation* ; the symbol of too resembling that just given for five. 



In describing the later Oreek notation, we leave out of view the 

 xtooaiona made by the nmthomaUcuuis, the principle of which ia 

 described in ARITHMETIC. It appear* moat distinctly that the system 

 wa* made either at a time when the Oreek alphabet was in possession 

 of more lettera than it permanently retained, or that it waa introduced 

 into Greece by communication with aome nation (the Phnjmoian, per- 

 hapa) which had aome additional lettera. The fan of the Hebrew and 

 Phoenician, which stands for six, and is wanting in the Greek alphabet, 

 appear* in the Greek numeral system under the name of Maiuutr BoD, 

 and ia expressed by a symbol resembling r, not very unlike the Vau 

 turned the other way. The Koph and Tsaddi appear under the names 

 of lr(<rn.uo nor** and iw'untl"" avnt, with symbols expressed iu our 

 types by and g; but the former is one behind iu place in numeral 

 signification, being 90 among the Greeks and 100 in the East : the 

 latter takes the aame numeral place as the final Tsaddi in the Hebrew 



item. The word air will be a useful guide to the letters beginning 



several scales, as follows : 



t ; . - . 



Letter $ y S 



Numeral SifniBcatiem 1 S 3 4 



i A n 



10 20 30 40 



c 

 A 



V 

 JO 



f 



CO 



o 

 70 



80 



5 



90 



100 200 300 400 900 600 700 800 900 



, ft, y, *, , *, C i, , 



1000 2000 3000 4000 4000 6000 7000 8000 9000 



The Rtnn notation, including all the varieties which occur in 

 printed works, ia as follows : 



1 I 



2 II 



3 III 



4 1111,1V 



:. v 



6 VI 



7 VII 



8 VIII. IIX 



vim. is 



10 x 



11 XI 



l-J XII 



18 XIII, XI IV 

 14 XI1II, XIV 

 18 XV 



16 XVI 



17 XVII 



18 XVIII. XIIX 



19 XV11II, XIX 



20 XX 

 30 XXX 



40 XXXX, XL 



50 L 



60 LX 



70 LXX 



80 LXXX, XXC 



90 LXX XX, XC 

 100 C 

 200 CC 

 300 CCC 

 400 CCCC 

 500 D, Io 

 600 DC, IoC 

 700 DCC, lo 

 800 DCCC, loCCC 

 900 DCCCC, loCCCC 



1000 do, M, <x>. I 



2000 CIoCIo, IICIC, IIM 



5000 loo, V 

 10000 CClQO 



50000 lr>.i i 



100000 CCCIOOO 



1000000 ccccioooo 



The grammarian Priscian would have it that I denoted unity, 

 because the Greek word ftia, with M cut off, boa i for its first letter ; 

 that V is five, as being the fifth vowel, and X ten, as being the tenth 

 consonant, of the Greek alphabet. Any explanation of this system 

 which endeavours at an alphabetical deduction must, as far as has yet 

 been seen, fail entirely in giving a probable origin. The following 

 scheme, however, contains suggestions of some antiquity, and certainly, 

 as before remarked, U either a true explanation or a most extraordinary 

 coincidence. The account of it was revived in our day by Leslie, in 



I II III llfl 



o b e d 



CO 



CD D ID X 



an article in the 'Edinburgh Review,' and afterwards in his ' Philosophy 

 of Arithmetic.' But it may be found almost entire in the ' Cursus 

 MathemaUcua ' (1690, vol. i., p. 28) of Deoholes, who gives it as the 

 opinion of several of his time. And the earliest hints (by no means 

 complete, and mixed with some absurdities) which we have found are 

 in the ' Lie NumorU Libri Duo ' of John Noviomagus, Paris, 1539, 8vo. 



Imagine a person used to decimal counting by means of the fingers 

 having recourse to simple counting by making a mark for each suc- 

 cessive unit, as in a, b, c, IK. At ten he might be expected to make 

 some symbol that bin hand/id was completed, and the drawinga mark 

 through the whole ten unit symbols, as at t, would do v. 

 This he might abbreviate, as at /, retaining only the symbol of a unit 



and that of the line drawn across. The handful of tens, or one hun- 

 dred, might be represented by retaining only the unit symbol and two 

 ligatures, namely, that of the tons and that which made all the ten 

 tens into one symbol, as at g. The ten hundreds would require a unit 

 with three ligatures, or four strokes altogether, which, it they were 

 written without taken the pen off, might be made as at A, which might 

 degenerate into I; and finally into /. Again, by cutting / into two 

 halves, we have m or w, which might suggest themselves as proper 

 representatives of half a thousand, or five hundred. Similarly, 

 bisection of / ami ;/, or of o and 9, would suggest p and r as proper 

 gymbolii for the halves of ton and one hundred. The symbol a has a 

 resemblance to I, / to X, <j to C, A to M. m to I), p to V, and r to L. 



We cannot find any precise information upon the time of the com- 

 mencement of the principle of local value which prevails to a 

 extent throughout the Koman system, namely, that a smaller symbol 

 before a larger one, in numbers less than one hundred, denotes a sub- 

 traction, after it an addition. This principle does not appear in the 

 riuiMiician or Palmyrene notations, which otherwise much resemble 

 the Roman in their principle of notation, though they approximate to 

 pure vicenary scales, both adopting distinct symbols for twenty. 



The Chinese use three systems : the first, not very simple, and 

 ancient ; the second, intentionally complicated, and employing symbols 

 of words to denote numbers, is introduced in deeds and other instru- 

 ments, to render alteration difficult ; the third, a simplification of the 

 first, supposed to have been made by the Jesuit missionaries. 



For further information on the subject of this article, with abundant 

 references, see the article ' Arithmetic ' in the ' Enclopxdia Metropoli- 

 tana,' by Dr. Peacock. 



NUMERATION is a term generally applied to the art of repre- 

 senting numbers by distinct names and symbols, a sense in which the 

 word is used by the oldest writers. Every treatise on arithmetic must 

 necessarily begin with something on this art of counting and repre- 

 senting the result* of counting, on the goodness of which, slight and 

 easy as any method may appear to which we have been habituated 

 from childhood, the progress of the arts of life, to say nothing of the 

 mathematical sciences, is in no slight degree dependent. The time U' 

 gone by for a formal eulogy upon the benefits of any fundamental 

 methodAf expression; we will therefore content ourselves with quoting 

 a part of that which is found in the first English work on arithmetic, 

 Robert Recorde's 'Grounde of Artes' (1540). We quote this also 

 because it is on instance (the only one we ever met with in a mathe- 

 matical work) of the species of doggrcl comic verse afterwards in use 

 on the stage (see the ' Comedy of Errors '), which has a sort of measure 

 and rhyme, though printed in the form of simple prose in the work 

 from which we cite (we put the syllables which are meant to rhyme in 

 italics) : " Master. Wherefore in all great workes are clerks so much 

 desired ? Wherefore are auditors so richly fed } What causeth geo- 

 metricians so highly to be inkaumcd 1 Why are astronomers so greatly 

 advanced > Because that by number such things they nml. which else 



would for excel man's mind Mailer. Exclude number, and 



answer to this question : How many yean old are you I Sdtolar, 

 Mum. Matter. So that if number want, you answer all by miimmet 1 



How many miles to London? Schular. A poak full of filuuu 



Matter. If number be lacking, it maketh men dumb, so that to most 

 unctions they must answer mum. Scholar. This is the cause, sir, that 

 1 judged it so vile, because it is so common in talking every ichilt ; for 

 plenty is not dainty, as the common saying i'. Matter. No ; nor store 

 is no sore, perceive you thit I The more common that the thing U 

 being needfully required, the better is the thing and the more to be 

 dnired. But in nimibriug, as some of it is light and jtiain, so the most 

 port is difficult and not eaaie to attain." 



The earliest method of signifying a large number must have been 

 such a one as the scholar uses above, when he designates a large 

 number of miles as a " poak full of plums," namely, the similitude of 

 some visible or well-known c<>ll-tu>n of units. The fingers of the 

 hand, or of both bonds, or the united number of fingers and toes 

 furnished natural collections of reference on which the various quinary, 

 decimal, and vicenary scales in existence have proceeded. The tranmtion 

 from counting by tens to counting by dozens might have been caused 

 by the facility of subdivision which the number twelve possesses, 

 though wo rather doubt this explanation, at least unless we assume 

 that the division of the Roman A into twelve iincim is to be explained 

 on the same principle. From this we think came the method of 

 reckoning by dozens to be introduced throughout Europe, as would 

 that by thirteen*, if the Roman coin or weight hod been so divi 



Our present numerative system is stated by writers to employ the 

 words unit, ten, hundred, thnuwnd, million, billion, trillion, quadrillion, 

 quintillion, sextillioti, wptillion, octillion, nonilliou, &o. But the 

 greater part of this is pure statement; for the terms billion, trillion, 

 Ac., though defined by arithmetical writers, have never found tli<>ir 

 way into common use, the want of such large numbers having never 

 been experience!. The French indeed have naturalised the term 

 /, meaning one thousand millions, in matters of public debt and 

 KM line, which only shown liow little, the term billion has been used 

 among them, since, according to their writers, the milliard and billion 

 are the same things. Tonstal expressly says, that in his time 

 (Henry VIII.) the common reckoning from millions was made by 

 millions of millions, Ac., and the word millio is noted as a vulgarism 



