1005 



NUMERATION. 



NUMISMATICS. 



1006 



by him (neither is it among the recognised barbarisms of Ducange). 

 Recorde uses nothing more than millions repeated ; so that it seems 

 the billions and higher denominations were never anything but a fancy 

 of arithmetical writers, conceived after the time when elementary 

 works ceased to be written in Latin. The probability of this' is 

 increased by their meaning different things in different countries : 

 with us the billion is a million of millions, a trillion is a million of 

 billions, and each denomination is a million of times the one pre- 

 ceding. With the French and the other Continental nations (except 

 some of the older writers, at least, among the Italians), the 

 billion is a thousand millions, and each denomination is a thousand 

 times the preceding. According to English writers, the number 

 1,234567,891234,567891 is one trillion, 234567 billions, 891234 mil- 

 lions, and 567891 ; according to the French writers, it is one quin- 

 tillion, 234 quadrillions, 567 trillions, 891 billions, 234 millions, 567 

 thousands, and 891. For common purposes, the denominations higher 

 than a million may be abandoned, it being remembered that all the 

 figures on the left, after six are taken oft' on the right, are so many 

 millions, and all above twelve figures so many millions of millions. In 

 writing, round numbers of millions should be written as such ; thus, 

 638 millions, not 638,000,000 : in computation it is of course a different 

 thing. Some authors seem to think it very scientific to parade ciphers, 

 sometimes by the dozen ; and so it is, no doubt, since it shows they 

 know how many ciphers go to a million or a million of millions ; but 

 no reader likes to stop and examine 000,000,000,000, when the words 

 " million of millions " would have done equally well. 



The decimal system, made complete, supposes a point always to be 

 placed at the end of the units, to separate them from the fractions 

 which may follow. When there are no fractions, the point is useless, 

 as in 675' or 675'000, which is 675. The numbers on the right of the 

 point, successively denoting tenths, hundredths, thousandths, &c. of a 

 unit [FRACTIONS], are in denominations which have not received 

 distinct names. The modern French call them declines, centimes, &c. ; 

 and the attempt has before now been made (see Wybard's ' Tacto- 

 metria ' 1650) to introduce centesms, millesms, &c. into English, but 

 with no success. 



The principle of local value [ARITHMETIC], which distinguishes our 

 system of numeration from that of the Greeks and Rodtens, is 

 applicable to any system, whether decimal or not. If 10 stands for 

 ten, that is, if its units in the second column are ten times in value 

 those of the first column, nine numeral symbols besides the cipher are 

 requisite ; but if 1 had signified fifteen, it would have been necessary 

 to have fourteen distinct symbols of number besides the cipher, since 

 10, 11, &c. now stand for sixteen, seventeen, &c. In such an explana- 

 tion, the frame-work of our numerical language (being decimal) is not 

 well calculated to give an easy comprehension of the change : we 

 should rather invent a word for fifteen, or five and ten, say A ; whence 

 A-one, A-two, &c. would be the spoken sounds answering to what we 

 now call sixteen, seventeen, &c. ; while ten, eleven, twelve, thirteen, 

 and fourteen would require new names not connected in etymology 

 with ten. 



The method of reducing a number, decimally expressed, to another 

 in which the radix or base of the system (as ten is that of the common 

 one), ia a, is as follows : divide the number successively by a, 

 expressed in the decimal system ; the remainders give the units, as, 

 aa, &c. of the new expression. Thus if 12376 is to be expressed in 

 the quinary system, whose base is 5, we should have the following 

 procew: 



6)12376 Rem. Number required, 



t>2475 1 344001. 



? )24 ' 5 Decimal. 



SJ495 3x5= = 9375 



5)99 4x5* = 2500 



5)19" 4 4x5*= 500 



5)3 4 



3 



12376 



This exhibits both the reduction to the quinary system and the 

 restitution of the decimal expression ; but if the number had been 

 given in the quinary system, it might have been reduced to the 

 decimal system by the same rule, the new base ten being, in the old 

 or culinary system, represented by 20, and the rule of division being 

 performed by the use of five in the same manner as ten is used in the 

 decimal system. 



20)344001 Rem. Number required, 



20)14422 11 or 6 12376. 



12 or 7 Decimal. Quinary. 



3 Ix (10)' = 310000 



2x(10)'= 31000 

 3x(10)* = 2200 



1 7x10 = 240 



6 = 11 



344001 



The quinary being supposed the old system, as soon as we come tn 

 the remainder 11, we have to invent anew symbol (say 6), since 11, 



in the new system, is to stand for eleven. For further examples, sea 

 the ' Library of Useful Knowledge : Treatise on the study of 

 Mathematics.' 



In teaching the elements of numeration by the abacus [ABACUS], it 

 is desirable that exercise should be given in several different systems, 

 were it only to prevent the formation of that impression which so 

 many students long retain, that the decimal system is natural and 

 necessary. The want of words for the denominations will be the only 

 difficulty; this may be got over by using the letters A, n, c, &c., to 

 represent them. Thus if the system be quinary, A counts as one ball 

 on the second row or five on the first, B as one ball on the third row, 

 five on the second, or twenty-five on the first, and so on. All the 

 balls on the second row may be marked A, those on the third B, &c. 



NUMERATOR (or numberer), the part of a fraction which states 

 how many of the aliquot parts of a unit are taken, such as are 

 described by the denominator. Thus f being three, not of xmits, but 

 of sevenths of a unit, 3 is the numerator. 



NUMERICAL, as opposed to literal, in algebra, applies to an expres- 

 sion in which the coefficients of a letter are all numbers, and not 

 letters. As opposed to algebraical, it applies to the magnitude of a 

 quantity, considered independently of its sign. Thus 7 is said to be 

 numerically greater than 5, though algebraically less. [NOTHING.] 



NUMERICAL DIFFERENCE. (Logic.) Down to our own time, 

 logicians have distinguished one individual as differing numerically 

 from another, when both belong to the same species : thus one horse 

 is said to differ numerically from another. The idiom has never found 

 its way into common language, though given in our old dictionaries. 

 Nor is this to be wondered at, for number in English is multitude ; 

 and two regiments may differ numerically, but two men cannot. 



The wrong use of the English adjective arises from the Greek word 

 apiS/j.0!, which is used for the result of counting, being supposed to 

 signify the multitude counted. This it did only in the sense in which 

 summa (sum) came to signify multitude in English : thus 10 is a sum, 

 of which the unitat summa is unitas decima. The difference is, that 

 the word sum never had any other sense in English : while in Greek it 

 never lost its first meaning, that of the item of enumeration. Thus 

 of apifyio! ToC vta/taTas meant the limits of the body : Herodotus 

 cannot tell what multitude (irA.ij0os) each nation furnished to the 

 enumeration (is apiS^iv), And Aristotle distinctly lays it down that 

 /uovtis and apiS^6! differ neither in quantity nor in quality : meaning 

 that the thing which is /lords when it stands alone becomes 

 api6/i6s when it stands in an enumeration as one of the items. Hi'iicc 

 the Greek logicians could properly say that though a horse and a 

 sheep differ fiSei (in species), one horse and another only differ 

 dpifyiw (as different items of one species). The word monadic would 

 be better than numerical, for logical use. See more details in the 

 ' Transactions of the Philological Society.' 



NUMISMATICS. The term numismatic, derived from the Greek 

 vofAtriia, is applied to the study of ancient and modern coins. By the 

 writers of the 15th and 16th centuries it was called the science of 

 medals. It divides itself into the following principal branches : 

 The study of coins considered in reference to the monetary systems 

 to which they belong, or the study of the historical, mythological, 

 and geographical allusions of inscriptions and types, and the discri- 

 mination of true ancient coins from modern imitations or forgeries. 

 So vast is the extent of the subject that a perfect knowledge of all 

 branches has never been attained by one individual. The value 

 of ancient coins to the study of geography, history, and philology is 

 considerable, for as they are monuments issued by public authority 

 it must be assumed that the information they convey is correct; 

 and they consequently offer many corrections to the orthography, in 

 some instances they rectify the chronology, and even add to history and 

 geography names which have disappeared from literature. Considered 

 in reference to the arts they are valuable as showing their state at 

 different fixed periods, while they present invaluable portraits of 

 princes and illustrious personages, valuable representations of temples, 

 statues, and localities, and important hints as to mythology and his- 

 torical events. The study of them is however generally subjective, 

 and they require elucidation from rather than give it to literature. In 

 some instances indeed they hand down the knowledge of obscure sites, 

 or give a clue to the history of dynasties which have not been recorded 

 j on the pages of the historian. As religious monuments they transmit a 

 history of the local deities, and representations of their shrines and 

 statues and the religious ideas of the age in which they were struck, 

 and they afford considerable insight into the autonomous or free, and 

 imperial administrations of states and cities, and their political changes, 

 monetary systems, and dialects. But their great charm is their being 

 monuments of ages long past, ancient pictorial illustrations of classical 

 authors, undeniable evidences of a past state, and links which con- 

 nect the familiar coins of the day with the first struggles of invention 

 and the development of monetary systems. At the same time they are 

 a most microscopical branch of archaeology. 



There can be little doubt that the Romans made collections of 

 ancient coins, from the price paid for remarkable forgeries and the 

 fact of their being inlaid and valued during the lower empire as gemg 

 by the laws. Petrarch, in 1374, was the first in the middle ages 

 to collect them, Alphouso, king of Arragon, and Cosmo di Medici 

 continued in the next century to do the same, and in the ICtli 



