292 
N U M 
for difcovering the affinities of nations, and arranging to¬ 
gether the tribes of limilar defcent; and, accordingly, 
lifts of numerals clofely refemble one another in languages 
which are fundamentally diffimilar. In a curious volume 
called the Remains of Japhet, a work remarkable for pro- 
fufion of erudition and for indiftintlnefs of fyftem, a lift of 
the European numerals is given, and compared with thole 
of the ancients. We ftiall tranfcribe a part of it: 
Irijh. 
Weljh. 
French. 
Latin. 
Greeh. 
Sanfcrit. 
I. 
Aon 
Un 
Un 
Unus 
Heis 
Ek 
2 * 
Do 
Duy 
Deux 
Duo 
Duo 
Dwa 
3 ‘ 
Tri 
Tri 
Trois 
Tres 
Treis 
T raya 
4 - 
Keathair Pedwar 
Quatre 
Quatuor 
Teftares Khatur 
5 - 
Kuig 
Pymp 
Cinq 
Quinque 
Pente 
Pancha 
6 . 
She 
Chuech 
Six 
Sex 
Hex 
Shat 
7- 
Sheaghd 
Saith 
Sept 
Septem 
Hepta 
Sapta 
8. 
Ocht 
Uitli 
Huit 
O6I0 
Okto 
Afhta 
9 - 
Nyi 
Naw 
Neuf 
Novem 
Ennea 
Nova 
10. 
Deic 
Deg 
Dix 
Decetn 
Deka 
Dafa 
All thefe nations have evidently gone to the fame cipher- 
ing-fchool. The French, Irifti, and Wellh, learned their 
numbers of the Romans; the Romans, of the Greeks; 
and the Greeks learned theirs at Nineveh, which was 
probably the metropolis of the Sanfcrit .tongue. The 
languages themfelves, however, are not fo limilar as their 
numerals, and confift in unequal proportions of the pri¬ 
maeval Afiatic idioms. The Greek and Sanfcrit have a 
common bafis, which does not comprehend the Celtic. 
The Latin and French, on the contrary, include a Celtic 
bafis; and the Irilli and Wellh have a Cimbric bafis, not 
contained in the other four languages. The Gothic na¬ 
tions appear to have invented their own numbers ; fince 
their numeric terms are all fignificative. An, or one, de-. 
lcribes the virile limb. Duagan, or two, the eyes. Tre, 
or three, a tree ; the fork of the branches and the ftem 
fuggefting a tripartite idea; Fier, is contra&ed from fingers, 
and therefore fignifies four; Fcm, or five, is the hand ; 
Teen, or ten, means the toes ; but the words fix, feven, 
eight, and nine, have been borrowed from the Romans, 
inftead of the round-about expreffions previoully adopted, 
fuch as twice-three, or twice-four. Of all the primary 
numbers, feven appears to have been the moft difficult to 
invent; and hence it is almoft every-where a borrowed 
word, and is common to the Hebrew dialed! as well as to 
thofe already fpecified. In the Koriac, an Afiatic, in the 
Jalloft'e, an African, and in the Mexican, an American, 
language, five-and- one, five-and-two, five-and-three, five- 
and-four, Hand for the fecond half of the digits. 
In order to afcertain whether a given tribe invented, or 
borrowed, its arithmetical table, it is neceflary to com¬ 
pile a catalogue of thofe words which have commonly 
been made the bafis of numerical metaphor. Thus, if in 
the Madagafcar numbers the word leemo reprefents five, it 
muft be next inquired whether this fame word fignifies 
hand or foot in Madagaffian ; but tan gu e is their word for 
hands, and tamboo for feet; hence it may be fufpedted 
that the Madagaffians did not invent their own numbers. 
On examining farther, it appears that all the Malay na¬ 
tions count with w'ords clofely refembling the Madagaffian 
numbers ; and it may therefore be inferred that the Ma¬ 
dagaffians, among whom thele numbers are not autoch¬ 
thonous, have learned to reckon of the Malays. The 
Malay numbers prove a commercial interconrfe with the 
Malays; but they are ufed by many nations of wholly 
diftinft origin and defcent. 
The theory of numbers is a modern and very intereftirig 
branch of analyfis, which is direfled towards the invefti- 
gation of the feveral properties, forms, divifors, pro- 
duffs, &c. of integral numbers. This fubjeft was in¬ 
deed confidered by home of the ancient mathematicians, 
as by Ariftotle and Pythagoras, and particularly by Euclid 
and Diophantus; but, in confequence of the embarraffing 
notation of thofe early tjmes, and the total want of the 
algebraical analyfis, but little progrefs was made in this 
B E R. 
branch of fcience, till about the beginning of the feven- 
teenth century, when Bachet, a French analyft of confi- 
derable reputation, undertook the tranflation of Dio¬ 
phantus into Latin, retaining alfo the Greek text, which 
was publifhed by him in 1621, interfperfed with many 
marginal notes of his own, and which may be confidered 
as containing the firft germ of our prefent theory. Thefe 
were afterwards confiderably extended by the celebrated 
Fermat, in his edition of the fame work, publifhed after 
his death in 1670, where we fincl many of the moft elegant 
theorems in this branch of analyfis; but they are gene¬ 
rally left without demonftratjon, an omiffion which the 
author accounted for, by Hating that he was himfelf pre¬ 
paring a treatifeon the theory of numbers, which would 
contain “ multa varia et abftrufiffima numerorum myf- 
teria;” but, unfortunately, this work never appeared, and 
moft of his theorems remained without demonftration for 
a confiderable time. At length the fubje6I was again re¬ 
vived by Euler, Waring, and La Grange, three of the moft 
eminent analyfis of modern times. The former, befides 
what is contained in his Elements of Algebra, and his 
Analyfis Infinitorum, has feveral papers in the Peterfourgh 
Abls, in which are given the demonftrations of many of 
Fermat’s theorems. What has been done by Waring on 
this fubjedf, is contained in chap. v. of his Meditationes 
Algebraical. And La Grange, who has greatly extended 
the theory of numbers, has feveral interefting papers on this 
head in the Memoirs of Berlin, befides what is contained 
in his Additions to Euler’s Algebra. It is, however, but 
lately that this branch of analyfis has been reduced into 
a regular fyftem ; a talk that was firft performed by Le¬ 
gendre, in his Effai fur la Theorie des Nombres, Paris, 
1800; a fecond edition of which was publifhed in 1807, 
and a third in 1815 ; and nearly at the fame time that the 
firft edition appeared, Gaufs publifhed his Difquifitiones 
Arithmeticae. Thefe two works eminently difplay the 
talents of their refpeftive authors, and contain a com¬ 
plete developement of this interefting theory. The latter, 
in particular, has opened a new field of enquiry, by the 
application of the properties of numbers to the folution 
of binomial equations of the form x n —1=0; on the fo¬ 
lution of which depends the divifion of the circle into n 
equal parts, as was before known from the Cotefian theo¬ 
rem. Mr. Barlow, of the Royal Military Academy, has 
alfo publifhed a concife treatife on this fubjeft, entitled 
“ An Elementary Inveftigation of the Theory of Num¬ 
bers ;” and more recently (18.17), we have Mr. Profeffor 
Leflie’s “ Philofophy of Arithmetic,” the greater part of 
which,had, however, previoufiy appeared in the Supple¬ 
ment to the Encyclopaedia Britannica. To thefe works 
we flaali be greatly indebted in the progrefs of the prefent 
article, which we wifli to be confidered as a fupplement 
to that of Arithmetic, in our fecond volume. 
Of the Characters ufed by different Nations to exprefs 
Numbers. 
It is not agreed how the Roman numerals originally- 
received their value. It has been fuppofed that the Ro¬ 
mans ufed M to denote 1000, becaufe it is the firft letter, 
of inille, which is Latin for 1000; and C to denote 100, 
becaufe it is the firft letter of centum, which is Latin for 
100. It has alfo been fuppofed, that D, being formed by 
dividing the old M in the middle, was therefore appointed 
to Hand for 500, that is, half as much as the M itood for 
when it was whole ; and that L, being half a C, was, for 
the fame reafbn, ufed to denominate 50. But what rea- 
fon is there to fuppofe, that 1000 and 100 were the num¬ 
bers which letters were firft ufed to exprefs ? And what 
reafon can be affigned why D, the firft letter in the Latin 
word deceit 1, ten, Ihould not rather have been chofen to 
ftand for that number, than for 500, becaufe it had a rude 
refemblance to half an M ? But, if thefe queftions could 
be latisfaftorily anfwered, there are other numeral letters 
which have never yet been accounted for at all. Thele 
confiderations render it probable that the Romans did 
2 not. 
