A R I 
cm* pieces to a regular conftruCtion and unity ot table, and 
to a chaftenefs of manners and character,—to the abolition 
of the broad caricature and unnatural peripetia which dif- 
grace the modern ftage. 
ARISTOX'ENU-S, the mod ancient muHcal writer 6f 
whole works any traces are come down to us. He was 
born at Tarentmn, a city of Calabria. He was the ion of 
a nmfician, w horn fome call Mnrjias, others Spint'/nirus. He 
had his firft education at Mantinsea, a city of Arcadia, 
under his father, and Lampyrus of Erythrae; he nextflu- 
died under Xenophilus, the Pythagoraean; and laftfy un¬ 
der Arjftotle, in company with Theopliraftus. Suidas, 
from whom thefe particulars are transcribed,' adds, that 
Ariftoxenus, enraged at Ariftotle’s having bequeathed his 
fchool to Theophraflus, traduced him ever after. But 
Ariftocles the Peripatetic in Eufebius, exculpates Ariftox¬ 
enus in this particular. From the preceding account it 
appears that Ariftoxenus lived under Alexander the Great 
and his firft fucceflors. His Harmonics in three books, all 
that are come down to us, together with Ptolemy’s Har- 
monicSj were firft publilhed by Gogavinus, at Venice, 
1562, in 4to, with a Latin verfion. John Meurfis next 
tranfiated the three books of Ariftoxenus into Latin, from 
the MS. of Jof. Scaliger. With thefe he printed at Ley¬ 
den, 1616, 4to, Nichomaclms and Alypius, two other Greek 
writers on raulic. After this, Meibomius collected tliefe 
mufical writers together; to which he added Eluclid, Bac- 
chius fenior, Ariftides Quintilianus; and publilhed the 
whole, with a Latin verfion and notes from the elegant prefs 
of Elzevir, Amft. 1652. Ariftoxenus is laid by Suidas to 
have written 452 different works, among which thole on 
mulic are the moft efteemed. 
ARITH'MANCY,/. [from number, and pav- 
divination.] A foretelling future events by numbers. 
ARlTH'METIC,yi[a^/zo?,number,and u£T^u,tomea- 
Lire.] Theart and fcience of numbers; or,that part of mathe¬ 
matics which conliders their powers and properties, and 
teaches how to compute or calculate truly, and with eafeand 
expedition. It is by fome authors alfo denned the fcience of 
diferete quantity. Arithmetic confifts chiefly in the four 
principal rules or operations of Addition, Subtrattion, Mul¬ 
tiplication, and Divifion-, to which may perhaps be added 
Involution and Evolution, or railing of powers and extraction 
of roots. But befides thefe, for the facilitating and expe¬ 
diting of computations, mercantile, aftronomidal, &c. ma¬ 
ny other ufeful rules have been contrived, which are ap¬ 
plications of the former, fuch as, the rules of Proportion, 
Progr.eJJion, Alligation, Fdlowfnip, Inter eft, Batter, Equation 
of Payments, ReduElion, Tare 13 Tret, £zc. Belides the doc¬ 
trine of the curious and abftratt properties of numbers. 
Very little is known of the origin and invention of arith¬ 
metic. In faCt it mu ft have commenced with mankind, or 
as foon as they began to hold any fort of commerce toge¬ 
ther; and mull have undergone continual improvements, 
as occafion was given by the exienlion of commerce, and 
by the difeovery and-cultivation of other lciences. It is 
therefore very probable ,hat the art has been greatly in¬ 
debted to the Phoenicians or Tyrians; and indeed Proclus, 
in his Commentary on the firft book of Euclid, fays, that 
the Phoenicians, by reafen of their traffic and commerce, 
were accounted the firft inventors of arithmetic. From 
Afia the art palled into Egypt, whither it was carried by 
Abraham, according to the opinion of Jolephus. Here 
it was greatly cultivated and improved ; infomuch that a 
confiderable part of the Egyptian philofophy and theology 
feems to have turned altogether upon numbers. Hence 
thofe wonders related by them about unity, trinity, with 
•the numbers 4, 7, 9, &c- In effect, Kircher, in his Oedip. 
Algypt. Ihews, that the Egyptians explained every thing 
by numbers; Pythagoras himfelf affirming, that the na¬ 
ture of numbers .pervades the whole univerfe ; and that the 
knowledge of numbers is the knowledge of the deity. 
From Egypt arithmetic was tranfmitted to the Greeks, by 
means of Pythagoras and other travellers ; amongft whom 
was greatly cultivated and improved, as appears by the 
A R I i6j 
writings of Euclid-, Archimedes, and others • with thefe 
improvements it palfed to the Romans, and from them it 
has defeended to us. 
The nature of the arithmetic however that is now in 
life, is very different from that above alluded to ; this art 
having undergone a total alteration by the introduction of 
the Arabic notation, about 800 years fince, into Europe : 
fo that nothing now remains of life from the Greeks, but 
the theory and abftraCt properties of numbers, which have 
no dependence on the peculiar nature of any particular 
fcale or mode of notation. That ufed by the Hebrews, 
Greeks, and Romans, was chiefly by means of the letters 
of their alphabets. The twenty-four letters, taken ac¬ 
cording to their order, at firft denoted the numbers 1, 2, 
3 > 4 . 5 > 7 > 8, 9, 10, 20, 30, 40, 50, 60, 70, So, 100, 200, 
300, 400, 500, 600, 700, and 800; to Which they added 
the three following, r, S, "), to reprefent 6, 90, and 900. 
The Romans, befides the characters for each rank of claf- 
fes, introduced others for five, fifty, and five hundred. 
Their method is frill tiled for difringuilhing the chapters, 
dates, &c. of books, and for fome other purpofes. Their 
numeral letters and values are the following : 
I V X L C D M 
One, five, ten, fifty, one hundred, five hundred, one thoufand. 
Any number may be reprefented by repeating and com¬ 
bining thefe according to the following rules: 1. When 
the lame letter is repeated twice, or oftener, its value is 
reprefented as often; thus, II fignifies two, XXX thirty, 
CC two hundred. 2. When a numeral letter is placed af¬ 
ter one of greater value, their values are added; thus, 
XI fignifies eleven, LXV fixty-five, MDCCLXXXVIII 
one thoufand feven hundred and eighty-eight. 3. When 
a numeral letter is placed before one of greater value, the 
value of the lefs is taken from that of the greater; thus, 
IV fignifies four, XL forty, XC ninety, CD four hundred. 
Sometimes 1 3 is ufed inftead of D for 500, and the value 
is increafed ten times by annexing 3 to the right hand; 
Thus I3 fignifies 500. Alfo C13 is ufed for 1000 
133 5000 CCJ33 10000 
JdDO 50000 CCC1333 100000 
Sometimes thoufands are reprefented by drawing a line 
over the top of the numeral, V being ufed for five thou¬ 
fand, L for fifty thoufand, CC two hundred thoufand. 
Archimedes invented a peculiar fcale and notation of 
his own, which lie employed in his Arenarius, to compute 
the number of the fands. In the 2d century of Chriftia- 
nity lived Cl. Ptolemy, who, it is fuppofed, invented the 
fexagefimal divifion of numbers, with its peculiar notation 
and operations: a mode of computation ftill ufed in aftro- 
nomy, &c. for the fubdivifions of the degrees of circles. 
Thofe notations however were ill adapted to the practical 
operations of arithmetic: and hence it is that the art ad¬ 
vanced but very little in this part; for, letting afide Eu¬ 
clid, who has given many plain and ufeful properties of 
numbers in his Elements, and Archimedes, in his Arena¬ 
rius, they moftly confift in dry and tedious diftinctions and 
divifipns of numbers ; as appears from the treatifes of Ni- 
chomachus, fuppofed to be written in the 3d century of 
Rome, and publilhed at Paris in 153S; ns alfo that of 
Boethius, written at Rome in the 6th century of Chrift. 
A compendium of the ancient arithmetic, written in 
Greek, by Pfellus, in the 9th sentury, was publilhed in 
Latin by Xylander, in 1556. A fimilar work was written 
foon after in Greek by Jodocus Willichius; and a more 
ample work of the fame kind was written by Jordanus, in 
the year 1200, and publilhed with a comment by Faber 
Stapulenlis in 1430. 
Since the introduction of the Indian notation into Eu¬ 
rope, about the 10th century, arithmetic lias greatly chan¬ 
ged its form, the whole algorithm, or practical operations 
with numbers, being quite altered, as the notation requi¬ 
red ; and the authors of arithmetic have gradually become 
more and more numerous. This method was brought into 
Spain 
