Encyclopaedia Britannica > Volume 3, Anatomy-Astronomy

572 A E I A E I
Aristoxe- ARISTOXENUS, a Peripatetic philosopher, and native
nus of Tarentum. He studied under Lamprus, Xenophilus,
Arithmetic ant^ Aristotle 5 anc4 according to Suidas, was the author of
\ ^ ^ , 453 works on music, philosophy, and history. The only
one of these that is now known is his Elements of Harmonics,
in three books, of which Meibomius published an edition with
Latin translation and notes in his Antiquce Musicce Auctores,
Amsterdam, 1652. This is considered the best edition of
Aristoxenus’s work, although it is imperfect; for part of each
book is wanting, and great confusion occurs from transposi¬
tion of passages. The Abate J. Morelli, librarian of St Mark’s
library at Venice, published fragments of Aristoxenus’s Ele- Aristoxe-
ments of Rhythm from a MS. in that library, Venice, 1785. nus
By the term ap/xovla, the ancient Greeks did not mean har- II
mony in the modern sense of that word as applied to music, Arithmetic
but the system of sounds upon which melody for voice or
instrument was founded. It is remarkable that Aristoxenus,
after asserting that the ear is the sole judge of harmonic
[melodic] intervals, should have plunged into a system of
false calculations of these intervals. He, like the other Greek
writers on music, has given us no information regarding the
practical parts of the art, either in composition or performance.
ARITHMETIC
Is a science which explains the properties of numbers, and
shows the method or art of computing by them.
I.—HISTORY OF ARITHMETIC.
In the early periods of society a vast mound of earth, or
a huge block of stone, was the only memorial of any great
event; but after the simpler arts came to be known, efforts
were made to transmit to posterity the representations of
the objects themselves. Sculptures of the humblest kind
occur on monumental stones in all parts of the world, suffi¬
cient to convey tolerably distinct images of the usual occu¬
pation and employments of the personages so commemo¬
rated. The next step in the progress of society was to re¬
duce and abridge those rude sculptures, and thence form
combinations of figures approaching to the hieroglyphical
characters. At this epoch of improvement the first attempts
to represent numerals would be made. Instead of repeat¬
ing the same objects, it was an obvious contrivance to
annex to the mere individual the simpler marks of such re¬
petition. Those marks would of necessity be suited to the
nature of the materials on which they were inscribed, and
the quality of the instruments employed to trace them. In
the historical representations, for instance, which the Mexi¬
cans and certain Tartar hordes painted on skins, a small
coloured circle, as exhibiting the original counter, shell, or
pebble, was repeated to denote numbers. But on the Egyp¬
tian obelisks the lower numbers, at least, are expressed by
combined strokes. Curve lines were not admitted in the
earliest rudiments of writing. Even after the use of hiero¬
glyphics had been laid aside, and the artificial system of al¬
phabetic characters adopted, the rectilineal forms were still
preferred, as evidently appears in the Greek and Roman
capitals, which, being originally of the lapidary sort, are
much older than the small or current letters. One of the
most ancient sets of characters, the Runic, in which the
northern languages were engraved, combines almost exclu¬
sively simple strokes at different angles.
The primary numeral traces may, therefore, be regarded
as the commencement of a philosophical and universal cha¬
racter, drawn from nature itself, and alike intelligible to all
ages and nations. They are still preserved, with very little
change, in the Roman notation. Those forms, prior to the
adoption of the alphabet itself, were no doubt imported by
the Grecian colonies that settled in Italy, and gave rise to
the Latin name and commonwealth. Assuming a perpen¬
dicular stroke | to signify one, another such g | would ex¬
press tivo, the junction of a third g | | three, and so repeat¬
edly till the reckoner had reached ten. (See Plate LXXII.)
The first class was now completed, and to intimate this the
carver threw a dash across the stroke or common unit; that is,
he employed two decussating strokes X t° denote ten. He
next repeated this mark to express tiventy, thirty, and so
forth, till he finished the second class of numbers. Arrived
at an hundred, he would signify it by joining another dash
to the mark for ten, or by merely connecting three strokes
thus [L Again, the same spirit of invention might lead
him to repeat this character in denoting two hundred, three
hundred, and so forth, till the third class was completed.
A thousand, which begins the fourth class on the Denary
Scale, was therefore expressed by four combined strokes
M, and this was the utmost length to which the Romans
first proceeded by direct notation.
But the division of these marks afterwards furnished
characters for the intermediate numbers, and hence greatly
shortened the repetition of the lower ones. Thus, having
parted in the middle the two decussating strokes X denot¬
ing ten, either the under half fc, or the upper half V?
was employed to signify five. Next, the mark [L for an
hundred, consisting of a triple stroke, was largely divided
into [” and either of which represented fifty. Again,
the four combined strokes M, which originally formed
the character for a thousand, came afterwards, in the pro¬
gress of the arts, to assume a rounded shape fre¬
quently expressed thus QiO> by two disparted semicircles
divided by a diameter. This last form, by abbreviation on
either side, gave two portions cl and lo to represent five
hundred.
It was an easy process, therefore, to devise a universal Origin of
character for expressing numbers. But the task was very Roman nu-
different to reduce the exhibition of language in general mei'als-
to such concise philosophical principles. This attempt
seems accordingly to have been early abandoned by all
nations except the Chinese. The inestimable advantage
of uniting again the whole human race, in spite of the
diversity of tongues, by the same permanent system of
communication, was sacrificed for the easier attainment of
representing by artificial signs those elementary and fu¬
gitive sounds into which the words of each particular dia¬
lect could be resolved. Hence the Alphabet was in¬
vented, which had very nearly attained its present form at
the period when the Roman commonwealth was extending
its usurpation over Italy. About that epoch a sort of re¬
action seems to have arisen between the artificial and the
natural systems ; and the numeral strokes were finally dis¬
placed by such alphabetic characters as then most resem¬
bled them. (See Plate LXXII.) The ancient Romans
employed the letter I to represent the single stroke or
mark for one; they selected the letter V, since it resem¬
bles the upper half of the two decussating strokes or
symbol, for five ; the letter X exactly depicted the double
mark for ten ; again, the letter L was adopted as resem-