Numĕri
（
ἄριθμοι). Numbers; numerals. The use of signs to denote
numbers is older than writing; yet most of the existing numerical signs in
Greek and Latin are alphabetic modifications; because very primitive peoples, being able to
count no higher than ten or so, need few symbols of number, so that the characters for large
numbers are of late origin. The earliest visible signs were probably the extended fingers. The
early system was in fact one based upon five, the number of fingers on one hand, traces of
which survive in the Greek words
πεμπάζειν, πεμπαστής from
πέντε, “five” (cf.
Odyss.
iv. 412-415); and our denary system is due to the fact that the total number of our fingers is
ten (cf. our English “-teen” as a termination). Finger-counting was very
highly developed by the ancients, and many fairly complicated arithmetical operations could be
denoted by finger-signs, as is still done in the Oriental bazaars, where the venders can
reckon on their ten digits sums involving five places of figures. This system is fully
described by Nicolaüs Smyrnaeus, a Greek of the thirteenth century A.D., in a
treatise entitled
Ἔκφασις τοῦ Δακτυλικοῦ Μέτρου, which
was printed at Paris in 1636. Units and tens were represented by the fingers of the left hand,
and hundreds or thousands by the fingers of the right. The thumb and forefinger of the left
hand were devoted to tens, those of the right to hundreds; the remaining fingers of the left
hand belonged to the units, those of the right to thousands. The fingers might be straight
(
ἐκτεινόμενοι), bent (
συστελλόμενοι), or closed (
κλινόμενοι). In the
left hand, bending the fourth finger marked 1; bending the third and fourth, 2; the middle,
third, and fourth, 3; the middle and third only, 4; the middle only, 5; the third only, 6.
Closing the fourth finger gave 7; the fourth and third fingers, 8; the middle, third, and
fourth, 9. The same motions on the right hand indicated thousands, from 1000 to 9000. The
motions of the forefinger and thumb, in representing tens and hundreds, on the left and right
hands respectively, are more difficult to describe. Various combinations were also indicated
by placing the hands upon the breast, the hips, etc. This system was taught in the Greek and
Roman schools. (See Plut.
Apophth. 1746; Dio Cass. lxxi. 32;
Anth.
Pal. xi. 72; and the works cited below.) Reckoning was also performed by pebbles or
counters arranged in sets of ten—a system which was developed into the
calculating-instrument known as the
abacus, and still used by the
Chinese, who call it
swan-pan. See
Abacus.
For recording numbers, a system of single strokes was first used as the most obvious; but
this, of course, would be too cumbrous when applied to large numbers. Hence, additional
symbols came into use for 5, 10, 100, and 1000; and after alphabetical writing was invented
these signs were employed as numerals, either following the order of the letters, or taking
the initial letter of the word for its symbol. Thus, in Greek, the inscriptions give I for
“one,”
Π (
πέντε) for “five,”
Δ
(
δέκα) for “ten,” H (old sign for rough
breathing,
ἕκατον) for “one hundred,”
Χ (
χίλιοι) for
“one thousand,” and
Μ (
μύριοι) for “ten thousand.” (See the articles on
each letter of the alphabet in this Dictionary.) Then, a
Π
with a
Δ inscribed in it stood for 5X10=50, or with H
inscribed in it for 500, etc. The twenty-five letters of the Ionic alphabet were used also for
the simple numbers, 1 to 24. In the third century B.C. a new system called the Herodian
or Alexandrian was introduced, by which the cursive alphabet was divided into three groups, of
which the first did duty for the units, the second for the tens, and the third for the
hundreds. This required 27 instead of 24 letters; so the old characters
digamma (q. v.),
koppa (q. v.), and the old sibilant
sampi () were revived, the first representing 6, the second 90,
and the third 900. Intermediate numbers like 11 were represented by the sum of 10 and 1, etc.,
as
ιά. This gave a notation for all numbers up to 999, and by
a system of suffixes and indices the system was extended to represent numbers as high as
100,000,000. Until a comparatively late period these signs were used only to record results
got by the use of the abacus; but at last they were employed in actual operations like our
own. Long lists of multiples were learned by heart, and various operations of multiplication
and division were obtained by repeated addition (
σύνθεσις）
and subtraction (
ἀφαίρεσις). As late as the year 944 we find
a scholar multiplying 400 by 5 by means of addition. A few skilled mathematicians like Hero of
Alexandria and Theon multiplied as we do. Thus, to multiply 18 by 13 the operation was as
follows:
In the Alexandrian system thousands could be made by subscribing an
ι beneath the units; thus,
α=1000;
αωθα=1891. A sort of algebraic method was also used for very large
numbers—e. g.
βΜ=(2X10,000)=20,000.
The Roman system is, in its ordinary use, familiar to all readers. It is thought that
denotes the opening between the thumb and the forefinger; is two
with the angles together; is possibly for
centum, but probably
an original assimilated to may be for
mille, but
probably for modified; is from an old Chalcidian form of ,
inscribed for lapidary purposes as , and then simplified. Others believe to
be from a circle with a vertical stroke, and the from a circle with a cross
, from which last , and would also be derived. The Romans, like
the Greeks, used the system of finger-signs, and did arithmetical operations by aid of the
abacus. (See
Abacus.) Their
arrangement of the latter was more complete than the Greek, and allowed very elaborate
calculations; and, in fact, the Romans were, in general, better arithmeticians than the
Greeks. There is a book by one Victorius of the fifth century B.C. entitled
Calculus, which is a sort of “ready-reckoner” of sums,
differences, products, quotients, reductions, etc.
Fractions (
λεπτά) are variously represented in MS., but the
most common way is to write the denominator over the numerator (the reverse of our method), or
to write the numerator once with one accent and the denominator twice with two accents. Thus,
αα, or
ιζ᾽
κα" κα". The Romans treated fractions as did the Greeks, and attempted quite
difficult operations, which were often very inexactly performed (Pliny ,
Pliny H. N. vi. 38).
On ancient numerals and arithmetic, see Delambre,
Die Arithmetik der
Griechen, rev. by Hoffmann
(Mainz, 1817); Benloew,
Sur
l'Origine des Noms de Nombre (Giessen, 1861); Hoefer,
Histoire des
Mathématiques, 3d ed.
(Paris, 1886);
Martin, Les
Signes Numéraux, etc. (Rome, 1864); Friedlein,
Die Zahlzeichen, etc.
(Erlangen, 1869); Taylor,
The
Alphabet, ii. pp. 263-268
(London, 1883); and
Treutlein,
Geschichte unserer Zahlzeichen (1875).
As to the history of mathematics among the Greeks and Romans, it may be said that the
earliest Greek school of mathematics was that founded by Thales of Miletus (B.C. 640-550), who
studied astronomy and geometry in Egypt, and, after returning to Miletus, taught them to his
disciples. (See
Thales.) His geometrical teaching was
largely deductive, and the following theorems of Euclid are ascribed to him: i. 5; i. 15; i.
26; vi. 4 (vi. 2?); iii. 31. He also wrote a treatise on astronomy. His philosophical
follower, Anaximander, wrote a treatise on spherical geometry and constructional globes. This
school, known as the Ionian School, flourished till about B.C. 400. It really gave more
attention to astronomy than to geometry, which as an actual part of a liberal education, dates
from Pythagoras (B.C. 569-500), whose philosophy and even whose ethics rested on a
mathematical basis. He first arranged the leading problems of geometry in logical order, and
carried arithmetic beyond the mere needs of the trader. (See Hoffmann,
Der
pythagorische Lehrsatz [Mainz, 1821].) Archytas, a follower of Pythagoras (about B.C.
400) and head of the school, applied mathematics to mechanics, and also worked in astronomy,
teaching that the earth is a sphere revolving on its axis once in twenty-four hours. He
attacked one of the most famous problems in antiquity—to find the side of a cube
whose volume should be double that of a given cube. Two other well-known mathematical schools
were the Eleatic, whose great name was Zeno's (B.C. 495-435), and the Atomistic School of
Democritus of Abdera (B.C. 460-370). In the fourth century Athens became a great centre for
mathematical study, and the scholars Anaxagoras (B.C. 500-428), Hippocrates of Chios, Eudoxus,
Plato, and Theaetetus are among its greatest names. Hippocrates wrote the first text-book on
geometry, and attempted the quadrature of the circle. Eudoxus founded the School of Cyzicus,
to which Menaechmus and Aristaeus also gave distinction. Aristotle (B.C. 384-322) did much to
stimulate the study of mathematics, and especially of mechanics. The establishment of a great
university in Alexandria (see
Alexandrian
School) made that city an intellectual centre; and there three of the greatest
mathematicians of antiquity flourished—Euclid (B.C. 330?-275), Archimedes (B.C.
287-212), and Apollonius (B.C. 260-200). (See
Apollonius;
Archimedes;
Euclides.) After these came Hipparchus, the most
eminent of Greek astronomers (B.C. 160), whose work is preserved in Ptolemy's great treatise
known as the
Almagest. He determined the true length of the year, and placed
the study of astronomy on a truly scientific basis. Hero of Alexandria (about B.C. 125) did
the same for land-surveying and engineering. The Roman occupation of Egypt seriously
interrupted the studies of the Alexandrian School; and no mathematicians of equal eminence
with those already mentioned are afterwards found. The most original works subsequently
published are the treatise by Serenus (A.D. 70) on the plane sections of the cone and
cylinder, and that by Menelaüs on spherical trigonometry. About A.D. 100 the Jewish
scholar Nicomachus wrote an arithmetic which, in a Latin version, remained the standard
treatise on the subject for a thousand years. Ptolemy of Alexandria, who died in the year A.D.
168, was the author of a great work on astronomy; Pappus, in the third century, published a
useful synopsis of Greek mathematics. In the fourth century geometrical studies decline, and
algebra begins to be pursued, though possibly not unknown. It was probably at first what is
called “rhetorical algebra”—i. e. the problems were solved by a
process of reasoning expressed in words rather than in symbols. Diophantus of Alexandria
(probably a non-Greek) introduced a system of signs and abbreviations. He lived in the fourth
century A.D. (See Heath,
Diophantos of Alexandria [Cambridge, 1885].) His work
is called
Arithmetica, but is really an algebra. The last of the Alexandrian
mathematicians are the famous
Hypatia (q.v.) and
Theon , her father. In the fifth century there were some Athenian geometricians of repute,
such as Proclus, Damascius, and Eutocius; and in the sixth century the Roman Boëtius
forms the link between the mathematical studies of antiquity and those of the Middle Ages. He
wrote a geometry which contained the problems of the first book of Euclid and a few other
selected propositions, and an arithmetic founded on that of Nicomachus. Cassiodorus (A.D.
470-566), and Isidorus of Seville (A.D. 570-636), also wrote in an elementary way of the
various mathematical sciences.
See Hankel,
Zur Geschichte der Mathematik (Leipzig, 1874);
Hoefer,
Histoire des Mathématiques, 3d ed.
(Paris,
1886); Gow,
A Short History of Greek Mathematics (Cambridge,
1884); and Ball,
A Short History of Mathematics (London and New
York, 1888).
