(ἄριθμοι). Numbers; numerals. The use of signs to denote numbers is older than writing; [p. 1110] 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 Nicolas 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 tena 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 ([figure in text]) 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:
ιγXιη=(ι+γ) (ι+η) ||13X18=(10+3) (10+8) |
|=ι(ι+η)+γ(ι+η) ||=10(10+8)+3(10+8) |
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 numberse. g. βΜ=(2X10,000)=20,000.
The Roman system is, in its ordinary use, familiar to all readers. It is thought that [figure in text] denotes the opening between the thumb and the forefinger; [figure in text] is two [figure in text] with the angles together; [figure in text] is possibly for centum, but probably an original [figure in text] assimilated to [figure in text] may be for mille, but probably for [figure in text] modified; [figure in text] is from an old Chalcidian form of [figure in text], inscribed for lapidary purposes as [figure in text], and then simplified. Others believe [figure in text] to be from a circle with a vertical stroke, and the [figure in text] from a circle with a cross [figure in text], from which last [figure in text], and [figure in text] 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 Mathmatiques, 3d ed. (Paris, 1886); Martin, Les Signes Numraux, etc. (Rome, 1864); [p. 1111] 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 antiquityto 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 flourishedEuclid (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 Menelas 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 algebrai. 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 Botius 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 Mathmatiques, 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).