Page:EB1911 - Volume 19.djvu/898

 that is, any number from 1 to 15 can be made up, and in one way only, with the parts 1, 2, 4, 8; and similarly any number from 1 to 2𝑘−1 can be made up, and in one way only, with the parts 1, 2, 4, .. 2𝑘−1. A like formula is

$1−𝑥^{3}⁄𝑥. 1−𝑥$ · $1−𝑥^{9}⁄𝑥^{3}. 1−𝑥^{3}$ . $1−𝑥^{27}⁄𝑥^{9}. 1−𝑥^{9}$&emsp;$1−𝑥^{81}⁄𝑥^{27}. 1−𝑥^{27}$＝$1−𝑥^{81}⁄𝑥^{40}. 1−𝑥$&thinsp;;

that is,

𝑥−1+ 1 +𝑥.𝑥−3+1+ 𝑥3.𝑥−9+1+𝑥9.𝑥−27+1+𝑥27 ＝𝑥−40+x−39.+1+𝑥 +𝑥39+𝑥40,

showing that any number from −40 to +40 can be made up, and that in one way only, with the parts 1, 3, 9, 27 taken positively or negatively; and so in general any number from −(3𝑘−1) to +(3𝑘−1) can be made up, and that in one way only, with the parts 1, 3, 9, . . 3𝑘−1 taken positively or negatively.

See further.

NUMENIUS, a Greek philosopher, of Apamea in Syria, Neo-Pythagorean and forerunner of the Neo-Platonists, flourished during the latter half of the 2nd century He seems to have taken Pythagoras as his highest authority, while at the same time he chiefly follows Plato. He calls the latter an “Atticizing Moses.” His chief divergence from Plato is the distinction between the “first god” and the “demiurge.” This is probably due to the influence of the Valentinian Gnostics and the Jewish-Alexandrian philosophers (especially Philo and his theory of the Logos). According to Proclus (Comment. in Timaeum, 93) Numenius held that there was a kind of trinity of gods, the members of which he designated as  (“father,” “maker,” “that which is made,” i.e. the world), or , (which Proclus calls “exaggerated language”). The first is the supreme deity or pure intelligence ( ), the second the creator of the world ( ), the third the world ( ). His works were highly esteemed by the Neoplatonists, and Amelius is said to have composed nearly 100 books of commentaries upon them.

Fragments of his treatises on the points of divergence between the Academicians and Plato, on the Good (in which according to Origen, Contra Celsum, iv. 51, he makes allusion to Jesus Christ), and on the mystical sayings in Plato, are preserved in the Praeparatio Evangelica of Eusebius. The fragments are collected in F. G. Mullach, ''Frag. phil. Graec. iii.; see also F. Thedinga, De Numenio'' philosopho Platonico (Bonn, 1875); Ritter and Preller, ''Hist. Phil.'' Graecae (ed. E. Wellmann, 1898), § 624-7; T. Whittaker, The Neo-Platonists (1901).

NUMERAL (from Lat. numerus, a number), a figure used to represent a number. The use of visible signs to represent numbers and aid reckoning is not only older than writing, but older than the development of numerical language on the denary system; we count by tens because our ancestors counted on their fingers and named numbers accordingly. So used, the fingers are really numerals, that is, visible numerical signs; and in antiquity the practice of counting by these natural signs prevailed in all classes of society. In the later times of antiquity the finger symbols were developed into a system capable of expressing all numbers below 10,000. The left hand was held up flat with the fingers together. The units from 1 to 9 were expressed by various positions of the third, fourth, and fifth fingers alone, one or more of these being either closed on the palm or simply bent at the middle joint, according to the number meant. The thumb and index were thus left free to express the tens by a variety of relative positions, e.g. for 30 their points were brought together and stretched forward; for 50 the thumb was bent like the Greek and brought against the ball of the index. The same set of signs if executed with the thumb and index of the right hand meant hundreds instead of tens, and the unit signs if performed on the right hand meant thousands.

The fingers serve to express numbers, but to make a permanent note of numbers some kind of mark or tally is needed. A single stroke is the obvious representation of unity; higher numbers are indicated by groups of strokes. But when the strokes become many they are confusing, and so a new sign

must be introduced, perhaps for 5, at any rate for 10, 100, 1000, and so forth. Intermediate numbers are expressed by the addition of symbols, as in the Roman system ccxxxvi＝236. This simplest way of writing numbers is well seen in the Babylonian inscriptions, where all numbers from 1 to 99 are got by repetition of the vertical arrowhead ＝1, and a barbed sign ＝10. But the most interesting case is the Egyptian, because from its hieratic form sprang the Phoenician numerals, and from them in turn those of Palmyra and the Syrians, as illustrated in table 1. Two things are to be noted in this table—first, the way in which groups of units come to be joined by a cross line, and then run together into a single symbol, and, further, the substitution in the hundreds of a principle of multiplication for the mere addition of symbols. The same thing appears in Babylonia, where a smaller number put to the right of the sign for 100 is to be added to it, but put to the left gives the number of hundreds. Thus ＝1000, but ＝110. The Egyptians had hieroglyphics for a thousand, a myriad, 100,000 (a frog), a million (a man with arms stretched out in admiration), and even for ten millions. EB1911 - Numerals - Table I.jpg

Alphabetic writing did not do away with the use of numerical symbols, which were more perspicuous, and compendious than words written at length. But the letters of the alphabet themselves came to be used as numerals. One way of doing this was to use the initial letter of the name of a number as its sign. This was the old Greek notation, said to go back to the time of Solon, and usually named after the grammarian Herodian, who described it about 200. stood for 1, for 5,  for 10,  for 100, for 1000, and for 10,000;  with  in its bosom was 50, with in its bosom it was 500. Another way common to the Greeks, Hebrews, and Syrians, and which in Greece gradually displaced the Herodian numbers, was to make the first nine letters stand for the units and the rest for the tens and hundreds. 