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−𝑥3𝑥 . 1−𝑥 · 1−𝑥9𝑥3 . 1−𝑥3 . 1−𝑥27𝑥9 . 1−𝑥9 1−𝑥81𝑥27 . 1−𝑥27＝1−𝑥81𝑥40 . 1−𝑥 ;
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 −12(3𝑘−1) to +12(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 Combinatorial Analysis. (A. Ca.)
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 A.D. 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 πατήρ, ποιητής, ποίημα (“father,” “maker,” “that which is made,” i.e. the world), or πάππος, ἔκγονος, ἀπόγονος, (which Proclus calls “exaggerated language”). The first is the supreme deity or pure intelligence (νοῦς), the second the creator of the world (δημιουργός), the third the world (κόσμος). 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 Γ 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.[1]
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.
Table I.
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 A.D. 200. Ι stood for 1, Π for 5, Δ for 10, Η for 100, Χ for 1000, and Μ for 10,000; Π with Δ in its bosom was 50, with Η 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.
- ↑ The system is described by Nicolaus Rhabda of Smyrna (8th century A.D.), ap. N. Caussinus, De eloquentia sacra et humana (Paris, 1636). The Venerable Bede gives essentially the same system, and it long survived in the East; see especially Rödiger, “Über die im Orient gebräuchliche Fingersprache, &c.,” D.M.G. (1845), and Palmer in Journ. of Philology, ii. 247 sqq.
