Page:EB1911 - Volume 19.djvu/879

 (1905–1910), who has traced the Celtiberian town, the lines of Scipio and several other Roman camps dating from the Numantine Wars.

NUMA POMPILIUS, second legendary king of Rome (715–672 ), was a Sabine, a native of Cures, and his wife was the daughter of Titus Tatius, the Sabine colleague of Romulus. He was elected by the Roman people at the close of a year's interregnum, during which the sovereignty had been exercised by the members of the senate in rotation. Nearly all the early religious institutions of Rome were attributed to him. He set up the worship of Terminus (the god of landmarks), appointed the festival of Fides (Faith), built the temple of Janus, reorganized the calendar and fixed days of business and holiday. He instituted the flamens (sacred priests) of Jupiter, Mars and Quirinus; the virgins of Vesta, to keep the sacred fire burning on the hearth of the city; the Salii, to guard the shield that fell from heaven; the pontifices and augurs, to arrange the rites and interpret the will of the gods; he also divided the handicraftsmen into nine gilds. He derived his inspiration from his wife, the nymph Egeria, whom he used to meet by night in her sacred grove. After a long and peaceful reign, during which the gates of Janus were closed, Numa died and was succeeded by the warlike Tullus Hostilius. Livy (xl. 29) tells a curious story of two stone chests, bearing inscriptions in Greek and Latin, which were found at the foot of the Janiculum (181 ), one purporting to contain the body of Numa and the other his books. The first when opened was found to be empty, but the second contained fourteen books relating to philosophy and pontifical law, which were publicly burned as tending to undermine the established religion.

No single legislator can really be considered responsible for all the institutions ascribed to Numa; they are essentially Italian, and older than Rome itself. Even Roman tradition itself wavers; e.g. the fetiales are variously attributed to Tullus Hostilius and Ancus Marcius. The supposed law-books, which were to all appearance new when discovered, were clearly forgeries.

NUMBER (through Fr. nombre, from Lat. numerus; from a root seen in Gr.  to distribute), a word generally expressive of quantity, the fundamental meaning of which leads on analysis to some of the most difficult problems of higher mathematics.

1. The most elementary process of thought involves a distinction within an identity—the A and the not-A within the sphere throughout which these terms are intelligible. Again A may be a generic quality found in different modes Aa, Ab, Ac, &c.; for instance, colour in the modes, red, green, blue and so on. Thus the notions of “one,” “two,” and the vague “many” are fundamental, and must have impressed themselves on the human mind at a very early, period: evidence of this is found in the grammatical distinction of singular, dual and plural which occurs in ancient languages of widely different races. A more definite idea of number seems to have been gradually acquired by realizing the equivalence, as regards plurality, of different concrete groups, such as the fingers of the right hand and those of the left. This led to the invention of a set of names which in the first instance did not suggest a numerical system, but denoted certain recognized forms of plurality, just as blue, red, green, &c., denote recognized forms of colour. Eventually the conception of the series of natural numbers became sufficiently clear to lead to a systematic terminology, and the science of arithmetic was thus rendered possible. But it is only in quite recent times that the notion of number has been submitted to a searching critical analysis: it is, in fact, one of the most characteristic results of modern mathematical research that the term number has been made at once more precise and more extensive.

2. Aggregates (also called manifolds or sets).—Let us assume the possibility of constructing or contemplating a permanent system of things such that (1) the system includes all objects to which a certain definite quality belongs; (2) no object without this quality belongs to the system; (3) each object of the system is permanently recognizable as the same thing, and as distinct from all other objects of the system. Such a collection is called an aggregate: the separate objects belonging to it are called its elements. An aggregate may consist of a single element.

It is further assumed that we can select, by a definite process, one or more elements of any aggregate $$\mbox{A}$$ at pleasure: these form another aggregate $$\mbox{B}$$. If any element of $$\mbox{A}$$ remains unselected, $$\mbox{B}$$ is said to be a part of $$\mbox{A}$$ (in symbols, $$\mbox{B}<\mbox{A}$$): if not, $$\mbox{B}$$ is identical with $$\mbox{A}$$. Every element of $$\mbox{A}$$ is a part of $$\mbox{A}$$. If $$\mbox{B}<\mbox{A}$$ and $$\mbox{C}<\mbox{B}$$, then $$\mbox{C}<\mbox{A}$$.

When a correspondence can be established between two aggregates $$\mbox{A}$$ and $$\mbox{B}$$ in such a way that to every element of $$\mbox{A}$$ corresponds one and only one element of $$\mbox{B}$$, and conversely, $$\mbox{A}$$ and $$\mbox{B}$$ are said to be equivalent, or to have the same power (or potency); in symbols, $$\mbox{A} \sim \mbox{B}$$. If $$\mbox{A} \sim \mbox{B}$$ and $$\mbox{B} \sim \mbox{C}$$, then $$\mbox{A} \sim \mbox{C}$$. It is possible for an aggregate to be equivalent to a part of itself: the aggregate is then said to be infinite. As an example, the aggregates $$2, 4, 6, ... 2n$$, &c., and $$1, 2, 3, ... n$$, &c., are equivalent, but the first is only a part of the second.

3. Order.—Suppose that when any two elements $$a, b$$ of an aggregate $$\mbox{A}$$ are taken there can be established, by a definite criterion, one or other of two alternative relations, symbolized by $$ab$$, subject to the following conditions:-(1) If $$a>b$$, then $$ba$$; (2) If $$a>b$$ and $$b>c$$, then $$a>c$$. In this case the criterion is said to arrange the aggregate in order. An aggregate which can be arranged in order may be called ordinable. An ordinable aggregate may, in general, by the application of different criteria, be arranged in order in a variety of ways. According as $$ab$$ we shall speak of a as anterior or posterior to $$b$$. These terms are chosen merely for convenience, and must not be taken to imply any meaning except what is involved in the definitions of the signs $$>$$ and $$<$$ for the particular criterion in question. The consideration of a succession of events in time will help to show that the assumptions made are not self-contradictory. An aggregate arranged in order by a definite criterion will be called an ordered aggregate. Let $$a, b$$ be any two elements of an ordered aggregate, and suppose $$a<b$$. All the elements $$c$$ (if any) such that $$a<c<b$$ are said to fall within the interval $$(a, b)$$. If an element $$b$$, posterior to $$a$$, can be found so that no element falls within the interval $$(a, b)$$, then $$a$$ is said to be isolated from all subsequent elements, and $$b$$ is said to be the element next after $$a$$. So if $$b'<a$$, and no element falls within the interval $$(b', a)$$, then $$a$$ is isolated from all preceding elements, and $$b'$$ is the element next before $$a$$. As will be seen presently, for any assigned element $$a$$, either, neither, or both of these cases may occur.

An aggregate $$\mbox{A}$$ is said to be well-ordered (or normally ordered) when, in addition to being ordered, it has the following properties: (1) $$\mbox{A}$$ has a first or lowest element a which is anterior to all the rest; (2) if $$\mbox{B}$$ is any part of $$\mbox{A}$$, then $$\mbox{B}$$ has a first element. It follows from this that every part of a well-ordered aggregate is itself well-ordered. A well-ordered aggregate may or may not have a last element.

Two ordered aggregates $$\mbox{A}, \mbox{B}$$ are said to be similar ($$\mbox{A}\simeq \mbox{B}$$) when a one-one correspondence can be set up between their elements in such a way that if $$b, b'$$ are the elements of B which correspond to any two elements $$a, a'$$ of A, then $$b>b'$$ or $$ba'$$ or $$a<a'$$. For example, $$(1,3,5, ...)\simeq(2,4,6, ...)$$, because we can make the even number $$2n$$ correspond to the odd number $$(2n-1)$$ and conversely.

Similar ordered aggregates are said to have the same order-type. Any definite order-type is said to be the ordinal number of every aggregate arranged according to that type. This somewhat vague definition will become clearer as we proceed.