1001 



NUMBERS, THE BOOK OF. 



NUMERAL CHARACTERS. 



lOOi 



not a little singular that the two great writers on this subject should 

 have been the men who, independently of each other'introduced the 

 method of LEAST SQUARES. 



The beginner in algebra may obtain some command over equations 

 of a simple character, not exceeding the second or third degree, by a 

 method which is, we believe, due to Playfair, or which, at least, is 

 published in the collection of his works. Let the equation be, for 

 instance, ay- + bxy + a a = 2 2 , in which x, y, and z are to be whole 

 numbers. Throw the equation into a form which admits of both sides 

 being reducible into factors ; for instance, 



) = (z-x) (z+x). 

 If then zx-vy, we have z + x=(ay + bx) : 11, which equations give 



Assuming at pleasure, z may be easily taken so as to make both x 

 and y whole numbers ; and the same method will succeed in many 

 equation*, 



Euclid's geometry, assuming only the use of his three celebrated 

 postulates, enables him, a linear unit being given, to construct the 

 length represented by any algebraical expression which involves only 

 additions, subtractions, multiplications, divisions, extraction of the 

 square root, or combinations of all these. But a cube or fifth root is 

 beyond the power of the system. Again, from the theory of equations 

 it is soon made obvious that the solution of the equation x* 1 = 

 and the division of a circle into n equal parts, are one and the same 

 problem. One solution of the preceding is .r = cos 6 + V 1. sin 0, 

 where 9 is the nth part of four right angles. [ROOT.] Euclid, in his 

 fourth book, shows how to cut a circle into three, four, five, and fifteen 

 equal parts ; and analysis shows that the sines and cosines of the angles 

 so involved can be obtained by formula; which involve no roots except 

 the square. But except into halves; thirds, fifths, or fifteenths, or parts 

 obtainable from these by one or more bisections, Euclid was not able 

 to cut a circle into equal parts. 



So the matter rested for about 2000 years, until Gauss, in his ' Dis- 

 quisitiones Arithmctica; ' (1801), not only pointed out how to extend 

 Euclid's conclusions, but also in a manner how to account for them. 

 The statement of his results, even without demonstration, is instructive 

 to the learner, and we shall give it accordingly : referring for the 

 demonstration to the works of Gauss or Legendre, or to Murphy's 

 ' Theory of Equations.' 



The expression a + V', and b being rational, is the solution of a 

 quadratic equation with rational co-efficients. But if a and b them- 

 selves have the form c + \?d, in which c and d have themselves the 

 same form, and so on; then a+\<b is the solution of a quadratic 

 equation in which the coefficients are themselves the solutions of 

 quadratic equations, and so on. Consequently, any equation, the root 

 of which is capable of construction by Euclid's postulates, must be 

 reducible to a system of quadratics ; and the converse. Now if be 

 a prime number, n 1 is an even number, and therefore has factors. 

 Let its prime factors be 2, a, b, c, &c., and let them severally enter p, 

 q, r, >, &c., times : so that 



n l = 2< ! aib' c* .... 



Gauss succeeded in showing that when n is a prime number, the solu- 

 tion of the equation x* 1 = 0, can be made to depend upon the solu- 

 tion of p equations of the second degree, r/ of the ath degree, r of the 

 4th degree, and so on. Consequently, whenever 2 is the only prime 

 factor of 1, or when n 1 = 2" , n being prime, or when 2? + 1 is a 

 prime number, the solution of x* 1 is. reducible to that of p quadratic 

 equations, and the division of the circle into n equal parts can be 

 accomplished by Euclid's geometry. And further, it is easily demon- 

 strated that 2' + 1 can never be a prime number, except when p itself 

 is a power of 2 (2" included) though 2* 1 + 1 is not then always prime. 

 Nor has it been shown that other divisions are impossible : Gauss's 

 theorem merely point* out cases in which the thing can be done, with- 

 out pronouncing the exclusion of others. Gauss, indeed, does assert 

 that he can demonstrate all other eases to be impossible to be con- 

 structed by geometry, that is, reducible to quadratic equations : and 

 the thing is highly probable. If we now construct the series 2 l + l, 

 V ,+ 1, 2' + 1, 2" + 1, &c., among which all our chances lie, we have 3, 5, 

 17, 257, 65537, 4294967297, &c. The first five are prime numbers: 

 Kuclid has disposed of the two first divisions ; Gauss has added that a 

 circle can be geometrically divided into 17, 257, and G5537 equal parts. 

 But 4291967207 is not a prime number, being divisible by 641. 



NUMBKHS, THE HOOK. OF, one of the books of the Pentateuch. 

 In Hebrew it has two titles, ~>3T1, and lie spake, which is the first 

 word of the book, and ~l2nC3, in the desert, which is the fifth word 

 in the first verse, and which applies to the whole book, inasmuch as 

 the events which it records took place in the desert. Its title in the 

 Si'ptimgint is 'Apifljuol, Numbers, because it contains the censuses of the 

 of Israel (chap*, i.-iii., and xxvi.). 



lirst four chapters of this book consist of separate accounts of 

 commands given by God to Moses, while the Israelites were encamped 

 at the i'"t >.f Sinai, n-Mp. r-ting the census and the classification of the 

 people, and the duties of the priests and Levites. The succeeding 



chapters (v.--x. 10) contain various laws, most of which are additions 

 to those before given in the books of Exodus and Leviticus ; and the 

 i-est of the book is occupied with the narrative of the journeys of the 

 Israelites, from the time of their leaving Sinai to their second arrival 

 at the Jordan, and their encampment in the plains of Moab. The 

 time over which the book extends is from the first day of the second 

 month of the second year after the departure from Egypt, to the first 

 day of the eleventh month of the fortieth year of the same epoch. 

 This part of the book also contains various enactments. 



We learn from the last verse of the last chapter that this book was 

 written by Moses "in the plains of Moab by Jordan near Jericho," 

 and consequently just before his death. Vater has attempted to show 

 that it is composed of short narratives written by different persons 

 (vol. iii., p. 452, &c.), and De Wette adduces several passages which 

 appear to disagree with each other, and with the parallel passages in 

 the book of Exodus (' Lehrbuch d. Hist. Krit. Einleitung in d. A. T.') ; 

 but the more carefully the facts and allusions contained in it are 

 examined, the more satisfactorily they confirm the belief that it, as 

 well as the other books of the Pentateuch, was the production of 

 Moses, and its genuineness rests upon the same points of evidence as 

 to its origin and its historical truth as the other books. Even 

 De Wette admits that parts of it belong to the Mosaical age. 



This book is quoted or referred to in the New Testament (compare 

 Numb. xx. 11 with 1 Cor. x. 4, and Numb. xxi. with John iii. 14). 

 The passage in chap. xxiv. 17-19, is generally understood as a pre- 

 diction of the Messiah. 



(Rosenmuller's Schulia in Vet. Test. ; the Introduction* of Eichhorn, 

 Jahn, De Wette, and Horne ; Graves's and Vater's Commentaries on the 

 Pentateuch.) 



NUMERAL CHARACTERS. There are three simple and ob- 

 vious modes of constructing symbols of number. 1. By arbitrary 

 invention. 2. By the choice of letters of the alphabet. 3. By a sys- 

 tem of repetitions of a single unit, as I, II, III, IIII, &c., with marks 

 of abbreviation. Some may doubt whether the first and third were 

 ever really employed ; but it is not known that we can assign to the 

 Indian numerals any other origin than the first, and the third explains 

 the Roman system with a degree of consistency which is most extra- 

 ordinary, if it be only accidental coincidence. 



Distinct numeral characters are found to have existed or to exist 

 among the Chinese, Indians, and Arabs, &c., Phoenicians, Palmyrenes, 

 Hebrews, Egyptians, Greeks, and Romans ; and others are given as in 

 ancient use amons; the Mexicans. We shall here confine ourselves to 

 the simplest explanation of those systems which will be wanted by the 

 student of ancient literature. Of these, as it should seem, the Indian 

 system may belong (though it may be doubted) to the first class ; the 

 Hebrew and the common Greek system to the second ; the Roman, 

 Phoenician, Palrnyrene, ancient Greek, Egyptian, and Chinese, to the 

 third class. 



The system received from the Hindus through the Arabs, and now 

 adopted throughout Europe, has been gradually much altered in the 

 forms of the symbols. The symbols we now use, especially in their 

 old manuscript forms, 1, 2, 3, 4, 8, may be explained with reasonable 

 probability upon the third system ; but not 5, 6, 7, 9. [ARITHMETIC.] 



The Hebrews used the'letters of their own alphabet, giving the 

 finals a separate and particular value, as follows : 



Letter N 

 Numeral Signification 1 



20 80 40 SO 60 70 80 



p 

 100 



200 300 400 



i n 



500 COO 



7 n v 



700 800 900 



The use of the final letters as signifying numbers is of newer date 

 than the rest ; the old system required the junction of subordinate 

 numbers to express 500, 600, &c. Numbers not expressed above were 

 made by juxta-position of letters denoting other numbers, according to 

 a decimal subdivision, as among the Greeks ; the only exception being 

 15, which, as 10 + 5 or !T made a word signifying the Creator, they 

 wrote as 9 + 6, or 1t3. In a language like the Hebrew it would be 

 impossible to prevent every combination of numbers from also stand- 

 ing for a word or words ; and the Oriental nations accordingly have 

 frequently expressed dates by sentences. Thus " Hooshung Shah is 

 no more," rendered into Persian, expresses, in the numeral force of 

 the letters, the year 837 of the Hegira, the date of the death of that 

 prince. 



The Greeks, in some enumerations, have three distinct methods of 

 expressing numbers ; but the first of them, which consists in the use 

 of the letters of the alphabet to denote the successive books of a work, 

 us in the ' Iliad,' is as much a method of naming as of counting. Some- 

 thing more to the point is the old system which occurs on inscriptions, 

 in which the unit is represented by a single mark, five by n (the initial 

 of riENTE), ten by A (that of AEKA), and 100 by H (that of HEKATON). 

 And in all cases five of any symbol are written by inclosing the symbol 

 inH: tbus|A|isfivtjtens,and|HJisfive hundreds. Thus 879is || HHH 

 |A| AAnilll, This ancient Greek method, as found on inscriptions 



