PRIME AXD ULTIMATE RATIOS. 



ritixcirAi.. 





x-r (a k* x >, but no reason has yet been given why A U 1, and H 

 ,- the common logarithm, we find that the numb, r of 

 prime* less than f is such a proportion of x a* '4342945 is of log x 

 4706S8, nearly. Thus, of all numbers less than a million of million 

 of millions, only one out of 40 is prime, while the number of prime* 

 under the square of that number I* one out of 82. 



It thus appear* that we might name a succession of number* begin- 

 ning with one so high, that a million, or any other number however 

 great, of number* should pass without containing one prime number. 

 Nevertheless there cannot be an end to the prime numbers ; for if so, 

 let p be the hut prime number, and let x be the product of all the 

 prime numbers, 2, S, 6, .... ,p. Now every number is either prime 

 or divisible by a prime ; but N + 1 is not divisible by 2, 3, 6, .... or 

 f, since it leaves a remainder 1 in every such division. It is tl. 

 prime, or there is a prime number y + l, greater than the greatest 

 prime number p, which !s absurd. The following are among the pro- 

 perties of prime numbers : (!.) Every prime number (except 2) is 

 odd, or of the form 'it + 1. (2.) Every pnme is of the form 4.r + 1 or 

 4> -f 3, and a prime of the form 4 x + 1 is always the sum of two 

 square*. (3.) Every prime is of the form 6> + l or e.r+5. (4.) No 

 algebraical formula can always represent a prime number ; but some 

 formulic show a long succession of primes: thus i? + x + 41 is prime 

 from j- = to x = 39, both inclusive. (5.) If 2x + l be a prime num- 

 ber, and x any number which it does not divide, either x* 1 or 

 x' + l must be divisible by r+l. (6.) If M and x be two prime 

 numbers, and if M = 2x + 1, N = 2# + 1, then if xandybe both odd 

 number*, either (M' 1) : x and (H* 1) : M are both whole num- 

 bers, or (M' + 1) : x and (x* + 1) : M are both whole numbers. But if 

 x sod y be not both odd, then either (M 1) : x and (x* +1) : M are 

 both whole numbers, or else (M + 1J : X and (N 1) : M are both 

 whole numbers. This theorem is of considerable importance in the 

 theory of numbers, and has been termed the law of reciprocity of 

 prime numbers. 



Two numbers are said to be prime to one another, when they have 

 not any common measure except unity : as 36 and 55. 



The prime factors of a number are those prime numbers which 

 divide it. Thus, 360 being 2 x $> x 6, its prime factors are 2, 3, 5, of 

 which the first enters three times, the second twice, and the third 

 once. If A, B, c, .... be the prime factors of a number, and a, b, c, 

 . . . the number of times which they severally enter, the number is 



A* x B* x t' c x and the number of divisors which it admits 



of (unity and itself included) is (u + 1) x (6+1) x (r-t-l)x 



Let A* B 1 .... be x : then the number of numbers less than N, and 

 prime to N, is 



o-l 6-1 e-1 



v y - X ^^ X w 



' ;o 6 c *' 



PRIME AXD ULTIMATE RATIOS. [RATIOS, PRIMK AHD 

 ULTIMATE; DIFFKBKXTIAI. CALCULUS.] 



l'l;l M KltO. a game at cards, so called from a situation in the game. 

 He who holds the prime (primero), that is, a sequence of the best cards 

 and a good trump, is sure to be successful over his adversaries ; hence 

 its denomination. Primero, prime, and primaruta were one and the 

 same game. 



Primero appears to have been one of the earliest games at cords 

 played in England, and continued to be the most fashionable game 

 throughout the reigns of Henry VIII., Edward VI., Mary, Eli 

 and James. In the Earl of Northumberland's letters relating to the 

 i plot, we find that Josceline Percy was playing at primero on 

 Sunday, when his uncle, the conspirator, called on him at Essex-house. 



Shakspere speak* of Henry VI II. playing at primero with the Duke 



(Singer's 'Researches into the History of Playing Cards,' 4to, 



H:iM<",i:\ITi:UE may be defined to be that rule of English law 

 by which a title of dignity or an estate in land comes to a person in 

 respect of hi* being an eldest male. If a man dies seised of real estate, 

 of which he had the absolute ownership, without having made any 

 diiipoaition of it by hi* hut will, the whole descend* to the heir at law, 

 or customary heir ; and the heir at law is such by virtue of being the 

 eldest male person of those who are in the same degree of kindred to 

 the person dying, or the representative of such eldest male. [DKSCEXT.] 

 This is a case in which primogeniture operates. A common example 

 of primogeniture is where a father dies absolutely entitled to real 

 e*Ute, and without disposing of it by will, in which case his eldest son 

 takes it all. If land i* settled or entailed on a man and his male issue, 

 the eldest son take* the land by two titles, first as being a male, and 

 m-xtas being the eldest son. The law of primogeniture then only 

 applim in the case of land when the owner dies without having made 

 any duponition of it by will, or where the land in settled on a man and 

 hi* male issue. It does not apply when the interest in land is a 

 chattel interest, or a term of years, whatever may be its duration ; nor 

 doe* it apply when real estatt descends to daughters a* coparceners. 



At present those who are the absolute owners of large landed estates 

 seldom die without making a disposition of them by will. In the case 

 of land* which are settled, the person in possession is generally tenant 

 for life, and the inheritance i* entailed on the eldest son. When the 

 Idest son is about to marry, it is usual for the father and son to take 



d steps (which they can do as soon as the s. 

 ueeiitate and obtain the absolute ownership. Tiny then 



estate, making the father tenant for life aa 



son, who was In-fore tenant in tail, U also made only a tenant for life ; 

 and the inheritance U settled, aa before, mi tin' eldest son of 

 tended marriage. Such eldest son takes the i.-t.-it<-, not as heir, and 

 re not by the law of primogeniture, but he takes it as the person 

 designated by the deed of scttlem< 



\\ i.. n a man happens to be tenant in tail, hp usually taken the legal 

 steps necessary (which be can do as soon . ,-ei to aeon 



absolute ownership of the pro|x.-rly, which In- then g< net.iily settles 

 again by deed or will, or disposes of absolutely. 



It is usual in England to settle all large estates, and tin- object 

 settlement i* to keep the estates together, and to ]>< "tu in 



one family ; but there is .1 limit to this power of settlement. A man 

 cannot, either by deed or will, settle his land, so as to prevent the 

 absolute ownership of it from being obtained, for a lon# T |H rioil th:m 

 a life or lives of persons in existence at the time when the sett I 

 takes effect, and twenty-one years more. 



Lands in UAVM.KIXD and IJoiiouc.ii Kxi.usu are an exception to the 

 general rule of law as to the descent of land. 



The law of primogeniture then only operates in the cases already 

 explained ; and the system of settlements by which property U kept 

 together in large mummi U quite distinct in principle from the law . !" 

 primogeniture. It is not the result of a law which favours pritno 



Citure, but it is the result of the legal power which an owner of 

 i has over it, and of the habits of the people. The various reasons 

 which have laid the foundation of this habit, and which perpetuate it, 

 an- foreign to the cmi-Mi T.I'MII of primogeniture aa a rule of law. 



In Virginia, after the Revolution, an Act was passed for converting 

 estates tail into fee-simple, and at the same time the law M' primo- 

 geniture was abolished. Tlie.se laws have so far been in ace. 

 with or have acted on public opinion, that a parent by his will now 

 generally makes the same disposition of his property as the law makes 

 in case he dies intestate. (Tucker's ' Life of Jefferson,' i. Uti, &c.) 



(Remark* un Primogeniture and UntaiU; Hayes, Introduction tu C"- 

 reyanciii:/.} 



PRIMULIN. A crystalliuable non-azotised matter of unknown 

 composition, found in the root of the primrose. 



PRIMUM MO'BILK, the name given in the old astronomy to the 

 imaginary sphere by the motion of which diurnal motion was given t . 

 the heavens. [PTOI.IM ur SvsTKM.J 



PRINCE is the Latin word priiiefps, which was originally i. 

 denote the person who was entitled princeps xenatux in the I, 

 State. He seems to have been originally the custos of th< 

 and his office was one of importance. Subsequently it In . Mine a title 

 of dignity, and the princeps was named by the censors. (Liv. xx\ 

 In the senate he gave his opinion first after the magistratus. Angn-- 

 tus and his successors adopted the title of princeps, as a name tli.it 

 carried no odium with it : and this became henceforward the clinx 

 teristic title of the master of the Roman world. Accordingly the 

 constitutions of the emperors are called principum (Qaius i. -'> or 

 principales. The word princeps is formed similarly to aneeps, mnni- 

 ceps, &c., and contains the same element as " primus." In the 

 of time the word prince, which is derived from prim:e|>*, has DO! 

 be applied to persons having personal pre-eminence, and especially to 

 certain sovereigns of smaller states possessing either pet feet in, I, 

 ence of aU others or enjoying under some superior high politic il 

 rights. Of the first kind were the old sovereigns of Wales, who, under 

 the name of princes, enjoyed the same right and power which I 

 to kings ; and of the second, the heads of certain states of <' 

 comprehended in the great Germanic confederation. Hut the v,,,i,l 

 seems not to have acquired so definite a sense as that which belongs to 

 king, duke, marquess, earl, and some others of the class; but rather t.. 

 denote persons of eminent rank in certain states, as in Prussia, Russia, 

 Italy, and other continental states, where no sovereignty, prop- 

 called, comes along with it, or persons who are junior mem 

 sovereign houses, as the younger members of the ducal famil 

 Saxo and Auh.ilt. Kiequeiitly, however, such members have specific 

 titles given, such as duke, count, &c., but never prince. 



In England it has sometimes been the practice of the heralds to 

 speak of a duke as the high and mighty prince; but the word seems 

 rather to be restricted among us in its application to persons who are 

 of the blood-royal, that is, a son, grandson, or nephew of a king ; and 

 it would probably be extended to the remote male posterity, though 

 no such case has arisen in the course of the last three centuries. Hut 

 in its application it is merely a term of common parlance, not 

 conferred, like the title of duke, in any formal manner ; and even the 

 precedence which is given to blood-royal has respect to birth, and 'not. 

 to the enjoyment of this word as a title of honour. The eldest son 

 'ivereign is however made Prince of Wales by a special act of 

 creation; the younger sons are princes until they Lave ol 

 conferred ; the daughters continue princesses, 



I'll I M 'I"S METAL. A kind of brass composed of 75 ports of 

 and 25 parts of zinc. 



PHI NCI PAL. [AuEXT.j 



PRINCIPAL, the name of a stop or row of metal pipes in an organ, 

 tuned an octave higher than the diapason, an octave lower than the 



