611 



POLYGON OF FORCES. 



POLYTHEISM. 



one. This is the reason of the stability of solid figures ; were it not 

 for this, a box, for example, would require internal cross-pieces to 

 support the sides. This remarkable property is assumed by Euclid as 

 a part of n definition, and that improperly ; since it is a new axiom. 



A proof of the axiom implied in the above was given by M. Cauchy 

 in the article already cited, and will be found in the notes to Legeudre's 

 . try. It is sufficient, but depends on considerations foreign to 

 the subject as usually considered. 



For the remarkable division of equal solids into symmetrically and 

 unsymmetrically equal, see SYMMETRICAL : for the more general view 

 of the nature of polygons, suggested by modern geometry, see TRANS- 

 VERSAL : see also TRIANGLE, REGULAR FIGURES and KEGULAH SOLIDS, 

 TRIGONOMETRY, Ac. 



To explain the meaning of the term polygonal number [NUMBERS, 

 APPELLATIONS OF], let us take as an instance the pentagonal number. 

 Take any pentagon B A !>, and construct a set of pentagons, \cc, ADd, 

 &c., double, treble, Ac., of A B b in linear dimension. Divide the sides 

 of each pentagon into parts, each equal to the corresponding side of 

 AH f>. Then if we begin with A, which U one point, and afterwards 

 take in all the point* of the first pentagon, we have 1 + 4, or 5 points. 

 If we now add all the additional points of the second pentagon (in- 

 cluding subdivision-points), we have 1 + 4 + 7 or 12 points. Take in 

 tin- next pentagon, and we have 1 + 4 + 7 + 1 0, or 22 points. Hence 

 the series 1, 5, 12, 22, &c., is called the series of pentagonal numbers; 

 and ,1 set of numbers is thus pointed out which may be as justly called 

 pentagonal as the get 1, 4, 9, Ac., may be called square. It must be 

 'd that the various sets of polygonal numbers were suggested by 

 the square numbers. 



POLYGON OF FORGES. A well-known proposition under this 



attributed to Leibnitz. If any number of forces act upon a 



and we take a polygon, one of whose sides is formed by the 



line representing one of the forces, and the other sides in succession 



by lines representing the other forces in magnitude, and parallel to 



their directions, then the line which completes the polygon will 



represent the resultant of the whole. 



POtiYGON, REGULAR. [REGULAR FIGURES ; REGULAR SOLIDS.] 



POLYGONAL NUMBERS. [NUMBERS, APPELLATIONS or.] 



POLYHEDRON. [POLYGON AND POLYHEDRON.] 



POLYMKIJIDES. Chemical compounds which have the i,ame per- 

 centage composition, but a different molecular arrangement. Thus, 

 acetate of methyl and formiate of ethyl are polymerides : 



Acetate or methjrl 

 Fonnlatc of ethyl . 



C.H.O, 



C.H.0 4 



POLYNOMIAL, an algebraical word meaning an expression which 

 has several terms, being the general term under which are included, 

 liinnmi.-ils, trinomials, quadrinomials, Ac., or expressions of two, three. 

 four, Ac. terms. The /><////'//i/ theorem ineani the theorem by which 

 a i*>lynomial expression is raised to its several powers. For an easy 

 way of doing this, see ' Library of Useful Knowledge,' " DifC Calc.," 

 PP. :;-J.3-337. 



l'i >LYPUS is a morbid growth attached to the interior of one or 

 other of the mucous canals, by a more or leas narrow pedicle. Polypi 

 occur most frequently near the orifices of external communication of 

 tin- mucous canals, as in the uterus, the fauces, the larynx, and the 

 A description of those that grow in the last of these situation* 

 may serve for all. 



Polypi of the nose are of four kinds, which are named respectively, 

 vehicular, gelatinous, fibrous, and malignant. Vesicular polypi are 



r yellowish transparent vesicles, containing a clear watery fluid 



with a little mucus; they are very soft, easily break down when they 



are pressed, and are liable to considerable increase of size when the 



nation from them is decreased in damp weather. They appear to 



t of enlarged mucous follicles. Gelatinous polypi are more solid 

 growths, consisting of dull white masses attached to the mucous mem- 

 brane of the nose, and themselves covered by a fine but rather tough 



rane. They seem composed of an excessive growth of the 

 mucous membrane infiltrated with fluid, and having a few fibres 

 running through its substance. Fibrous polypi are still more solid 



In ; they consist of a dense fibrous and vascular tissue, which it 



n 'iiilicult to cut through, and which is sometimes converted 



into bone. They ore usually intimately connected with the deeper 



parts of the mucous membrane. Malignant polypi are growths of a 



cancerous nature from the mucous membrane, which have received 



me of polypi rather from having the same situation as the pre- 

 ceding than from their similarity of form. They may assume the 

 characters of simple or schirrhous cancer, but more commonly they 

 have those of the soft or medullary variety [CANCER] which is fre- 

 quently called fungus htcmatodes. 



common symptoms of all polypi of the nose ore that the 

 patient is unable to breathe through the nostril in which they are 



I , and has a nasal voice ; there is usually a discharge of watery 



-, a loss of smell, and a diminution of the power of taste. If 

 situated far back in the nose, they may cause deafness, or obstruct the 

 swallowing of food. They even render the breathing difficult. The 



softer kinds seldom produce worse effects than these, for their growth 

 is restrained by the firm tissues of the nostrils ; but the two last and 

 tir:ii varieties will continue to increase, expanding the bones and other 

 tissues of the nostrils and face into huge and hideous swellings, and 

 producing death either by their ulceration, or by their pressure on 

 the vessels of the brain, or on the brain itself through the base of 

 the skull. 



The most usual situation of polypi is in the upper and back parts 

 of the nostrils ; and almost always on the outer wall, the septum 

 being very rarely affected. In their increase they mould them- 

 selves to the form of the passages of the nose, and at last protrude 

 either forwards through the anterior openings or backwards into the 

 fauces. 



The growth of vesicular polypi is usually connected with a generally 

 disordered state of the health. They seldom require to be removed 

 by operation ; the patient should take mild alterative and tonic 

 medicines with purgatives ; and lotions containing alum or sulphate of 

 zinc, or some equally powerful astringent, should be frequently 

 injected into the nostrils. When the polypi are thus destroyed, 

 ointment containing nitrate of mercury, or savine powder, or some 

 other stimulant, should be rubbed on the membrane where they were 

 seated. Gelatinous polypi are not usually benefited by the preceding 

 plan of treatment ; although after their removal it is very useful to 

 prevent them from growing again. They must bo pulled from their 

 attachment by forceps, which should have rough blades, and, if 

 possible, be placed around the pedicle of the growth, and withdrawn 

 with a kind of twisting motion. All the polypi being thus removed, 

 the mucous membrane should have some powerfully astringent lotion 

 or ointment applied to it. The fibrous polypi usually require a more 

 difficult operation for their removal ; when they can be withdrawn by 

 the forceps, it is only with much difficulty, from their being seated 

 far back in the nostrils, and with some violence and danger of hemorr- 

 hage. In some cases a ligature may be tied round the base of such a 

 polypus, so as to make it slough off; in others they must be cut out 

 with the knife or scissers. The choice of these operations must be 

 made according to the circumstances of each case. After their 

 removal, the same means should be adopted to prevent their return as 

 in the preceding cases. The malignant growths in the nose, like those 

 in other parts of the body, where their early and complete removal 

 cannot certainly be accomplished, had better be treated only by 

 palliative measures. 



PO'LYTHEISM (wo\iis, much, many, 9ij, a god), the doctrine or 

 worship of a plurality of gods. This form of the word is not found 

 in Greek writers. Philo-Judaous conveys the notion by rb mKiteor, 

 and TroXvOtas 8<f{o, and Justin Martyr by oAu0e<fT7)s. Philo also 

 speaks of polyarchy (iroXuopx'o), with which he contrasts monarchy 

 (iua>ff\la.), meaning respectively polytheism and the unity of God. 

 St. Athanasius uses woAvdcfa as the opposite to norapx'"- 



Polytheism differs from idolatry in this respect : the former refers 

 to a plurality of gods, without including necessarily the notion of 

 fomu, real or imaginary ; the latter refers either to one god only, 

 under some one visible form, or to any number of gods, under as many 

 visible forms. 



No one who takes the Bible for his authority will deny that in the 

 earliest ages of mankind the unity of God was implicitly believed. 

 Wlifii the idea of a plurality of gods was first entertained, it is im- 

 possible to say. The writings of Moses contain no evidence that men 

 were polytheists before the flood ; but it is plain that in the time of 

 Moses (B.C. 1500) polytheism prevailed. It is probable that the Chal- 

 dieans, of whom Abraham wis one, were addicted to polytheism before 

 his time (B.C. 1900); but it is certain that Abraham was not a poly- 

 theist. His contemporaries, Melchizedek, king of Salem, and Abime- 

 loch, king of the Philistines, both Canaanites, were worshippers of 

 one God, and the God of Abraham himself. The ancient Persians 

 might have been free from polytheism in Abraham's days ; but they 

 became, first, Sabians, or worshippers of the host of heaven, and 

 afterwards Magians, or worshippers of fire. As Magians, they held 

 that there are two principles, one the cause of all good, and the other 

 the cause of all evil ; the one they represented by darkness, and the 

 other by light ; the former, according to Plutarch (' De Isid. et Osir.,' 

 46), was called Oromasdes, and the latter Arimanius. [AHIMANKS.] 

 Among the ancient Arabians we find Job and his friends, and Jethro, 

 the contemporary of Moses, worshippers of one God ; but yet, in the 

 time of Job, the worship of the heavenly bodies was practised, as is 

 plainly indicated in Job xxxi. 2(5, 27. The Egyptians, in Abraham's 

 days, seem to have had a religion not different from his ; but in the 

 time of Moses, polytheism had become established among them ; for 

 many provisions of his law refer to this fact, and are intended es- 

 pecially to guard the Israelites from imitating the polytheism and 

 idolatry of Egypt. Cudworth is of opinion (c. 4, 18) that poly- 

 tliei.-m originated in Egypt, but such an assertion is incapable of 

 proof. The Greeks and Romans acknowledged one being, under the 

 names of Zeus (ZeiJt) and Jupiter, as supreme over all other gods ; 

 and they believed further, that this one being was under the absolute 

 power of Fate. Yet these inferior beings were worshipped as gods, 

 and the system was essentially polytheistic. 



Cudworth has written at great length to show that the ancient 

 philosophers in general, though they seem to have been polytheists, 



