MEAUX

meaux, a town of northern france, capital of an arrondissement in the department of seine-et-marne, and chief town of the agricultural region of brie, 28 m. e.n.e. of paris by rail. pop. (1906), 11,089. the town proper stands on an eminence on the right bank of the marne; on the left bank lies the old suburb of le marche, with which it is united by a bridge of the 16th century. two rows of picturesque mills of the same period are built across the river. the cathedral of st stephen dates from the 12th to the 16th centuries, and was restored in the 19th century. of the two western towers, the completed one is that to the north of the facade, the other being disfigured by an unsightly slate roof. the building, which is 275 ft. long and 105 ft. high, consists of a short nave, with aisles, a fine transept, a choir and a sanctuary. the choir contains the statue and the tomb of bossuet, bishop from 1681 to 1704, and the pulpit of the cathedral has been reconstructed with the panels of that from which the "eagle of meaux" used to preach. the transept terminates at each end in a fine portal surmounted by a rose-window. the episcopal palace (17th century) has several curious old rooms; the buildings of the choir school are likewise of some archaeological interest. a statue of general raoult (1870) stands in one of the squares. meaux is the centre of a considerable trade in cereals, wool, brie cheeses, and other farm-produce, while its mills provide much of the flour with which paris is supplied. other industries are saw-milling, metal-founding, distilling, the preparation of vermicelli and preserved vegetables, and the manufacture of mustard, hosiery, plaster and machinery. there are nursery-gardens in the vicinity. the canal de l'ourcq, which surrounds the town, and the marne furnish the means of transport. meaux is the seat of a bishopric dating from the 4th century, and has among its public institutions a sub-prefecture, and tribunals of first instance and of commerce. in the roman period meaux was the capital of the meldi, a small gallic tribe, and in the middle ages of the brie. it formed part of the kingdom of austrasia, and afterwards belonged to the counts of vermandois and champagne, the latter of whom established important markets on the left bank of the marne. its communal charter, received from them, is dated 1179. a treaty signed at meaux in 1229 after the albigensian war sealed the submission of raymond vii., count of toulouse. the town suffered much during the jacquerie, the peasants receiving a severe check there in 1358; during the hundred years' war; and also during the religious wars, in which it was an important protestant centre. it was the first town which opened its gates to henry iv. in 1594. on the high-road for invaders marching on paris from the east of france, meaux saw its environs ravaged by the army of lorraine in 1652, and was laid under heavy requisitions in 1814, 1815 and 1870. in september 1567 meaux was the scene of an attempt made by the protestants to seize the french king charles ix., and his mother catherine de' medici. the plot, which is sometimes called the "enterprise of meaux," failed, the king and queen with their courtiers escaping to paris. this conduct, however, on the part of the huguenots had doubtless some share in influencing charles to assent to the massacre of st bartholomew. mecca (arab. _makkah_),[1] the chief town of the hejaz in arabia, and the great holy city of islam. it is situated two camel marches (the resting-place being bahra or hadda), or about 45 m. almost due e., from jidda on the red sea. thus on a rough estimate mecca lies in 21° 25 n., 39° 50 e. it is said in the koran (_sur._ xiv. 40) that mecca lies in a sterile valley, and the old geographers observe that the whole haram or sacred territory round the city is almost without cultivation or date palms, while fruit trees, springs, wells, gardens and green valleys are found immediately beyond. mecca in fact lies in the heart of a mass of rough hills, intersected by a labyrinth of narrow valleys and passes, and projecting into the tehama or low country on the red sea, in front of the great mountain wall that divides the coast-lands from the central plateau, though in turn they are themselves separated from the sea by a second curtain of hills forming the western wall of the great wadi marr. the inner mountain wall is pierced by only two great passes, and the valleys descending from these embrace on both sides the mecca hills. holding this position commanding two great routes between the lowlands and inner arabia, and situated in a narrow and barren valley incapable of supporting an urban population, mecca must have been from the first a commercial centre.[2] in the palmy days of south arabia it was probably a station on the great incense route, and thus ptolemy may have learned the name, which he writes makoraba. at all events, long before mahomet we find mecca established in the twofold quality of a commercial centre and a privileged holy place, surrounded by an inviolable territory (the haram), which was not the sanctuary of a single tribe but a place of pilgrimage, where religious observances were associated with a series of annual fairs at different points in the vicinity. indeed in the unsettled state of the country commerce was possible only under the sanctions of religion, and through the provisions of the sacred truce which prohibited war for four months of the year, three of these being the month of pilgrimage, with those immediately preceding and following. the first of the series of fairs in which the meccans had an interest was at okaz on the easier road between mecca and taif, where there was also a sanctuary, and from it the visitors moved on to points still nearer mecca (majanna, and finally dhul-majaz, on the flank of jebel kabkab behind arafa) where further fairs were held,[3] culminating in the special religious ceremonies of the great feast at 'arafa, quzah (mozdalifa), and mecca itself. the system of intercalation in the lunar calendar of the heathen arabs was designed to secure that the feast should always fall at the time when the hides, fruits and other merchandise were ready for market,[4] and the meccans, who knew how to attract the bedouins by hospitality, bought up these wares in exchange for imported goods, and so became the leaders of the international trade of arabia. their caravans traversed the length and breadth of the peninsula. syria, and especially gaza, was their chief goal. the syrian caravan intercepted, on its return, at badr (see mahomet) represented capital to the value of £20,000, an enormous sum for those days.[5] the victory of mahommedanism made a vast change in the position of mecca. the merchant aristocracy became satraps or pensioners of a great empire; but the seat of dominion was removed beyond the desert, and though mecca and the hejaz strove for a time to maintain political as well as religious predominance, the struggle was vain, and terminated on the death of ibn zubair, the meccan pretendant to the caliphate, when the city was taken by hajjaj (a.d. 692). the sanctuary and feast of mecca received, however, a new prestige from the victory of islam. purged of elements obviously heathen, the ka'ba became the holiest site, and the pilgrimage the most sacred ritual observance of mahommedanism, drawing worshippers from so wide a circle that the confluence of the petty traders of the desert was no longer the main feature of the holy season. the pilgrimage retained its importance for the commercial well-being of mecca; to this day the meccans live by the hajj--letting rooms, acting as guides and directors in the sacred ceremonies, as contractors and touts for land and sea transport, as well as exploiting the many benefactions that flow to the holy city; while the surrounding bedouins derive support from the camel-transport it demands and from the subsidies by which they are engaged to protect or abstain from molesting the pilgrim caravans. but the ancient "fairs of heathenism" were given up, and the traffic of the pilgrim season, sanctioned by the prophet in _sur._ ii. 194, was concentrated at mina and mecca, where most of the pilgrims still have something to buy or sell, so that mina, after the sacrifice of the feast day, presents the aspect of a huge international fancy fair.[6] in the middle ages this trade was much more important than it is now. ibn jubair (ed. wright, p. 118 seq.) in the 12th century describes the mart of mecca in the eight days following the feast as full of gems, unguents, precious drugs, and all rare merchandise from india, irak, khorasan, and every part of the moslem world. the hills east and west of mecca, which are partly built over and rise several hundred feet above the valley, so enclose the city that the ancient walls only barred the valley at three points, where three gates led into the town. in the time of ibn jubair the gates still stood though the walls were ruined, but now the gates have only left their names to quarters of the town. at the northern or upper end was the bab el ma'la, or gate of the upper quarter, whence the road continues up the valley towards mina and arafa as well as towards zeima and the nejd. beyond the gate, in a place called the hajun, is the chief cemetery, commonly called el ma'la, and said to be the resting-place of many of the companions of mahomet. here a cross-road, running over the hill to join the main medina road from the western gate, turns off to the west by the pass of kada, the point from which the troops of the prophet stormed the city (a.h. 8).[7] here too the body of ibn zubair was hung on a cross by hajjaj. the lower or southern gate, at the masfala quarter, opened on the yemen road, where the rain-water from mecca flows off into an open valley. beyond, there are mountains on both sides; on that to the east, commanding the town, is the great castle, a fortress of considerable strength. the third or western gate, bab el-omra (formerly also bab el-zahir, from a village of that name), lay almost opposite the great mosque, and opened on a road leading westwards round the southern spurs of the red mountain. this is the way to wadi fatima and medina, the jidda road branching off from it to the left. considerable suburbs now lie outside the quarter named after this gate; in the middle ages a pleasant country road led for some miles through partly cultivated land with good wells, as far as the boundary of the sacred territory and gathering place of the pilgrims at tanim, near the mosque of ayesha. this is the spot on the medina road now called the omra, from a ceremonial connected with it which will be mentioned below. the length of the sinuous main axis of the city from the farthest suburbs on the medina road to the suburbs in the extreme north, now frequented by bedouins, is, according to burckhardt, 3500 paces.[8] about the middle of this line the longitudinal thoroughfares are pushed aside by the vast courtyard and colonnades composing the great mosque, which, with its spacious arcades surrounding the ka'ba and other holy places, and its seven minarets, forms the only prominent architectural feature of the city. the mosque is enclosed by houses with windows opening on the arcades and commanding a view of the ka'ba. immediately beyond these, on the side facing jebel abu kobais, a broad street runs south-east and north-west across the valley. this is the mas'a (sacred course) between the eminences of safa and merwa, and has been from very early times one of the most lively bazaars and the centre of meccan life. the other chief bazaars are also near the mosque in smaller streets. the general aspect of the town is picturesque; the streets are fairly spacious, though ill-kept and filthy; the houses are all of stone, many of them well-built and four or five storeys high, with terraced roofs and large projecting windows as in jidda--a style of building which has not varied materially since the 10th century (mukaddasi, p. 71), and gains in effect from the way in which the dwellings run up the sides and spurs of the mountains. of public institutions there are baths, ribats, or hospices, for poor pilgrims from india, java, &c., a hospital and a public kitchen for the poor. the mosque is at the same time the university hall, where between two pilgrim seasons lectures are delivered on mahommedan law, doctrine and connected branches of science. a poorly provided public library is open to the use of students. the madrassehs or buildings around the mosque, originally intended as lodgings for students and professors, have long been let out to rich pilgrims. the minor places of visitation for pilgrims, such as the birthplaces of the prophet and his chief followers, are not notable.[9] both these and the court of the great mosque lie beneath the general level of the city, the site having been gradually raised by accumulated rubbish. the town in fact has little air of antiquity; genuine arab buildings do not last long, especially in a valley periodically ravaged by tremendous floods when the tropical rains burst on the surrounding hills. the history of mecca is full of the record of these inundations, unsuccessfully combated by the great dam drawn across the valley by the caliph omar (_kutbeddin_, p. 76), and later works of mahdi.[10] the fixed population of mecca in 1878 was estimated by assistant-surgeon 'abd el-razzaq at 50,000 to 60,000; there is a large floating population--and that not merely at the proper season of pilgrimage, the pilgrims of one season often beginning to arrive before those of the former season have all dispersed. at the height of the season the town is much overcrowded, and the entire want of a drainage system is severely felt. fortunately good water is tolerably plentiful; for, though the wells are mostly undrinkable, and even the famous zamzam water only available for medicinal or religious purposes, the underground conduit from beyond arafa, completed by sultan selim ii. in 1571, supplies to the public fountains a sweet and light water, containing, according to 'abd el-razzaq, a large amount of chlorides. the water is said to be free to townsmen, but is sold to the pilgrims at a rather high rate.[11] medieval writers celebrate the copious supplies, especially of fine fruits, brought to the city from taif and other fertile parts of arabia. these fruits are still famous; rice and other foreign products are brought by sea to jidda; mutton, milk and butter are plentifully supplied from the desert.[12] the industries all centre in the pilgrimage; the chief object of every meccan--from the notables and sheikhs, who use their influence to gain custom for the jidda speculators in the pilgrim traffic, down to the cicerones, pilgrim brokers, lodging-house keepers, and mendicants at the holy places--being to pillage the visitor in every possible way. the fanaticism of the meccan is an affair of the purse; the mongrel population (for the town is by no means purely arab) has exchanged the virtues of the bedouin for the worst corruptions of eastern town life, without casting off the ferocity of the desert, and it is hardly possible to find a worse certificate of character than the three parallel gashes on each cheek, called tashrit, which are the customary mark of birth in the holy city. the unspeakable vices of mecca are a scandal to all islam, and a constant source of wonder to pious pilgrims.[13] the slave trade has connexions with the pilgrimage which are not thoroughly clear; but under cover of the pilgrimage a great deal of importation and exportation of slaves goes on. since the fall of ibn zubair the political position of mecca has always been dependent on the movements of the greater mahommedan world. in the splendid times of the caliphs immense sums were lavished upon the pilgrimage and the holy city; and conversely the decay of the central authority of islam brought with it a long period of faction, wars and misery, in which the most notable episode was the sack of mecca by the carmathians at the pilgrimage season of a.d. 930. the victors carried off the "black stone," which was not restored for twenty-two years, and then only for a great ransom, when it was plain that even the loss of its palladium could not destroy the sacred character of the city. under the fatimites egyptian influence began to be strong in mecca; it was opposed by the sultans of yemen, while native princes claiming descent from the prophet--the hashimite amirs of mecca, and after them the amirs of the house of qatada (since 1202)--attained to great authority and aimed at independence; but soon after the final fall of the abbasids the egyptian overlordship was definitely established by sultan bibars (a.d. 1269). the turkish conquest of egypt transferred the supremacy to the ottoman sultans (1517), who treated mecca with much favour, and during the 16th century executed great works in the sanctuary and temple. the ottoman power, however, became gradually almost nominal, and that of the amirs or sherifs increased in proportion, culminating under ghalib, whose accession dates from 1786. then followed the wars of the wahhabis (see arabia and wahhabis) and the restoration of turkish rule by the troops of mehemet 'ali. by him the dignity of sherif was deprived of much of its weight, and in 1827 a change of dynasty was effected by the appointment of ibn 'aun. afterwards turkish authority again decayed. mecca is, however, officially the capital of a turkish province, and has a governor-general and a turkish garrison, while mahommedan law is administered by a judge sent from constantinople. but the real sovereign of mecca and the hejaz is the sherif, who, as head of a princely family claiming descent from the prophet, holds a sort of feudal position. the dignity of sherif (or grand sherif, as europeans usually say for the sake of distinction, since all the kin of the princely houses reckoning descent from the prophet are also named sherifs), although by no means a religious pontificate, is highly respected owing to its traditional descent in the line of hasan, son of the fourth caliph 'ali. from a political point of view the sherif is the modern counterpart of the ancient amirs of mecca, who were named in the public prayers immediately after the reigning caliph. when the great mahommedan sultanates had become too much occupied in internecine wars to maintain order in the distant hejaz, those branches of the hassanids which from the beginning of islam had retained rural property in arabia usurped power in the holy cities and the adjacent bedouin territories. about a.d. 960 they established a sort of kingdom with mecca as capital. the influence of the princes of mecca has varied from time to time, according to the strength of the foreign protectorate in the hejaz or in consequence of feuds among the branches of the house; until about 1882 it was for most purposes much greater than that of the turks. the latter were strong enough to hold the garrisoned towns, and thus the sultan was able within certain limits--playing off one against the other the two rival branches of the aristocracy, viz. the kin of ghalib and the house of ibn'aun--to assert the right of designating or removing the sherif, to whom in turn he owed the possibility of maintaining, with the aid of considerable pensions, the semblance of his much-prized lordship over the holy cities. the grand sherif can muster a considerable force of freedmen and clients, and his kin, holding wells and lands in various places through the hejaz, act as his deputies and administer the old arabic customary law to the bedouin. to this influence the hejaz owes what little of law and order it enjoys. during the last quarter of the 19th century turkish influence became preponderant in western arabia, and the railway from syria to the hejaz tended to consolidate the sultan's supremacy. after the sherifs, the principal family of mecca is the house of shaibah, which holds the hereditary custodianship of the ka'ba. _the great mosque and the ka'ba._--long before mahomet the chief sanctuary of mecca was the ka'ba, a rude stone building without windows, and having a door 7 ft. from the ground; and so named from its resemblance to a monstrous _astragalus_ (die) of about 40 ft. cube, though the shapeless structure is not really an exact cube nor even exactly rectangular.[14] the ka'ba has been rebuilt more than once since mahomet purged it of idols and adopted it as the chief sanctuary of islam, but the old form has been preserved, except in secondary details;[15] so that the "ancient house," as it is titled, is still essentially a heathen temple, adapted to the worship of islam by the clumsy fiction that it was built by abraham and ishmael by divine revelation as a temple of pure monotheism, and that it was only temporarily perverted to idol worship from the time when 'amr ibn lohai introduced the statue of hobal from syria[16] till the victory of islam. this fiction has involved the superinduction of a new mythology over the old heathen ritual, which remains practically unchanged. thus the chief object of veneration is the black stone, which is fixed in the external angle facing safa. the building is not exactly oriented, but it may be called the south-east corner. its technical name is the black corner, the others being named the yemen (south-west), syrian (north-west), and irak (north-east) corners, from the lands to which they approximately point. the black stone is a small dark mass a span long, with an aspect suggesting volcanic or meteoric origin, fixed at such a height that it can be conveniently kissed by a person of middle size. it was broken by fire in the siege of a.d. 683 (not, as many authors relate, by the carmathians), and the pieces are kept together by a silver setting. the history of this heavenly stone, given by gabriel to abraham, does not conceal the fact that it was originally a fetish, the most venerated of a multitude of idols and sacred stones which stood all round the sanctuary in the time of mahomet. the prophet destroyed the idols, but he left the characteristic form of worship--the _tawaf_, or sevenfold circuit of the sanctuary, the worshipper kissing or touching the objects of his veneration--and besides the black stone he recognized the so-called "southern" stone, the same presumably as that which is still touched in the tawaf at the yemen corner (_muh. in med._ pp. 336, 425). the ceremony of the tawaf and the worship of stone fetishes was common to mecca with other ancient arabian sanctuaries.[17] it was, as it still is, a frequent religious exercise of the meccans, and the first duty of one who returned to the city or arrived there under a vow of pilgrimage; and thus the outside of the ka'ba was and is more important than the inside. islam did away with the worship of idols; what was lost in interest by their suppression has been supplied by the invention of spots consecrated by recollections of abraham, ishmael and hagar, or held to be acceptable places of prayer. thus the space of ten spans between the black stone and the door, which is on the east side, between the black and irak corners, and a man's height from the ground, is called the _multazam_, and here prayer should be offered after the tawaf with outstretched arms and breast pressed against the house. on the other side of the door, against the same wall, is a shallow trough, which is said to mark the original site of the stone on which abraham stood to build the ka'ba. here the growth of the legend can be traced, for the place is now called the "kneading-place" (ma'jan), where the cement for the ka'ba was prepared. this name and story do not appear in the older accounts. once more, on the north side of the ka'ba, there projects a low semicircular wall of marble, with an opening at each end between it and the walls of the house. the space within is paved with mosaic, and is called the hijr. it is included in the tawaf, and two slabs of _verde antico_ within it are called the graves of ishmael and hagar, and are places of acceptable prayer. even the golden or gilded _mizab_ (water-spout) that projects into the hijr marks a place where prayer is heard, and another such place is the part of the west wall close to the yemen corner. the feeling of religious conservatism which has preserved the structural rudeness of the ka'ba did not prohibit costly surface decoration. in mahomet's time the outer walls were covered by a veil (or _kiswa_) of striped yemen cloth. the caliphs substituted a covering of figured brocade, and the egyptian government still sends with each pilgrim caravan from cairo a new kiswa of black brocade, adorned with a broad band embroidered with golden inscriptions from the koran, as well as a richer curtain for the door.[18] the door of two leaves, with its posts and lintel, is of silver gilt. the interior of the ka'ba is now opened but a few times every year for the general public, which ascends by the portable staircase brought forward for the purpose. foreigners can obtain admission at any time for a special fee. the modern descriptions, from observations made under difficulties, are not very complete. little change, however, seems to have been made since the time of ibn jubair, who describes the floor and walls as overlaid with richly variegated marbles, and the upper half of the walls as plated with silver thickly gilt, while the roof was veiled with coloured silk. modern writers describe the place as windowless, but ibn jubair mentions five windows of rich stained glass from irak. between the three pillars of teak hung thirteen silver lamps. a chest in the corner to the left of one entering contained korans, and at the irak corner a space was cut off enclosing the stair that leads to the roof. the door to this stair (called the door of mercy--bab el-rahma) was plated with silver by the caliph motawakkil. here, in the time of ibn jubair, the _maqam_ or standing stone of abraham was usually placed for better security, but brought out on great occasions.[19] the houses of ancient mecca pressed close upon the ka'ba, the noblest families, who traced their descent from kosai, the reputed founder of the city, having their dwellings immediately round the sanctuary. to the north of the ka'ba was the dar el-nadwa, or place of assembly of the koreish. the multiplication of pilgrims after islam soon made it necessary to clear away the nearest dwellings and enlarge the place of prayer around the ancient house. omar, othman and ibn jubair had all a share in this work, but the great founder of the mosque in its present form, with its spacious area and deep colonnades, was the caliph mahdi, who spent enormous sums in bringing costly pillars from egypt and syria. the work was still incomplete at his death in a.d. 785, and was finished in less sumptuous style by his successor. subsequent repairs and additions, extending down to turkish times, have left little of mahdi's work untouched, though a few of the pillars probably date from his days. there are more than five hundred pillars in all, of very various style and workmanship, and the enclosure--250 paces in length and 200 in breadth, according to burckhardt's measurement--is entered by nineteen archways irregularly disposed. after the ka'ba the principal points of interest in the mosque are the well zamzam and the maqam ibrahim. the former is a deep shaft enclosed in a massive vaulted building paved with marble, and, according to mahommedan tradition, is the source (corresponding to the beer-lahai-roi of gen. xvi. 14) from which hagar drew water for her son ishmael. the legend tells that the well was long covered up and rediscovered by 'abd al-mot[t.]alib, the grandfather of the prophet. sacred wells are familiar features of semitic sanctuaries, and islam, retaining the well, made a quasi-biblical story for it, and endowed its tepid waters with miraculous curative virtues. they are eagerly drunk by the pilgrims, or when poured over the body are held to give a miraculous refreshment after the fatigues of religious exercise; and the manufacture of bottles or jars for carrying the water to distant countries is quite a trade. ibn jubair mentions a curious superstition of the meccans, who believed that the water rose in the shaft at the full moon of the month shaban. on this occasion a great crowd, especially of young people, thronged round the well with shouts of religious enthusiasm, while the servants of the well dashed buckets of water over their heads. the maqam of abraham is also connected with a relic of heathenism, the ancient holy stone which once stood on the ma'jan, and is said to bear the prints of the patriarch's feet. the whole legend of this stone, which is full of miraculous incidents, seems to have arisen from a misconception, the maqam ibrahim in the koran meaning the sanctuary itself; but the stone, which is a block about 3 spans in height and 2 in breadth, and in shape "like a potter's furnace" (ibn jubair), is certainly very ancient. no one is now allowed to see it, though the box in which it lies can be seen or touched through a grating in the little chapel that surrounds it. in the middle ages it was sometimes shown, and ibn jubair describes the pious enthusiasm with which he drank zamzam water poured on the footprints. it was covered with inscriptions in an unknown character, one of which was copied by fakihi in his history of mecca. to judge by the facsimile in dozy's _israeliten te mekka_, the character is probably essentially one with that of the syrian safa inscriptions, which extended through the nejd and into the hejaz.[20] _safa and merwa._--in religious importance these two points or "hills," connected by the mas'a, stand second only to the ka'ba. safa is an elevated platform surmounted by a triple arch, and approached by a flight of steps.[21] it lies south-east of the ka'ba, facing the black corner, and 76 paces from the "gate of safa," which is architecturally the chief gate of the mosque. merwa is a similar platform, formerly covered with a single arch, on the opposite side of the valley. it stands on a spur of the red mountain called jebel kuaykian. the course between these two sacred points is 493 paces long, and the religious ceremony called the "sa'y" consists in traversing it seven times, beginning and ending at safa. the lowest part of the course, between the so-called green milestones, is done at a run. this ceremony, which, as we shall presently see, is part of the omra, is generally said to be performed in memory of hagar, who ran to and fro between the two eminences vainly seeking water for her son. the observance, however, is certainly of pagan origin; and at one time there were idols on both the so-called hills (see especially azraqi, pp. 74, 78). _the ceremonies and the pilgrimage._--before islam the ka'ba was the local sanctuary of the meccans, where they prayed and did sacrifice, where oaths were administered and hard cases submitted to divine sentence according to the immemorial custom of semitic shrines. but, besides this, mecca was already a place of pilgrimage. pilgrimage with the ancient arabs was the fulfilment of a vow, which appears to have generally terminated--at least on the part of the well-to-do--in a sacrificial feast. a vow of pilgrimage might be directed to other sanctuaries than mecca--the technical word for it (_ihlal_) is applied, for example, to the pilgrimage to manat (_bakri_, p. 519). he who was under such a vow was bound by ceremonial observances of abstinence from certain acts (e.g. hunting) and sensual pleasures, and in particular was forbidden to shear or comb his hair till the fulfilment of the vow. this old semitic usage has its close parallel in the vow of the nazarite. it was not peculiarly connected with mecca; at taif, for example, it was customary on return to the city after an absence to present oneself at the sanctuary, and there shear the hair (_muh. in med._, p. 381). pilgrimages to mecca were not tied to a single time, but they were naturally associated with festive occasions, and especially with the great annual feast and market. the pilgrimage was so intimately connected with the well-being of mecca, and had already such a hold on the arabs round about, that mahomet could not afford to sacrifice it to an abstract purity of religion, and thus the old usages were transplanted into islam in the double form of the omra or vow of pilgrimage to mecca, which can be discharged at any time, and the hajj or pilgrimage at the great annual feast. the latter closes with a visit to the ka'ba, but its essential ceremonies lie outside mecca, at the neighbouring shrines where the old arabs gathered before the meccan fair. the omra begins at some point outside the haram (or holy territory), generally at tanim, both for convenience sake and because ayesha began the omra there in the year 10 of the hegira. the pilgrim enters the haram in the antique and scanty pilgrimage dress (ihram), consisting of two cloths wound round his person in a way prescribed by ritual. his devotion is expressed in shouts of "labbeyka" (a word of obscure origin and meaning); he enters the great mosque, performs the tawaf and the sa'y[22] and then has his head shaved and resumes his common dress. this ceremony is now generally combined with the hajj, or is performed by every stranger or traveller when he enters mecca, and the ihram (which involves the acts of abstinence already referred to) is assumed at a considerable distance from the city. but it is also proper during one's residence in the holy city to perform at least one omra from tanim in connexion with a visit to the mosque of ayesha there. the triviality of these rites is ill concealed by the legends of the sa'y of hagar and of the tawaf being first performed by adam in imitation of the circuit of the angels about the throne of god; the meaning of their ceremonies seems to have been almost a blank to the arabs before islam, whose religion had become a mere formal tradition. we do not even know to what deity the worship expressed in the tawaf was properly addressed. there is a tradition that the ka'ba was a temple of saturn (shahrastani, p. 431); perhaps the most distinctive feature of the shrine may be sought in the sacred doves which still enjoy the protection of the sanctuary. these recall the sacred doves of ascalon (philo vi. 200 of richter's ed.), and suggests venus-worship as at least one element (cf. herod i. 131, iii. 8; ephr. syr., _op. syr._ ii. 457). to the ordinary pilgrim the omra has become so much an episode of the hajj that it is described by some european pilgrims as a mere visit to the mosque of ayesha; a better conception of its original significance is got from the meccan feast of the seventh month (rajab), graphically described by ibn jubair from his observations in a.d. 1184. rajab was one of the ancient sacred months, and the feast, which extended through the whole month and was a joyful season of hospitality and thanksgiving, no doubt represents the ancient feasts of mecca more exactly than the ceremonies of the hajj, in which old usage has been overlaid by traditions and glosses of islam. the omra was performed by crowds from day to day, especially at new and full moon.[23] the new moon celebration was nocturnal; the road to tanim, the mas'a, and the mosque were brilliantly illuminated; and the appearing of the moon was greeted with noisy music. a genuine old arab market was held, for the wild bedouins of the yemen mountains came in thousands to barter their cattle and fruits for clothing, and deemed that to absent themselves would bring drought and cattle plague in their homes. though ignorant of the legal ritual and prayers, they performed the tawaf with enthusiasm, throwing themselves against the ka'ba and clinging to its curtains as a child clings to its mother. they also made a point of entering the ka'ba. the 29th of the month was the feast day of the meccan women, when they and their little ones had the ka'ba to themselves without the presence even of the sheybas. the central and essential ceremonies of the hajj or greater pilgrimage are those of the day of arafa, the 9th of the "pilgrimage month" (dhu'l hijja), the last of the arab year; and every moslem who is his own master, and can command the necessary means, is bound to join in these once in his life, or to have them fulfilled by a substitute on his behalf and at his expense. by them the pilgrim becomes as pure from sin as when he was born, and gains for the rest of his life the honourable title of hajj. neglect of many other parts of the pilgrim ceremonial may be compensated by offerings, but to miss the "stand" (_woquf_) at arafa is to miss the pilgrimage. arafa or arafat is a space, artificially limited, round a small isolated hill called the hill of mercy, a little way outside the holy territory, on the road from mecca to taif. one leaving mecca after midday can easily reach the place on foot the same evening. the road is first northwards along the mecca valley and then turns eastward. it leads through the straggling village of mina, occupying a long narrow valley (wadi mina), two to three hours from mecca, and thence by the mosque of mozdalifa over a narrow pass opening out into the plain of arafa, which is an expansion of the great wadi naman, through which the taif road descends from mount kara. the lofty and rugged mountains of the hodheyl tower over the plain on the north side and overshadow the little hill of mercy, which is one of those bosses of weathered granite so common in the hejaz. arafa lay quite near dhul-majaz, where, according to arabian tradition, a great fair was held from the 1st to the 8th of the pilgrimage month; and the ceremonies from which the hajj was derived were originally an appendix to this fair. now, on the contrary, the pilgrim is expected to follow as closely as may be the movements of the prophet at his "farewell pilgrimage" in the year 10 of the hegira (a.d. 632). he therefore leaves mecca in pilgrim garb on the 8th of dhu'l hijja, called the day of _tarwiya_ (an obscure and pre-islamic name), and, strictly speaking, should spend the night at mina. it is now, however, customary to go right on and encamp at once at arafa. the night should be spent in devotion, but the coffee booths do a lively trade, and songs are as common as prayers. next forenoon the pilgrim is free to move about, and towards midday he may if he please hear a sermon. in the afternoon the essential ceremony begins; it consists simply in "standing" on arafa shouting "labbeyka" and reciting prayers and texts till sunset. after the sun is down the vast assemblage breaks up, and a rush (technically _ifada_, _daf'_, _nafr_) is made in the utmost confusion to mozdalifa, where the night prayer is said and the night spent. before sunrise next morning (the 10th) a second "stand" like that on arafa is made for a short time by torchlight round the mosque of mozdalifa, but before the sun is fairly up all must be in motion in the second _ifada_ towards mina. the day thus begun is the "day of sacrifice," and has four ceremonies--(1) to pelt with seven stones a cairn (_jamrat al 'aqaba_) at the eastern end of w. mina, (2) to slay a victim at mina and hold a sacrificial meal, part of the flesh being also dried and so preserved, or given to the poor,[24] (3) to be shaved and so terminate the _ihram_, (4) to make the third _ifada_, i.e. go to mecca and perform the tawaf and sa'y (_'omrat al-ifada_), returning thereafter to mina. the sacrifice and visit to mecca may, however, be delayed till the 11th, 12th or 13th. these are the days of mina, a fair and joyous feast, with no special ceremony except that each day the pilgrim is expected to throw seven stones at the _jamrat al 'aqaba_, and also at each of two similar cairns in the valley. the stones are thrown in the name of allah, and are generally thought to be directed at the devil. this is, however, a custom older than islam, and a tradition in azraqi, p. 412, represents it as an act of worship to idols at mina. as the stones are thrown on the days of the fair, it is not unlikely that they have something to do with the old arab mode of closing a sale by the purchaser throwing a stone (biruni, p. 328).[25] the pilgrims leave mina on the 12th or 13th, and the hajj is then over. (see further mahommedan religion.) the colourless character of these ceremonies is plainly due to the fact that they are nothing more than expurgated heathen rites. in islam proper they have no _raison d'etre_; the legends about adam and eve on arafa, about abraham's sacrifice of the ram at thabii by mina, imitated in the sacrifices of the pilgrimage, are clumsy afterthoughts, as appears from their variations and only partial acceptance. it is not so easy to get at the nature of the original rites, which islam was careful to suppress. but we find mention of practices condemned by the orthodox, or forming no part of the moslem ritual, which may be regarded as traces of an older ceremonial. such are nocturnal illuminations at mina (ibn batuta i. 396), arafa and mozdalifa (ibn jubair, 179), and tawafs performed by the ignorant at holy spots at arafa not recognized by law (snouck-hurgronje p. 149 sqq.). we know that the rites at mozdalifa were originally connected with a holy hill bearing the name of the god quzah (the edomite koze) whose bow is the rainbow, and there is reason to think that the _ifadas_ from arafa and quzah, which were not made as now after sunset and before sunrise, but when the sun rested on the tops of the mountains, were ceremonies of farewell and salutation to the sun-god. the statistics of the pilgrimage cannot be given with certainty and vary much from year to year. the quarantine office keeps a record of arrivals by sea at jidda (66,000 for 1904); but to these must be added those travelling by land from cairo, damascus and irak, the pilgrims who reach medina from yanbu and go on to mecca, and those from all parts of the peninsula. burckhardt in 1814 estimated the crowd at arafa at 70,000, burton in 1853 at 50,000, 'abd el-razzak in 1858 at 60,000. this great assemblage is always a dangerous centre of infection, and the days of mina especially, spent under circumstances originally adapted only for a bedouin fair, with no provisions for proper cleanliness, and with the air full of the smell of putrefying offal and flesh drying in the sun, produce much sickness. literature.--besides the arabic geographers and cosmographers, we have ibn 'abd rabbih's description of the mosque, early in the 10th century (_'ikd farid_, cairo ed., iii. 362 sqq.), but above all the admirable record of ibn jubair (a.d. 1184), by far the best account extant of mecca and the pilgrimage. it has been much pillaged by ibn batuta. the arabic historians are largely occupied with fabulous matter as to mecca before islam; for these legends the reader may refer to c. de perceval's _essai_. how little confidence can be placed in the pre-islamic history appears very clearly from the distorted accounts of abraha's excursion against the hejaz, which fell but a few years before the birth of the prophet, and is the first event in meccan history which has confirmation from other sources. see noldeke's version of tabari, p. 204 sqq. for the period of the prophet, ibn hisham and wakidi are valuable sources in topography as well as history. of the special histories and descriptions of mecca published by wustenfeld (_chroniken der stadt mekka_, 3 vols., 1857-1859, with an abstract in german, 1861), the most valuable is that of azraqi. it has passed through the hands of several editors, but the oldest part goes back to the beginning of the 9th christian century. kutbeddin's history (vol. iii. of the _chroniken_) goes down with the additions of his nephew to a.d. 1592. of european descriptions of mecca from personal observation the best is burckhardt's _travels in arabia_ (cited above from the 8vo ed., 1829). _the travels of aly bey_ (badia, london, 1816) describe a visit in 1807; burton's _pilgrimage_ (3rd ed., 1879) often supplements burckhardt; von maltzan's _wallfahrt nach mekka_ (1865) is lively but very slight. 'abd el-razzaq's report to the government of india on the pilgrimage of 1858 is specially directed to sanitary questions; c. snouck-hurgronje, _mekka_ (2 vols., and a collection of photographs, the hague, 1888-1889), gives a description of the meccan sanctuary and of the public and private life of the meccans as observed by the author during a sojourn in the holy city in 1884-1885 and a political history of mecca from native sources from the hegira till 1884. for the pilgrimage see particularly snouck-hurgronje, _het mekkaansche feest_ (leiden, 1880). (w. r. s.) footnotes: [1] a variant of the name makkah is bakkah (_sur._ iii. 90; bakri, 155 seq.). for other names and honorific epithets of the city see bakri, _ut supra_, azraqi, p. 197, yaqut iv. 617 seq. the lists are in part corrupt, and some of the names (kutha and 'arsh or 'ursh, "the huts") are not properly names of the town as a whole. [2] mecca, says one of its citizens, in waqidi (kremer's ed., p. 196, or _muh. in med._ p. 100), is a settlement formed for trade with syria in summer and abyssinia in winter, and cannot continue to exist if the trade is interrupted. [3] the details are variously related. see biruni, p. 328 (e. t., p. 324); asma'i in yaqut, iii. 705, iv. 416, 421; azraqi, p. 129 seq.; bakri, p. 661. jebel kabkab is a great mountain occupying the angle between w. naman and the plain of arafa. the peak is due north of sheddad, the hamlet which burckhardt (i. 115) calls shedad. according to azraqi, p. 80, the last shrine visited was that of the three trees of uzza in w. nakhla. [4] so we are told by biruni, p. 62 (e. t., 73). [5] waqidi, ed. kremer, pp. 20, 21; _muh. in med._ p. 39. [6] the older fairs were not entirely deserted till the troubles of the last days of the omayyads (azraqi, p. 131). [7] this is the cross-road traversed by burckhardt (i. 109), and described by him as cut through the rocks with much labour. [8] istakhri gives the length of the city proper from north to south as 2 m., and the greatest breadth from the jiyad quarter east of the great mosque across the valley and up the western slopes as two-thirds of the length. [9] for details as to the ancient quarters of mecca, where the several families or septs lived apart, see azraqi, 455 pp. seq., and compare ya'qubi, ed. juynboll, p. 100. the minor sacred places are described at length by azraqi and ibn jubair. they are either connected with genuine memories of the prophet and his times, or have spurious legends to conceal the fact that they were originally holy stones, wells, or the like, of heathen sanctity. [10] baladhuri, in his chapter on the floods of mecca (pp. 53 seq.), says that 'omar built two dams. [11] the aqueduct is the successor of an older one associated with the names of zobaida, wife of harun al-rashid, and other benefactors. but the old aqueduct was frequently out of repair, and seems to have played but a secondary part in the medieval water supply. even the new aqueduct gave no adequate supply in burckhardt's time. [12] in ibn jubair's time large supplies were brought from the yemen mountains. [13] the corruption of manners in mecca is no new thing. see the letter of the caliph mahdi on the subject; wustenfeld, _chron. mek._, iv. 168. [14] the exact measurements (which, however, vary according to different authorities) are stated to be: sides 37 ft. 2 in. and 38 ft. 4 in.; ends 31 ft. 7 in. and 29 ft.; height 35 ft. [15] the ka'ba of mahomet's time was the successor of an older building, said to have been destroyed by fire. it was constructed in the still usual rude style of arabic masonry, with string courses of timber between the stones (like solomon's temple). the roof rested on six pillars; the door was raised above the ground and approached by a stair (probably on account of the floods which often swept the valley); and worshippers left their shoes under the stair before entering. during the first siege of mecca (a.d. 683), the building was burned down, the ibn zubair reconstructed it on an enlarged scale and in better style of solid ashlar-work. after his death his most glaring innovations (the introduction of two doors on a level with the ground, and the extension of the building lengthwise to include the hijr) were corrected by hajjaj, under orders from the caliph, but the building retained its more solid structure. the roof now rested on three pillars, and the height was raised one-half. the ka'ba was again entirely rebuilt after the flood of a.d. 1626, but since hajjaj there seem to have been no structural changes. [16] hobal was set up within the temple over the pit that contained the sacred treasures. his chief function was connected with the sacred lot to which the meccans were accustomed to betake themselves in all matters of difficulty. [17] see ibn hisham i. 54, azraki p. 80 ('uzza in batn marr); yakut iii. 705 (otheyda); bar hebraeus on psalm xii. 9. stones worshipped by circling round them bore the name _dawar_ or _duwar_ (krehl, _rel. d. araber_, p. 69). the later arabs not unnaturally viewed such cultus as imitated from that of mecca (yaqut iv. 622, cf. dozy, _israeliten te mekka_, p. 125, who draws very perverse inferences). [18] the old _kiswa_ is removed on the 25th day of the month before the pilgrimage, and fragments of it are bought by the pilgrims as charms. till the 10th day of the pilgrimage month the ka'ba is bare. [19] before islam the ka'ba was opened every monday and thursday; in the time of ibn jubair it was opened with considerable ceremony every monday and friday, and daily in the month rajab. but, though prayer within the building is favoured by the example of the prophet, it is not compulsory on the moslem, and even in the time of ibn batuta the opportunities of entrance were reduced to friday and the birthday of the prophet. [20] see de vogue, _syrie centrale: inscr. sem._; lady anne blunt _pilgrimage of nejd_, ii., and w. r. smith, in the _athenaeum_, march 20, 1880. [21] ibn jubair speaks of fourteen steps, ali bey of four, burckhardt of three. the surrounding ground no doubt has risen so that the old name "hill of safa" is now inapplicable. [22] the latter perhaps was no part of the ancient omra; see snouck-hurgronje, _het mekkaansche feest_ (1880) p. 115 sqq. [23] the 27th was also a great day, but this day was in commemoration of the rebuilding of the ka'ba by ibn jubair. [24] the sacrifice is not indispensable except for those who can afford it and are combining the hajj with the omra. [25] on the similar pelting of the supposed graves of abu lahab and his wife (ibn jubair, p. 110) and of abu righal at mughammas, see noldeke's translation of tabari, 208. mechanics. the subject of mechanics may be divided into two parts: (1) theoretical or abstract mechanics, and (2) applied mechanics. 1. theoretical mechanics historically theoretical mechanics began with the study of practical contrivances such as the lever, and the name _mechanics_ (gr. [greek: ta mechanika]), which might more properly be restricted to the theory of mechanisms, and which was indeed used in this narrower sense by newton, has clung to it, although the subject has long attained a far wider scope. in recent times it has been proposed to adopt the term _dynamics_ (from gr. [greek: dynamis] force,) as including the whole science of the action of force on bodies, whether at rest or in motion. the subject is usually expounded under the two divisions of _statics_ and _kinetics_, the former dealing with the conditions of rest or equilibrium and the latter with the phenomena of motion as affected by force. to this latter division the old name of _dynamics_ (in a restricted sense) is still often applied. the mere geometrical description and analysis of various types of motion, apart from the consideration of the forces concerned, belongs to _kinematics_. this is sometimes discussed as a separate theory, but for our present purposes it is more convenient to introduce kinematical motions as they are required. we follow also the traditional practice of dealing first with statics and then with kinetics. this is, in the main, the historical order of development, and for purposes of exposition it has many advantages. the laws of equilibrium are, it is true, necessarily included as a particular case under those of motion; but there is no real inconvenience in formulating as the basis of statics a few provisional postulates which are afterwards seen to be comprehended in a more general scheme. the whole subject rests ultimately on the newtonian laws of motion and on some natural extensions of them. as these laws are discussed under a separate heading (motion, laws of), it is here only necessary to indicate the standpoint from which the present article is written. it is a purely empirical one. guided by experience, we are able to frame rules which enable us to say with more or less accuracy what will be the consequences, or what were the antecedents, of a given state of things. these rules are sometimes dignified by the name of "laws of nature," but they have relation to our present state of knowledge and to the degree of skill with which we have succeeded in giving more or less compact expression to it. they are therefore liable to be modified from time to time, or to be superseded by more convenient or more comprehensive modes of statement. again, we do not aim at anything so hopeless, or indeed so useless, as a _complete_ description of any phenomenon. some features are naturally more important or more interesting to us than others; by their relative simplicity and evident constancy they have the first hold on our attention, whilst those which are apparently accidental and vary from one occasion to another arc ignored, or postponed for later examination. it follows that for the purposes of such description as is possible some process of abstraction is inevitable if our statements are to be simple and definite. thus in studying the flight of a stone through the air we replace the body in imagination by a mathematical point endowed with a mass-coefficient. the size and shape, the complicated spinning motion which it is seen to execute, the internal strains and vibrations which doubtless take place, are all sacrificed in the mental picture in order that attention may be concentrated on those features of the phenomenon which are in the first place most interesting to us. at a later stage in our subject the conception of the ideal rigid body is introduced; this enables us to fill in some details which were previously wanting, but others are still omitted. again, the conception of a force as concentrated in a mathematical line is as unreal as that of a mass concentrated in a point, but it is a convenient fiction for our purpose, owing to the simplicity which it lends to our statements. the laws which are to be imposed on these ideal representations are in the first instance largely at our choice. any scheme of abstract dynamics constructed in this way, provided it be self-consistent, is mathematically legitimate; but from the physical point of view we require that it should help us to picture the sequence of phenomena as they actually occur. its success or failure in this respect can only be judged a posteriori by comparison of the results to which it leads with the facts. it is to be noticed, moreover, that all available tests apply only to the scheme as a whole; owing to the complexity of phenomena we cannot submit any one of its postulates to verification apart from the rest. it is from this point of view that the question of relativity of motion, which is often felt to be a stumbling-block on the very threshold of the subject, is to be judged. by "motion" we mean of necessity motion relative to some frame of reference which is conventionally spoken of as "fixed." in the earlier stages of our subject this may be any rigid, or apparently rigid, structure fixed relatively to the earth. if we meet with phenomena which do not fit easily into this view, we have the alternatives either to modify our assumed laws of motion, or to call to our aid adventitious forces, or to examine whether the discrepancy can be reconciled by the simpler expedient of a new basis of reference. it is hardly necessary to say that the latter procedure has hitherto been found to be adequate. as a first step we adopt a system of rectangular axes whose origin is fixed in the earth, but whose directions are fixed by relation to the stars; in the planetary theory the origin is transferred to the sun, and afterwards to the mass-centre of the solar system; and so on. at each step there is a gain in accuracy and comprehensiveness; and the conviction is cherished that _some_ system of rectangular axes exists with respect to which the newtonian scheme holds with all imaginable accuracy. a similar account might be given of the conception of time as a measurable quantity, but the remarks which it is necessary to make under this head will find a place later. the following synopsis shows the scheme on which the treatment is based:-- _part 1.--statics._ 1. statics of a particle. 2. statics of a system of particles. 3. plane kinematics of a rigid body. 4. plane statics. 5. graphical statics. 6. theory of frames. 7. three-dimensional kinematics of a rigid body. 8. three-dimensional statics. 9. work. 10. statics of inextensible chains. 11. theory of mass-systems. _part 2.--kinetics._ 12. rectilinear motion. 13. general motion of a particle. 14. central forces. hodograph. 15. kinetics of a system of discrete particles. 16. kinetics of a rigid body. fundamental principles. 17. two-dimensional problems. 18. equations of motion in three dimensions. 19. free motion of a solid. 20. motion of a solid of revolution. 21. moving axes of reference. 22. equations of motion in generalized co-ordinates. 23. stability of equilibrium. theory of vibrations. part i.--statics § 1. _statics of a particle._--by a _particle_ is meant a body whose position can for the purpose in hand be sufficiently specified by a mathematical point. it need not be "infinitely small," or even small compared with ordinary standards; thus in astronomy such vast bodies as the sun, the earth, and the other planets can for many purposes be treated merely as points endowed with mass. a _force_ is conceived as an effort having a certain direction and a certain magnitude. it is therefore adequately represented, for mathematical purposes, by a straight line ab drawn in the direction in question, of length proportional (on any convenient scale) to the magnitude of the force. in other words, a force is mathematically of the nature of a "vector" (see vector analysis, quaternions). in most questions of pure statics we are concerned only with the _ratios_ of the various forces which enter into the problem, so that it is indifferent what _unit_ of force is adopted. for many purposes a gravitational system of measurement is most natural; thus we speak of a force of so many pounds or so many kilogrammes. the "absolute" system of measurement will be referred to below in part ii., kinetics. it is to be remembered that all "force" is of the nature of a push or a pull, and that according to the accepted terminology of modern mechanics such phrases as "force of inertia," "accelerating force," "moving force," once classical, are proscribed. this rigorous limitation of the meaning of the word is of comparatively recent origin, and it is perhaps to be regretted that some more technical term has not been devised, but the convention must now be regarded as established. [illustration: fig. 1.] the fundamental postulate of this part of our subject is that the two forces acting on a particle may be compounded by the "parallelogram rule." thus, if the two forces p,q be represented by the lines oa, ob, they can be replaced by a single force r represented by the diagonal oc of the parallelogram determined by oa, ob. this is of course a physical assumption whose propriety is justified solely by experience. we shall see later that it is implied in newton's statement of his second law of motion. in modern language, forces are compounded by "vector-addition"; thus, if we draw in succession vectors [->hk], [->kl] to represent p, q, the force r is represented by the vector [->hl] which is the "geometric sum" of [->hk], [->kl]. by successive applications of the above rule any number of forces acting on a particle may be replaced by a single force which is the vector-sum of the given forces: this single force is called the _resultant_. thus if [->ab], [->bc], [->cd] ..., [->hk] be vectors representing the given forces, the resultant will be given by [->ak]. it will be understood that the figure abcd ... k need not be confined to one plane. [illustration: fig. 2.] if, in particular, the point k coincides with a, so that the resultant vanishes, the given system of forces is said to be in _equilibrium_--i.e. the particle could remain permanently at rest under its action. this is the proposition known as the _polygon of forces_. in the particular case of three forces it reduces to the _triangle of forces_, viz. "if three forces acting on a particle are represented as to magnitude and direction by the sides of a triangle taken in order, they are in equilibrium." a sort of converse proposition is frequently useful, viz. if three forces acting on a particle be in equilibrium, and any triangle be constructed whose sides are respectively parallel to the forces, the magnitudes of the forces will be to one another as the corresponding sides of the triangle. this follows from the fact that all such triangles are necessarily similar. [illustration: fig. 3.] as a simple example of the geometrical method of treating statical problems we may consider the equilibrium of a particle on a "rough" inclined plane. the usual empirical law of sliding friction is that the mutual action between two plane surfaces in contact, or between a particle and a curve or surface, cannot make with the normal an angle exceeding a certain limit [lambda] called the _angle of friction_. if the conditions of equilibrium require an obliquity greater than this, sliding will take place. the precise value of [lambda] will vary with the nature and condition of the surfaces in contact. in the case of a body simply resting on an inclined plane, the reaction must of course be vertical, for equilibrium, and the slope [alpha] of the plane must therefore not exceed [lambda]. for this reason [lambda] is also known as the _angle of repose_. if [alpha] > [lambda], a force p must be applied in order to maintain equilibrium; let [theta] be the inclination of p to the plane, as shown in the left-hand diagram. the relations between this force p, the gravity w of the body, and the reaction s of the plane are then determined by a triangle of forces hkl. since the inclination of s to the normal cannot exceed [lambda] on either side, the value of p must lie between two limits which are represented by l1h, l2h, in the right-hand diagram. denoting these limits by p1, p2, we have p1/w = l1h/hk = sin ([alpha] - [lambda])/cos ([theta] + [lambda]), p2/w = l2h/hk = sin ([alpha] + [lambda])/cos ([theta] - [lambda]). it appears, moreover, that if [theta] be varied p will be least when l1h is at right angles to kl1, in which case p1 = w sin ([alpha] - [lambda]), corresponding to [theta] = -[lambda]. [illustration: fig. 4.] just as two or more forces can be combined into a single resultant, so a single force may be _resolved_ into _components_ acting in assigned directions. thus a force can be uniquely resolved into two components acting in two assigned directions in the same plane with it by an inversion of the parallelogram construction of fig. 1. if, as is usually most convenient, the two assigned directions are at right angles, the two components of a force p will be p cos [theta], p sin [theta], where [theta] is the inclination of p to the direction of the former component. this leads to formulae for the analytical reduction of a system of coplanar forces acting on a particle. adopting rectangular axes ox, oy, in the plane of the forces, and distinguishing the various forces of the system by suffixes, we can replace the system by two forces x, y, in the direction of co-ordinate axes; viz.-- x = p1 cos [theta]1 + p2 cos [theta]2 + ... = [sigma](p cos [theta]), } y = p1 sin [theta]1 + p2 sin [theta]2 + ... = [sigma](p sin [theta]). } (1) these two forces x, y, may be combined into a single resultant r making an angle [phi] with ox, provided x = r cos [phi], y = r sin [phi], (2) whence r2 = x2 + y2, tan [phi] = y/x. (3) for equilibrium we must have r = 0, which requires x = 0, y = 0; in words, the sum of the components of the system must be zero for each of two perpendicular directions in the plane. [illustration: fig. 5.] a similar procedure applies to a three-dimensional system. thus if, o being the origin, [->oh] represent any force p of the system, the planes drawn through h parallel to the co-ordinate planes will enclose with the latter a parallelepiped, and it is evident that [->oh] is the geometric sum of [->oa], [->an], [->nh], or [->oa], [->ob], [->oc], in the figure. hence p is equivalent to three forces pl, pm, pn acting along ox, oy, oz, respectively, where l, m, n, are the "direction-ratios" of [->oh]. the whole system can be reduced in this way to three forces x = [sigma] (pl), y = [sigma] (pm), z = [sigma] (pn), (4) acting along the co-ordinate axes. these can again be combined into a single resultant r acting in the direction ([lambda], [mu], [nu]), provided x = r[lambda], y = r[mu], z = r[nu]. (5) if the axes are rectangular, the direction-ratios become direction-cosines, so that [lambda]2 + [mu]2 + [nu]2 = 1, whence r2 = x2 + y2 + z2. (6) the conditions of equilibrium are x = 0, y = 0, z = 0. § 2. _statics of a system of particles._--we assume that the mutual forces between the pairs of particles, whatever their nature, are subject to the "law of action and reaction" (newton's third law); i.e. the force exerted by a particle a on a particle b, and the force exerted by b on a, are equal and opposite in the line ab. the problem of determining the possible configurations of equilibrium of a system of particles subject to extraneous forces which are known functions of the positions of the particles, and to internal forces which are known functions of the distances of the pairs of particles between which they act, is in general determinate. for if n be the number of particles, the 3n conditions of equilibrium (three for each particle) are equal in number to the 3n cartesian (or other) co-ordinates of the particles, which are to be found. if the system be subject to frictionless constraints, e.g. if some of the particles be constrained to lie on smooth surfaces, or if pairs of particles be connected by inextensible strings, then for each geometrical relation thus introduced we have an unknown reaction (e.g. the pressure of the smooth surface, or the tension of the string), so that the problem is still determinate. [illustration: fig. 6.] [illustration: fig. 7.] the case of the _funicular polygon_ will be of use to us later. a number of particles attached at various points of a string are acted on by given extraneous forces p1, p2, p3 ... respectively. the relation between the three forces acting on any particle, viz. the extraneous force and the tensions in the two adjacent portions of the string can be exhibited by means of a triangle of forces; and if the successive triangles be drawn to the same scale they can be fitted together so as to constitute a single _force-diagram_, as shown in fig. 6. this diagram consists of a polygon whose successive sides represent the given forces p1, p2, p3 ..., and of a series of lines connecting the vertices with a point o. these latter lines measure the tensions in the successive portions of string. as a special, but very important case, the forces p1, p2, p3 ... may be parallel, e.g. they may be the weights of the several particles. the polygon of forces is then made up of segments of a vertical line. we note that the tensions have now the same horizontal projection (represented by the dotted line in fig. 7). it is further of interest to note that if the weights be all equal, and at equal horizontal intervals, the vertices of the funicular will lie on a parabola whose axis is vertical. to prove this statement, let a, b, c, d ... be successive vertices, and let h, k ... be the middle points of ac, bd ...; then bh, ck ... will be vertical by the hypothesis, and since the geometric sum of [->ba], [->bc] is represented by 2[->bh], the tension in ba: tension in bc: weight at b as ba: bc: 2bh. [illustration: fig. 8.] the tensions in the successive portions of the string are therefore proportional to the respective lengths, and the lines bh, ck ... are all equal. hence ad, bc are parallel and are bisected by the same vertical line; and a parabola with vertical axis can therefore be described through a, b, c, d. the same holds for the four points b, c, d, e and so on; but since a parabola is uniquely determined by the direction of its axis and by three points on the curve, the successive parabolas abcd, bcde, cdef ... must be coincident. § 3. _plane kinematics of a rigid body._--the ideal _rigid body_ is one in which the distance between any two points is invariable. for the present we confine ourselves to the consideration of displacements in two dimensions, so that the body is adequately represented by a thin lamina or plate. [illustration: fig. 9.] the position of a lamina movable in its own plane is determinate when we know the positions of any two points a, b of it. since the four co-ordinates (cartesian or other) of these two points are connected by the relation which expresses the invariability of the length ab, it is plain that virtually three independent elements are required and suffice to specify the position of the lamina. for instance, the lamina may in general be fixed by connecting any three points of it by rigid links to three fixed points in its plane. the three independent elements may be chosen in a variety of ways (e.g. they may be the lengths of the three links in the above example). they may be called (in a generalized sense) the _co-ordinates_ of the lamina. the lamina when perfectly free to move in its own plane is said to have _three degrees of freedom_. [illustration: fig. 10.] by a theorem due to m. chasles any displacement whatever of the lamina in its own plane is equivalent to a rotation about some finite or infinitely distant point j. for suppose that in consequence of the displacement a point of the lamina is brought from a to b, whilst the point of the lamina which was originally at b is brought to c. since ab,