in geometry, denotes the contact between any curve and its osculatory circle; that is, the circle of the same curvature with the given curve, at the point of contact or of osculation. See Involution in this Suppl.

Osculation also means the point of concourse of two branches of a curve which touch each other. For example, if the equation of a curve be $y = \sqrt{x} + \sqrt{-x}$, it is easy to see that the curve has two branches touching one another at the point where $x = 0$, because the roots have each the signs $+$ and $-$.

OUADELIM and LABDELEBA, two tribes of Arabs inhabiting the Sahara or Great Desert of Africa, of whom almost nothing was known to Europeans till the publication of Brosson's narrative of his shipwreck and captivity among the latter tribe. He describes the Ouadelim and Labdelleba as the most formidable of all the interior tribes of Arabia, and as often extending their ravages to the very gates of Morocco. "Their hordes (he says) are frequently intermingled with those of the Roufege, Rathidium, Chelus, Tucanois, and Ouadeli tribes, as they have no distinct boundaries, and change their habitations as the desert affords pasturage and water. They are tall, handsome, stout, and vigorous men. Their hair is bristled, and their nails, which they often use in battle, as long as claws; large hanging ears and a long beard give them a stern ferocious air. The Ouadelim in particular are fierce, arrogant, and warlike, but soon dispirited by obstinate resistance, especially when they have not a decided superiority in numbers. In their hordes they lodge by families, in tents which are covered with a thick cloth of camel's hair, which the women spin and weave upon a loom so small, that they work sitting on the ground. The furniture of their tents consist of two large sacks of leather, in which they keep old clothes and pieces of old iron, three or four goat skins for holding milk and water, two large stones for grinding their barley, a smaller one for driving the pins of their tents, an oxier containing which serves for a bed, a thick carpet for a covering, a small kettle, and some wooden dishes, with pack-saddles for their camels. The person who, besides these articles, possesses a few horses, camels, sheep, and goats, is reckoned wealthy, as there are many Arabs who only possess sheep and goats. Except for eyes and the cholera, they are subject to few endemic diseases. The first disorder is caused by the reflection of light from the burning sands of the desert, the other proceeds from the verdigrase which contaminates all their victuals. Their kettles are not tinned, and never washed, so that they are quite crouded over with verdigrase, the violence of which is probably diminished by the quantity of milk they use. When they reside long in one place, they sometimes plough the spots which are moistened by the rain, and sprinkle them with seed in a careless manner. Plentiful crops are often thus produced; but instead of waiting till the grain attains maturity, they cut it down, and dry it over hot cinders. Treachery and perfidy are the innate vices of the Arabs; affiliations are frequent; no man trusts the promise of another; no man makes a written agreement, as the poignard cancels all bonds and obligations. The men often relate their exploits to each other; the embellishing of a story is succeeded by a charge of falsehood, and the poignard solves every difficulty. The ancient rites of hospitality, however, are practised among these tribes in their utmost extent. The Arab, who in the field is a rapacious plunderer, becomes liberal and generous as soon as he enters his tent. War is only a species of rapine, and the victory is decided at the first shock. The Arab is devoid of languorous courage; he attacks only to plunder, and never thinks that booty is to be put in competition with his life. When the battle is ended, each party makes graves for the slain, and enclose the tombs with mounds of stones. The ages of the warriors are denoted by the space of ground which the grave occupies, and the funeral procession is closed by the howls of the females.

"The women never assume the name of their husbands, and never eat with them at meals. They are faithful to their husbands, and cannot be divorced except by the decree of the seniors of the horde. The Arabs display their opulence by the ornaments of their women, whose ears, arms, and legs, are generally adorned with rings of gold and silver. An Arab beauty must have long teeth shooting out of her mouth, a body extremely thick, and limbs of the longest size. At the birth of a son, every woman, to testify her joy, black- Oua [320] Oua

Oudelimsens her face for 40 days. At the birth of a daughter, she only daubs the half of her face during the space of 20 days. A mother treats her son with the same respect as her husband, almost as soon as he is able to walk; she prepares his food, serves him, and eats when he has finished his repast. In the education of their young men, the most important acquisitions are dexterity in the use of the poignard, skill in embowelling their enemies with their long nails, and a plausible air in uttering a falsehood. More rude and ferocious than the tribes whose territories lie upon the shore of the sea, the Labdeffeba and Oudelims Arabs are also more confined and illiberal in their ideas, not only believing that they are the first nation in the world, but fancying that the sun rises only for them. Briffon relates, that some of them expressed this idea in unequivocal terms. "Behold (said they) that luminary, which is unknown in thy country. During the night, thou art not enlightened, as we are, by that heavenly body, which regulates our days and our nights. His children (the stars) point out to us the hours of prayer. You have neither trees nor camels, sheep, goats, nor dogs. Are your women similar to ours?" "How long didst thou remain in the womb of thy mother (said another)?"

"As long (replied Briffon) as thou in that of thine." Indeed (said a third, counting the fingers and toes of the Frenchman) he is made like us; he differs only in his colour and language." "Do you sow barley in your houses?" said the Arabs, alluding to the ships of the Europeans. "No (said Briffon), we sow our fields almost in the same season as you." "How! (cried several) do you inhabit the earth? we believed that you were born and lived upon the sea!" These Arabs, according to the Turkish proverb, believe that all the world is like their father's house: unacquainted with the manners of other nations, and unaccustomed to reflect upon the causes of national character, every variation from their own customs appears not only ridiculous, but monstrous; every difference of opinion not only absurd, but criminal. This ignorance of the Arabs, conjoined with their local and religious prejudices, enables us to account for the insulting treatment which Briffon and his companions received, without having recourse to inherent depravity of nature." That treatment was indeed shocking.

Briffon had surrendered himself, on his shipwreck, to Sidi Mahomet, a Talbe or priest of the tribe of Labdeffeba. During the absence of the priest, the Labdeffeba, who guarded the captives, were attacked and maltreated by a party of the Oudelims, and during the battle which ensued, Briffon had almost lost his life. Instead of compassionating his forlorn situation, the women threw sand into his eyes, as they said, to dry his eyelids. The Arabs, into whose hands he had fallen, had only come down to the sea-coast to gather wild grain, three days before the shipwreck; and to preserve their booty, they immediately retreated to the interior part of the desert. A guide preceded the hordes, to place at intervals small pyramids of stone, to direct their course, at a distance from every hostile tribe. After passing some very high mountains, wholly covered with small greyish pebbles as sharp as flints, they descended into a sandy plain overspread with thorns and thistles. When Briffon was unable to walk, on account of the bleeding of his feet, he was mounted on a camel; the bristly hair and hard feet of which Oude soon excoriated him so much, that the blood run copiously down its flanks. By throwing heated stones into a wooden vessel, filled with barley meal, diluted with water procured on the seashore, preserved in a goat's skin, and mixed with pitch to prevent putrefaction, the Arabs prepared a kind of soup, which they kneaded with their hands, and ate unchewed. They roasted a goat in heated sand, ate its fat raw; and, after having devoured the flesh, gnawed the bones, and scraped them with their nails, threw them to Briffon and his companions, desiring them to eat quickly, and load the camels, that the journey might not be impeded. Proceeding eastward, they crossed a vast plain, covered with small stones white as snow, round and flat as a lentil, where not a single plant was produced. The earth beneath their feet resounded dull and hollow, and the small stones pricked them like sparks of fire. The reflection of the rays of the sun from the sand was scorching; the atmosphere was loaded with a red vapour, and the country appeared as if filled with flaming volcanoes. Neither birds nor insects could be seen in the air. The profound silence was frightful. If a gentle breeze ever arose, it produced extreme languor, chapping of the lips, burning heat of the skin, with small irritating pimples. This plain was even traversed by wild beasts. After traversing this plain, they entered another, where the wind had thrown up furrows the sand, which was of a reddish colour. On the tops of the furrows grew a few sweet-scented plants, which were devoured by the camels. On quitting this sandy plain, they entered a valley surrounded by mountains, where the soil was white and flinty, and where they found water of a noxious smell, covered with green moss, and soon after discovered a horde of the friendly tribe Rouffye.

After another journey of fifteen days, they arrived at the tents of the Labdeffeba horde, to which Sidi Mahomet belonged. The tents pitched among thick bushy trees, and the numerous flocks feeding along the sides of the hills, presented at a distance an aspect of happiness and pastoral simplicity. On approaching near, the trees of beautiful green foliage proved to be only old gnarled stumps, almost void of branches, so encircled with thorns that their shade was inaccessible. The women approached, with loud cries and the most fawning servility, to welcome their tyrants, to throw stones at the Christians, and spit in their faces, while the children imitated the example of their mothers. Briffon, who endeavoured to ingratiate himself with his master's favourite, not only failed in this, but incurred her implacable resentment, through his irritability, which to the Arab women seemed extremely to resemble petulance. During his residence with Sidi Mahomet, the hardships he endured were almost incredible. With the excessive heat, the milk of the sheep, goats, and camels, diminished, and then the dogs fared better than the Christians, who were forced to subsist on wild herbs and raw snails. When the rains fell, and the least prelude made the water to spring up through the sandy soil, the Christians slept behind a bush, sheltered on the bare ground. Briffon and his master sometimes reasoned about religion; when the latter always answered the harangues of the former by declaring, that he preferred a bowl of churned milk to such absurdities.