one of the simple earths. See Chemistry in this Supplement.

AMICABLE NUMBERS have been defined, and the first pair of them given in the Encyclopaedia. The second pair of amicable numbers are 17296 and 18416; and the third pair are 936584 and 9437056.

Dr Hutton informs us, that these three pairs of amicable numbers, with the properties from which they re- To find the first pair, he puts $4x$ and $4yz$, or $a^2x$ and $a^2yz$ for the two numbers where $a = 2$; then making each of these equal to the sum of the aliquot parts of the other, gives two equations, from which are found the values of $x$ and $y$, and consequently assuming a proper value for $y$, the two amicable numbers themselves $4x$ and $4yz$.

In like manner for the other pairs of such numbers; in which he finds it necessary to assume $16x$ and $16yz$, or $a^3x$ and $a^3yz$ for the second pair, and $128x$ and $128yz$ or $a^4x$ and $a^4yz$ for the third pair.

Schooten then gives this practical rule, from Descartes, for finding amicable numbers, viz. assume the number 2, or some power of the number 2, such that if unity or 1 be subtracted from each of these three following quantities, viz. from 3 times the assumed number, also from 6 times the assumed number, and from 18 times the square of the assumed number, the three remainders may be all prime numbers; then the last prime number being multiplied by double the assumed number, the product will be one of the amicable numbers sought, and the sum of its aliquot parts will be the other.

That is, if $a$ be put = the number 2, and $n$ some integer number, such that $3a^n - 1$, and $6a^n - 1$, and $18a^{2n} - 1$ be all three prime numbers; then is $18a^{2n} - 1 \times 2a^n$ one of the amicable numbers; and the sum of its aliquot parts is the other.

AMSTERDAM and St. Paul, are two islands in the South Sea, lying in the same degree of longitude, and generally confounded with each other. The Dutch navigators have given the name of Amsterdam to the northern, and of St. Paul to the southern island, and Captain Cook conforms to that appellation. Most other English navigators, and particularly Messrs Cox and Mortimer, with Sir George Staunton, reverse the names, calling the southern island Amsterdam, and the other St. Paul. At this southern island the Lion man of war stopped on her voyage to China with Lord Macartney, the late ambassador to the court of Pekin, which gave an opportunity to the men of science in the train of the ambassador to examine the island with more skill and attention than probably it had ever been examined before.

Dr Gillan, who was appointed physician to the embassy, as well for his knowledge of chemistry as for his medical skill, is confident that the island of Amsterdam is the product of subterraneous fire, as it bears in every part of it evident marks of volcanic eruption. "On the west and south-west sides (says he) there are four small cones regularly formed, with craters in their centres, in which the lava and other volcanic substances have every appearance of recent formation. The heat continues still so great, and such a quantity of elastic vapours issues through numberless crevices, that there can be no doubt of their having been very lately in a state of eruption. In a thermometer placed upon the surface, the quicksilver rose constantly to 180 degrees, and when sunk a little into the ashes, it advanced to 212 degrees. It certainly would have risen still higher; but the scale being graduated only to the point of boiling water, and the length of the tube proportioned to that extent, the thermometer was immediately withdrawn, lest the increasing expansion of the quicksilver should burst the glass. The ground was felt tremulous under the feet; a stone thrown violently upon it returned a hollow sound; and the heat was so intense for a considerable distance around, that the foot could not be kept for a quarter of a minute in the same position without being scorched. But the great crater on the eastern side, now full of water, is by far the largest here, or perhaps elsewhere, and is of an astonishing size, considerably exceeding in diameter those of Etna or Vesuvius. The quantity of matter to be thrown up, which required to wide an orifice for its passage, and the force with which such matter was impelled, in order to overcome the resistance of the superincumbent earth and sea, must have been indeed prodigious.

This vast crater, according to the usual method of computing the antiquity of volcanoes, must have been formed at a very remote period. The lava all around its sides is much decomposed, and has mouldered into dust, which lies on the surface in many parts to a considerable depth. The decomposition has supplied a rich soil for the long grass growing on the sides of the crater, and has even spread over most parts of the island. The fibrous roots of the grass, extending in all directions through the decomposed lava and volcanic ashes, and mixed in a decaying state with the vegetable mould, produced from the annual putrefaction of the leaves and stalks, have formed a layer of soil several feet deep all over the island. But as it has nothing except its own weight to compress it together, it is of a light spongy texture, with very little cohesion, and in many places furrowed and intersected by the summer rains, and the torrents occasioned by the melting of the snow which lies upon it in the winter, from three to four feet thick, in all those places where the subterraneous heat is not great enough to prevent its accumulation. In some parts these furrows and cavities are deeper than the level of the common channel; hence they serve the purpose of small natural reservoirs. The water flows into them from all the neighbouring ground; and as their sides are shaded, and almost covered over by the leaves of the long grass, growing from their edges in opposite directions, the rays of the sun are excluded, and very little is lost by evaporation. These reservoirs, however, are very small, and but few in number; the largest could not contain more than three or four hogsheads of water; and there is none else to be found, except in the springs on the sides of the large crater.

The soil everywhere being light and spongy, and full of holes, formed in it by tea-birds for nests, is very troublesome to walk upon; the foot breaks through the surface, and sinks deep at every step; a circumstance which renders the journey across the island uncommonly fatiguing, although it be scarcely three miles from the edge of the great crater to the opposite west side. There is one place near the centre of the island, extending about 200 yards in length, and somewhat less in breadth, where particular caution is necessary in walking over it. From this spot a hot fresh spring is supposed to derive its source, finding its way through the interstices of the lava to the great crater, and bursting out a little above the water covering its bottom. The heat in this upper spot is too great to admit of vegetation. The surface is covered with a kind of mud or paille formed from the ashes, moistened by steam constantly rising from below. When the mud is removed, the vapour issues forth with violence, and in some parts copiously. This mud is so hot, that a gentleman who inadvertently stepped into it, had his foot severely scalded by it. The same causes which have prevented vegetation on this spot, have had the same effect on the four cones recently thrown up. Their surfaces are covered with ashes only; nor is there the least appearance even of moss on the surrounding lava, for the production of which there does not appear to have elapsed a sufficient length of time since the cones were formed: but this is not the case with the lava of the great primary crater; for in those parts of it where the edges are more perpendicular, and where consequently the mouldering decomposed earth, having no basis to support it, slides down the sides of the rock, pretty long moss was generally found growing upon it. All the springs or reservoirs of hot water, except one only, were brackish. One spring derives its source from the high ground and ridges of the crater. The water in it, instead of boiling upwards through the stones and mud, as in the other springs, flows downward with a considerable velocity, in a small collected stream. Its temperature has been found not to exceed 122 degrees. The hand could be easily kept in it for a considerable time. It is a pretty strong chalybeate. The sides of the rock whence it issues, and of the cavity into which it falls, are incrusted with ochre deposited from it.

"When the great crater is viewed from the high ground, it appears to have been originally a perfect circle, but to have been encroached upon by the sea on the eastern side, where the flood tide strikes violently. The rocks of lava which formed the edge of the crater on that side have fallen down. The depth of the water in the crater is about 170 feet, rendering the whole height of the crater, from the bottom to its upper ridge, nearly if not quite 900 feet. The lofty rocks forming this ridge are the highest parts of the island, which seems to have been originally produced by the melted lava flowing down on all sides from hence. Thus there is a gradual slope from the edges of the crater to the sea; and the lava, though very irregular, and lying in mixed rain and confusion immediately around the crater, assumes a more uniform appearance at some distance, layer resting regularly upon layer, with a gradual declivity the whole way down to the sea. This disposition of the layers is particularly observable in the west side, where they happen to terminate in an abrupt precipice. The eruptions that took place at different periods appear here distinctly marked by the different layers that are found with regular divisions between them; the glassy lava being undermost, the compact next, the cellular lava next above, over it the volcanic ashes and lighter substances, and a layer of vegetable mould covering the whole."

The island appears indeed in such a state of volcanic inflammation, that from the ships decks at night were observed, upon the heights of the island, several fires issuing out of the crevices of the earth, more considerable, but in other respects resembling somewhat the nightly flames at Pietra Mala, in the mountains between Florence and Bologna, or those near Bradley in Lancashire, occasioned by some of the coal-pits having taken fire. In the day nothing more than smoke could be perceived.

The length of the island from north to south is upwards of four miles, its breadth from east to west about two miles and a half, and its circumference eleven miles, comprehending a surface of about eight square miles, or 5,120 acres, almost the whole of which is covered with a fertile soil. The island is inaccessible except on the east side, where the great crater forms a harbour, the entrance to which is deepening annually, and might by the aid of art be made fit for the passage of large ships. The tides run in and out at the rate of three miles an hour, and rise perpendicularly eight or nine feet on the full and change of the moon; a northerly wind making the highest tide. The water is eight or ten fathoms deep close to the edge of the crater; and in the basin formed by the crater itself, the variation of the compass was found to be nineteen degrees and fifty minutes westward of the north pole.

On the island, which has no native inhabitants, were found three Frenchmen and two natives of England, who at the end of the American war had emigrated to Botton. The whole five had come last from the Isle of France in the Indian Ocean, and had been left on the island of Amsterdam, about five months before the arrival of the Lion, for the purpose of procuring a cargo of 25,000 seal-skins for the Canton market, which, as they had already procured 8000, they hoped to complete in about ten months more. The vessel which brought them from the Isle of France was gone to Nootka Sound, with a view of bringing a quantity of sea-otter skins to China; and afterwards of calling for the cargo of seal skins at this place, to be carried to China likewise; proceeding thus alternately to Nootka and Amsterdam island as long as the owners should find their account in it.

The seals, whose skins are thus an article of commerce, are found here in greater numbers in the summer than in the winter, when they generally keep in deep water, and under the weeds, which shelter them from the inclemency of the weather. In the summer months they come ashore, sometimes in droves of 800 or 1000 at a time, out of which about 100 are destroyed, that number being as many as five men can skin and peg down to dry in the course of a day. Little of the oil which these animals might furnish is collected, for want of casks to put it in; part of the belt is boiled, and serves those people instead of butter. The seal of Amsterdam is the phoca urina of Linnaeus. The female weighs usually from 70 to 120 pounds, and is from three to five feet in length, but the male is considerably larger. In general they are not shy; sometimes they plunge into the water instantly upon any one's approach, but at other times remain steadily on the rocks, bark, and rear themselves up in a menacing posture; but the blow of a stick upon the nose seemed sufficient to dispatch them. As the skins alone were the objects wanted, the carcases were left on the ground to putrefy at leisure, strewed in such numbers as to render it difficult to avoid treading on them in walking along. The people thus employed were remarkable for the squalor and filth of their persons, clothes, and dwelling; yet none of them seemed delirious of leaving the place before the benefits they came upon should be completed. One of them, an Englishman, who had been a considerable time upon the island on a former adventure, gave but an unfavourable account of the weather during the winter months, which are always boisterous. boisterous, with hail and snow; but in summer he acknowledged it to be very fine.

The tea supplies this island with great varieties of excellent fish, particularly a kind of cod, which was equally relished whether fresh or salted. Cray fish were in such abundance on the bar across the entrance into the crater, that at low water they might be taken with the hand; and at the anchorage of the ships, when baskets, in which were proper baits, were let down into the sea, they were in a few minutes drawn up filled with cray fish. This circumstance is the more extraordinary, that in the same place were found abundance of sharks and dog fish of uncommon size, which are known to be voracious and such enemies to all other fish. The basin of the crater abounds with tench, bream, and perch; and the person who with a hook and line has caught any of these fish in the cold water of the basin, may with a slight motion of his hand let them drop into the adjoining hot spring already mentioned, in which they will be boiled and rendered fit for eating in the space of fifteen minutes. This was often practised by the gentlemen of the embassy, and furnished them at once with a singular amusement and a highly relished repast.

Of all the birds which frequent this island, so extraordinary in its origin, formation, and appearance, not one is common to the same degree of latitude in the northern hemisphere. Of the larger kind were several species of the albatross; on examining one of which, distinguished by the name of exulant, it was found, that instead of having only the rudiments of a tongue, as naturalists generally suppose, it had one equalling half the length of the bill. Another large bird is likewise common here, called the great black petrel, or procellaria equinoctialis of Linnaeus. It is the determined enemy of the albatross, as well as of the blue petrel of Amsterdam, or procellaria fortleri. This blue petrel, which is about the size of a pigeon, constitutes the principal food of the seal-catchers on the island. During the day-time they hide themselves in the ground, in order to escape, if possible, their destroyer the black petrel. At night they come abroad, and thence are termed night birds by the people at Amsterdam; but being fond of flocking to any light, they fall into another snare laid for them by the seal-catchers, who kindle torches to attract them, and then kill them in multitudes. The prettiest of the feathered tribe, inhabiting or visiting Amsterdam, is the silver bird, or sternia biruncula, about the size of a large swallow or swift, with a forked or swallow tail. The bill and legs are of a bright crimson colour, the belly white, and the back and wings of a bluish ash colour. This bird subsists chiefly on small fish, which it picks up as they are swimming over the surface of the water.

This singular island lies in 38° 42' S. Lat. and in 76° 54' E. Long. from Greenwich. St Paul's, or the island lying in sight and to the northward, differed in appearance materially from Amsterdam. It presented no very high land, or any rising in a conic form; and seemed to be overspread with shrubs or trees of a middling size. It was said to abound with fresh water, but to have no good anchorage near it, nor any place of easy landing.—Sir George Staunton's Account of an Embassy to the Emperor of China.

ANACLASTIC curves, a name given by M. de Mairan to certain apparent curves formed at the bottom of a vessel full of water, to an eye placed in the air; or Anaphora the vault of the heavens, seen by refraction through the atmosphere.