place (Comment. de B. G. lib. vi.), he speaks of it as 'the in a mine open to the day. On the borders of the region | people about the present town of Rheims). In another towards the west there are some rich iron mines. The celebrated mineral waters of Spa issue from these slaty rocks. The country of the Ardennes is in general sterile; and even in the best part of it, which constitutes the French department of Ardennes, there is only about a third of the land in cultivation. There are vast heaths and extensive marshes which can only be approached in the three driest months of the year. These heaths are called Fagnes, and the most elevated part of the region on the south-east is called Les Hautes Fagnes. There are extensive forests of oak and beech; more rarely, of alder, ash, and birch. Pines and firs occur but seldom. The people of Belgium, living on the borders of the Ardennes, call them the Neur-Paï, that is, NoirPays, 'black country,' because it contains no limestone, and because the only grains cultivated are rye and dwarf oats. Around the villages there are patches of land which have been brought into cultivation by means of a process of paring and burning, called essartage: it consists in taking off the turf and burning it on the ground, and by this process the soil is rendered capable of yielding three successive crops; the first year, rye, generally of a very good quality; the second year, oats; and the third year, potatoes; but after these crops have been got off the land, it must lie fallow for six, twelve, or even twenty years. Meadows and regularly cultivated lands occur only in the valleys. The rearing of cattle, sheep, and horses, is carried on to a great extent. The mutton is celebrated for its excellence, but the wool is not in such high repute. A great deal of ewe-milk cheese is made. The oxen, sheep, and horses are of a small breed. The hardy and valuable Ardennes ponies and little horses appear to be indigenous. They were as highly esteemed in antient times as they are in the present day; for at the time of the invasion of Gaul by the Romans, the cavalry of the Treviri, in which this particular breed was employed, was esteemed the best in Gaul. ARDENNES, a department in the north of France, on the frontier. It is bounded on the N. and N.E. by the kingdom of Belgium, E. by the department of Meuse, W. by that of Aisne, and S. by that of Maine. Its length is about sixty-five miles N. and S., and its breadth sixty miles from E. to W. Its superficial extent is 1955 English square miles, and the population about 282,000, being about 144 to a square mile. This department is traversed by ridges which may be regarded as remote branches of the Vosges, and which separate the waters of the basin of the Meuse from those of the Seine. The streams which flow from the N.E. slopes of these ridges fall into the Meuse; the Bar (which is navigable for several miles) just below Donchery, the Vence, and the Sormonne, near Mezieres, and the Faux and the Viroin, a considerable way farther down. The Meuse itself traverses the department in a direction S.E. and N.W, nearly parallel to and not very far from the Belgian frontier; it then turns more towards the N., and waters a portion of the French territory which projects into the kingdom of Belgium. It receives the abovementioned streams on its left bank: on the right it receives the Semoy, which has the greater part of its course in the Belgian territory. The Aisne forms an arc in the southern part of the department, flowing in a direction which may be described as, on the whole, E.S.E. and W.N.W.; it receives the Vaux on its left bank from the range of heights above alluded to; and falling into the Oise, far beyond the limits of the department, ultimately joins the Seine. Its navigation begins at Château Portien, a little before it leaves this department. Some of the other feeders of the Oise rise just on the eastern border of Ardennes. The elevations in this department appear from their steep declivities and rugged summits to be more lofty than they really are. They afford excellent slates, equal in quality to those of Angers, though not so deep in colour. (Encyc. Méthod.; Géog. Physique, Art. ARDOISES.) Slate and stone are quarried to a considerable extent. Coal, iron, and some lead, are also worked: the great quantity of wood which the department produces, furnishes fuel for considerable iron works. These heights were once covered with an immense forest. Cæsar (Comment. de B. G. lib. v.) describes it as spreading in vast extent through the middle of the country of the Treviri (people of the diocese of Trèves, now included in the Prussian Grand Duchy of the Lower Rhine), from the river Rhine to the beginning of the territory of the Remi' (the The chief manufactures carried on in this department are of cloth and woollen stuffs, at Sedan and Rethel; cutlery, hardwares, nails, and fire-arms, at Charleville; leather. which is in good repute; hosiery, hats, serges, &c. The chief towns are Mézières, the capital (population 4000), Rethel (population 6000), Rocroy (population 3500 or 4000), Sedan (population 13,000), and Vouziers (popu. lation under 2000); all which are seats of sub-prefects, Charleville (population 8000), which is separated from Mézières only by the Meuse; and Charlemont, with Givet, Notre Dame, and Givet St. Hilaire, which form one town with a population of about 4000. Several of these being on the frontier are fortified,-Mézières, Rocroy, Sedan, and Charlemont with the Givets. For a further account of these towns, see CHARLEVILLE, Charlemont, Meʼzie`res, RETHEL, ROCROY, SEDAN. This department is included in the archbishopric of Rheims, and is under the jurisdiction of the Cour Royale (Assize Court) of Metz. It sends three members to the Chamber of Deputies. (Malte Brun; Balbi; Encyc. Méthodique; Diction. Géog. de la France, &c.) ARDESHIR. [See SASSANIDE] ARDFERT, called antiently ARDART, ARDFEARTBRENN, or ARDBREINN, a decayed city of Ireland, in the barony of Clanmaurice, county of Kerry; 184 English miles S.W. by W. from Dublin, and about four N.N.W. of Tralee. Although now much reduced, its former importance and its episcopal rank entitle it to notice. The see of Ardfert was erected in the fifth century, and was so early united with that of Aghadoe that they now form but one diocese, comprehending the county of Kerry and part of Cork, and containing eighty-eight parishes, and fortynine benefices. In 1663, the united sees were added to that of Limerick, but without incorporation. The chapter of Ardfert consists of five dignitaries, viz., dean, precentor, chancellor, treasurer, and archdeacon, but no prebendaries; only the archdeacon of Aghadoe has a stall. The five dignitaries above-mentioned have the cure of souls in the parish of Ardfert, and contribute each one-fifth to the curate's salary. The parish church serves as the cathedral, and is the relic of a very extensive edifice, the rest of which was demolished in the wars of 1641. The ruins of the nave and choir are twenty-six yards long and ten broad. There are the remains of an aisle on the south side, and there was probably one on the north side, which was rebuilt not long before the demolition of the church in 1641. Towards the west end of the cathedral, there are two detached chapels, said to have belonged to the dignitaries of the cathedral, one of them bearing marks of remote antiquity. Opposite the west door was formerly one of the an As the Roman mile was about11-12ths of our mile, the distance will be about 458 English miles-which is impossible: for it is little more than 300 Roman miles from the Rhine at Strasburg to the coast of Flanders. And Strabo expressly says, the forest was not of the extent which some writers assign to it, mentioning 4000 stadia as the exaggerated dimensious, which are equal to Caesar's measurement. Eight stadia are equivalent to one Roman mile, Ardfert was once the capital of Kerry, and had a university of high repute. The bishops were antiently called bishops of Kerry. St. Brendan, or Brandon, to whom the cathedral was dedicated, erected a sumptuous abbey here in the sixth century, but it was burned, as well as the town, in 1089. The town suffered a similar fate again in 1151 and 1179, on which last occasion the abbey was entirely destroyed. tient round towers (see ANTRIM town) nearly a hundred and 20 broad in the clear-(250 feet long and 24 broad, feet high, and built mostly of a dark kind of marble; but according to Seward, Topog. Hibern., which are probably this fell in the year 1770 or 1771.. The area of the cathe-the exterior dimensions,)-is situated close by the harbour, drai is crowded with tombs, on one of which is the effigy of and washed by the sea on the north end and the east side. a bishop rudely sculptured in relief. On that side there are only spike-holes; but on the west side, or front, are sixteen arched stone doors, alternating with fifteen square windows; there are also three towers, two connected with the building, the third, now a little detached, but which probably at first constituted one extremity, as the remaining two towers occupy the centre and the other end of the building. The whole building has been divided into small apartments in two ranges, one over the other, with a staircase in the centre. The lower rooms are about seven feet high; the upper, six and a half; there is a small water-closet in each of the latter, the drain running down through the wall into the sea. The towers have each three rooms, ten feet square, with broad-flagged floors supported without any timbers. The building is surmounted with a battlement, at least on the side next the sea. Within the demesne formerly belonging to the earls of Glandore and barons of Ardfert (titles now extinct) are the remains of an antient monastery, forming a most picturesque addition to the grounds. These remains, according to Sir R. C. Hoare, who visited them in July, 1806, consist of the tower, nave, and a great part of the cloisters, which are in tolerable preservation. The architecture of the building does not bespeak a very antient date. There is some difference of opinion as to its origin; Smith (Nat. and Civ. Hist. of Kerry) ascribes its foundation to Thomas, Lord of Kerry, in 1253, in which he is followed by Archdale (Monast. Hibern.); others ascribe it to a baron of Kerry, in 1389. It is thought to occupy the site of the former monastery founded by St. Brendan, and was destroyed when the town was burnt in the years 1089 and 1179. There are three fairs in the year. The population amounted, in 1821, to 629 in the town, or 2481 in the whole parish. It was a parliamentary borough before the Union, and sent two members to the Irish House of Commons; it is still governed by a portreeve and twelve burgesses. In 1821, there was an Hibernian Society school of forty boys and twenty-one girls. 52° 19' N. lat., long. 9° 39 W. from Greenwich. Ardfert is so near the sea, that single trees, or even rows, are destroyed by the wind; yet there are fine plantations in the grounds of the late Earl of Glandore. ARDGLASS, a town in Ireland, in the barony of Lecale, county of Down, a short distance E. by N. of the town of Killough, which is 100 miles N. by E. of Dublin. It lies upon the east side of the tongue of land which separates the bay of Killough from that of Ardglass; the road between the two towns leads round the head of the first-named of these bays, a distance of about five miles, but this may be very much shortened by crossing the sands when the tide permits. Ardglass lies on a small rocky bay or creek about 150 fathoms wide, and extending, at high water, 500 fathoms inland, with three or four sandy coves along its shores, divided from each other by rocky ledges. The outer of these ledges on the west side has been built up so as to form a kind of pier, at the extremity of which is a light-house; and as there are always three or four fathoms water at the entrance, it may be run for at night, even at low water. The harbour is, however, far from secure when the southeast wind, the most violent on this coast, sets in. (Report of the Commissioners of Irish Fisheries for 1822.) It is inhabited chiefly by fishermen. The population of the whole parish was only 976 in the year 1821, the inhabitants of the town not being discriminated. It is the centre of one of the districts or stations into which the Irish fisheries are divided. In the year ending the 5th of April, 1830, there were employed within the district 208 sailing and 300 row-boats; 2441 fishermen, and probably about 300 other persons, as fish-curers, net-makers, coopers, sailmakers, and other artificers connected with the fisheries, and depending on them for support. In 1822 there were two packets to the Isle of Man. The harbour has been within a very few years substantially repaired by W. Ogilvy, Esq., and government have lately made a grant towards the erection of a pier. There is in the town a school on the foundation of Erasmus Smith, the school-house for which was built by Mr. Ogilvy: it contained in 1826 about 120 pupils, half of whom were boys and half girls. Ardglass was once a corporate town of considerable importance, both as a seat of commerce and a military post. In the time of Queen Elizabeth it was, next to Newry and Down, the principal place in the county. Some authorities make it the second town for trade in all Ulster, Carrickfergus being the first. Several remains attest its former strength and greatness. A range of buildings 234 feet long This singular erection is termed by the inhabitants the new works,' although they have no tradition as to its use, which, however, its construction seems sufficiently to point out. It appears to have been intended for the secure deposit and sale of the goods of some merchants who came from beyond sea. About ten feet from the south tower of this building is a square tower, forty feet by thirty, (we know not whether these are the inside or the outside dimensions, but we believe them to be the latter,) consisting of two stories, and called Horn Tower, from the quantity of horns of oxen and deer found about it. It is thought to have been the merchants' dining-hall and kitchen, from the fire-places and other marks about it. There are at Ardglass three castles, called King's Castle, Cowed (or Coud) Castle, and Jordan's Castle. The last, though not so large as King's Castle, is a finer building than any of the rest. In the great rebellion of Tyrone (in the reign of Elizabeth) it was defended by Simon Jordan, the owner, for three years, until the garrison was relieved by the Lord Deputy, Mountjoy. The parish of Ardglass is in the union (i.e. united parishes) of Ballyphilip, in the diocese of Down, and ecclesiastical province of Armagh; but it has been erected into a perpetual curacy, and a new church built. The old church of Ardholl was the parish church, but was desecrated by the dreadful massacre of the whole congregation at the Christmas midnight mass by the septs (clans) of the M'Cartanes. (Antient and Present State of the County of Down; Seward's Topogr. Hibernica; Parliamentary Papers.) ARDNAMURCHAN. [See ARGYLESHIRE.] ARDOCH, a village in Scotland, in the district of Strathallan, county of Perth, where there are the remains of a permanent Roman station, supposed to be in the most perfect preservation of any in the island, and the traces of three temporary Roman camps. The station is on the right of the great military road from Stirling through Crieff to the north Highlands, and close upon the little river Knaick or Knaig, a feeder of the Allan, which falls into the Forth. This station is supposed, by General Roy, to be the Lindum of Richard of Cirencester*; and to have been founded by Agricola in one of his northern campaigns, perhaps in the fourth. It was on a road carried by the Romans from the wall erected by them between the Firths of Forth and Clyde into Strathmore beyond the Tay, and which crosses the river Knaig immediately below the station. The accompanying plan, from General Roy's Military Antiquities of the Romans in Britain, will show the great pains taken to strengthen it. Its form, according to the general practice of the Romans, is rectangular; its dimensions are about 500 feet by 430 within the entrenchments; and its four sides nearly face the four cardinal points. On the north and east sides, where the works are most perfect, there are five ditches and six ramparts. From the nature of the ground the direction of the outer rampart varies, but the aggregate breadth of the works on the east side, where intersected by the line A B, is about 180 feet, and that of the works on the north side, where intersected by the line CD, is more than 270 feet. The prætorium, or general's quarter, is near the centre, but not in it; it is a rectangle, and almost a square, having its greater side about 70 feet, but its sides are not parallel to those of the station. On the south side of the latter the works have been much defaced by the process of cultivation, and A monk of Westminster, author of a History and Map of Roman Britain, written about A.D. 1338, the MS, of which was discovered in Denmark in 1757. ARD on the west by the modern military road from Stirling towards Inverness. Three of the gates remain. The entrance at the prætorian gate crosses the entrenchments, not at right angles, but obliquely. There is a road out of the camp on the south side; but whether it coincides with the remaining (decuman) gate is not clear from the plans. The Roman stations and camps had usually four gates: the Prætorian, in front of the prætorium or general's quarters; the Decuman, at the back of the same; and the right and left principal gates. From an inscription on a sepulchral stone dug up at this place, it appears that a body of Spanish auxiliary troops lay in garrison here. The west side of the camp is protected by the river Knaig, the banks of which, as the section shows, are very steep. The level of the camp is sixty feet above the river. The prætorium, which has from time immemorial been called Chapel Hill, has been at some time enclosed with a stone wall, and has the foundations of a house ten yards by seven. The whole station has been of late years enclosed with a high stone wall in order to preserve it. There is said to be on one side of the prætorium a subterraneous passage, supposed to extend under the bed of the river, but the entrance having been closed about 1720, to prevent hares, when pursued, from taking refuge there, it is not known where the passage is. Search has been made for it, but in yain. Previous to its being closed, a man who had been condemned in the baron court of some neighbouring lord, consented, upon condition of pardon, to explore it; but after bringing out some Roman spears, helmets, and bits of bridles and other things, he descended again and was killed by the foul air. The articles brought out were carried off by the duke of Argyle's soldiers after the battle of Shereffmuir in 1715, and were never recovered. The camps are a little way, north of the station on the way to Crieff, and are of different magnitudes. The largest of them has a mean length of 2800 feet, and a mean breadth of 1950, and was calculated to hold between 25,000 and The military road enters the camp by the 26,000 men. south gate, and has levelled half of the small work which covered it, leaving the other half of it standing. On the east rampart of this camp is a small redoubt, on a gentle eminence; the only thing of the kind in the temporary camps of Agricola in these parts. The area of this camp is marshy, and some parts of it appear to have been always so. The second camp is smaller, and its ramparts obliquely intersect those of the last. The north end and part of the east and west sides remain entire. Its length is 1910 feet, and its breadth 1340, and it would contain about 14,000 men, according to the Roman method of encamping. The area is drier than that of the great camp. These camps Roy supposes to have been formed and occupied by Agricola in his sixth campaign; the smaller one after the larger, when he had divided his forces. The part of the rampart of the first included within the second was not levelled. The lower parts of both, where they approach the river Knaig, are now demolished. The third camp is immediately adjacent to the station, and was probably an addition to it. Its mean length is 1060 feet, and its mean breadth 900, so that it would contain about 4000 men. It was stronger than the great camp, and was formed subsequently to it, the works of the great camp having been defaced by its rampart, and the part included within it has been levelled either by the Romans or others since their time. In this part of Scotland are the remains of two other Roman stations, but neither of them are so perfect as that at Ardoch. One of them, at Strageath or Strathgeth, on the river Earn, about six miles and a half N.N.E. of Ardoch, is thought to be the Hierna of Richard of Cirencester; and between this and Ardoch, about two miles and a half from the latter, is a small post called Kaim's Castle, supposed to have been a look-out for both stations, the remains of which are very perfect. The other station, of which only slight vestiges remain, is in the neighbourhood of West Dealgin Ross, near the junction of the rivers Ruagh Huil and Earn, about eight miles and a quarter N.N.W. from Ardoch, and eight and a half W.N.W. from Strageath. Near it are the remains of a small temporary camp, whereof great part of the intrenchments and the four gates (which are covered in a manner singularly curious) remain entire. This station General Roy supposes to be the Victoria of Richard of Cirencester, and the camp that of the ninth legion, which was attacked by the Caledonians in the sixth campaign of Agricola. About half a mile S.W. of Ardoch, at the Grinnan Hill of Keir, is a circular Roman work. (Roy's Military Antiquities of the Romans in North Britain; Sir John Sinclair's Statistical Account of Scotland.) About a mile west of Ardoch was a cairn of extraordinary dimensions, viz., 182 feet in length, 30 feet in sloping height, and 45 feet in breadth at the base. (Gordon's Itinerarium Septentrionale.) The stones have been now mostly carried away to form enclosures for the neighbouring farms; but a large stone coffin, in which was a skeleton seven feet long, has been preserved, together with a few large stones around it. (Sir John Sinclair's Statistical Account of Scotland.) ARDROSSAN, a sea-port and parish in the district of Cunningham, the most northern division of Ayrshire in Scotland. The harbour of Ardrossan was begun in 1806. The port had previously considerable natural advantages, being sheltered by a large island (Horse Island) off the coast. The works were carried on under the auspices of the late earl of Eglinton, who bestowed upon them much trouble and expense. The harbour was to form one outlet of a canal intended to connect the Clyde with this part of the coast, and the projectors seem to have hoped to render Ardrossan the port of Glasgow. The harbour has been for many years in a state to receive shipping, and is considered as one of the safest and most capacious and accessible on the west coast of Scotland. A circular pier of 900 yards* was finished in 1811; but the progress of the wet dock and other works was suspended by Lord Eglinton's death in 1820. The canal (begun in 1807) has never been finished. It has been carried from Glasgow past Paisley to the village of Johnston, a distance of eleven miles, at an expense of 90,000l. A rail-road has been commenced from Ardrossan to the canal, which will thus complete the communication, though not in the manner first designed. Baths have been constructed at Ardrossan, which render it somewhat attractive as a watering-place. The parish has a medium length of six miles. Its greatest breadth is about five miles, and its least not more than three. The kirk is close to the town of Saltcoats, part of which is in this parish. [See SALTCOATS.] The popuArdrossan is in the presbytery lation in 1831 was 3494. of Irvine, and the synod of Glasgow and Ayr. It gives the title of baron to the family of Montgomery, earls of Eglintoun. (Sinclair's Statistical Account of Scotland, &c.) ARDSTRAW, an extensive parish in Ireland, in the county of Tyrone. [See NEWTON STEWART.] ARE, the modern French measure of surface, forming part of the new decimal system adopted in that country after the revolution; it is obtained as follows:-the metre or measure of length, being the forty-millionth part of the whole meridian, as determined by the survey, is 3.2809167 English feet; and the are is a square, the side of which is 10 metres long. The following denominations are also used:Decare Hectare Chilare Myriare Deciare. Centiare Milliare The are is or or is 10 ares. The hectare is generally used in describing a quantity of land. It is 2:4711695 English acres, or 404 hectares make 1000 acres, which disagrees with the first result by less than 1 part out of 50,000. A'REA. This term is a Latin word, and means the same thing as superficies or quantity of surface, but is applied exclusively to plane figures. Thus we say, the surface of a sphere, the area of a triangle,' and the surface of a cube is six times the area of one of its faces. The word is also applied to signify any large open space, or the ground upon which a building is erected; whence, in modern built houses, the portion of the site which is not built upon is commonly called the area. Returning to the mathematical meaning of the term, the measuring unit of every area is the square described upon the measuring unit of length: thus, we talk of the square inches, square feet, square yards, or square miles, which an area contains. And two figures which are similar, as it is called in geometry, that is, which are perfect copies one of the other on different scales, have their areas proportional to the squares of their linear dimensions. That is, suppose a plan of the front of a house to be drawn so that a length of 500 feet would be represented in the picture by one of 3 feet. Then the area in the real front is to the area of the front in the picture in the proportion of 500 times 500 to 3 times 3, or of 250,000 to 9. Similarly, if the real height were 20 times as great as the height in the picture, or in the proportion of 20 to 1, the real area would be to that of the picture as 20 times 20 to once one, or as 400 to 1, that is, the first would be 400 times as great as the second. Any figure which is entirely bounded by straight lines may be divided into triangles, as in the adjoining diagram. B The area of every triangle may be measured separately by either of the following rules; in which the word in italics may mean inches, yards, miles, or any other unit, provided only that it stands for the same throughout. 1. Measure a side, A B, of the triangle ABC, and the perpendicular CD which is let fall upon it from the opposite vertex, both in units. Half the product of A B and C D is the number of square units in the triangle A BC. Thus, if A B be 30 yards, and C D 16 yards, the triangle contains 240 square yards. 2. Measure the three sides, AC, CB, BA, in units; take suspect some error. Two statements of the plans of Mr. Telford, the engineer, the half sum of the three, from it subtract each of the sides, give 600 yards as the intended length of this pier. multiply the four results together, and extract the square There are some ruins of an old castle, the remains of which indicate it to have been of considerable extent. It was in a great degree demolished by Cromwell, who used the stones of it for the erection of the fort of Ayr. This is the statement in the Ency. Britannica, last edition; but we root of the product; this gives the number of square units in the triangle. For instance, let the three sides be 5, 6, and 7 inches; the half sum is 9: which, diminished by the three sides respectively, gives 4, 3, and 2: 9, 4, 3, 2, multiplied together, give 216, the square root of which is 14 7, 14 very nearly. The triangle, therefore, contains about 14 square inches. C R T ACp1, 17q2, 2 q r 3, &c. considering Cp, pq, qr, &c. as straight lines. This will be equivalent to adding half the rectangle, DE, to the sum of the rectangles aforesaid. The practical rule is:-Add all the intermediate ordinates, 1 p, 2 q, &c. to the half sum of the extreme ordinates AC and B D: multiply the total by the common value of A 1, or 1 2, &c. This approximation is the first step of the method of QUADRATURES, which see. The mathematical process of finding the area carries the preceding approximation one step further, and finds what is the limit to which the sum of the inscribed parallelograms approaches nearer and nearer, as the number of divisions of A B is increased. This limit, it is easy to show, is an exact The following rules may be applied in the following cases expression for the area required. If x represent one of -for a parallelogram, multiply A B, a side, by CD, its the lines A1, A2, &c., and y the corresponding line 1 p, perpendicular distance from the opposite side-for a rect-2q, &c., the area of the curve is found by the process of the angie, multiply together adjoining sides, P Q and P R-for integral calculus thus represented a four-sided figure, in which RT and S V are parallel, but TV and R S converge; multiply R S, one of the converging sides, by Y Z, its perpendicular distance from the middle point of the other. When RT and SV are perpendicular to RS, then Y Z is half the sum of R T and Š V. D B P D B Q R To find the area of a circle, multiply the radius OA by itself and the result by 355; then divide by 113. To find the area of the sector OAD B, see ANGLE. To find the area of the portion ABD, find those of the sector O ADB, and the triangle OA B separately, and subtract the second from the first. In all these cases, the result is in the square units corresponding to the linear units in which the measurements were made. The area of a curvilinear figure can only be strictly found by mathematical processes too difficult to be here described, but the following method will give an idea of the principles employed. Let ACDB be a curvilinear figure bounded by the curve CD and the lines CA, A B, BD, of which the first and third are perpendicular to the second. Divide AB into any number of equal parts (eight is here supposed) by the points 1, 2, 3, &c. and construct the accompanying obvious figure by making A p, 1 q, &c. parallelograms. It is plain that the area sought, ACD B, is greater than the sum of the inscribed rectangles, denoted E 2 3 4 5 6 7 B by the letters or numbers at opposite corners, 1C, 2p, 3 g, 4r, 5s, 6 t, 7 u, Bv; and that it is less than the sum of the circumscribing rectangles Ap, 1q, 2r, 38, 4t, 5 u, 6 v, 7 D. Therefore the area sought does not differ from either of these sums by so much as they differ from one another; but the sums differ from one another by the sum of the rectangles Cp, pq, qr, ns, st, tu, uv, v D, which, placed under one another, give the rectangle DE, which is less than D7: consequently neither sum differs from the area sought by so much as D7. But by carrying the division of A B, with which we set out, to a sufficient degree, the area of D7 might have been reduced to any extent which might have been thought necessary; that is, name any fraction of a square inch, however small, and A B can be divided into such a number of equal parts that D7 shall be smaller than that fraction of a square inch. Hence the sum of the inscribed or circumscribed parallelograms may, by dividing the line A B sufficiently, be made as nearly equal to the area as any practical purpose can require. The accuracy of the preceding process will be increased by summing, not the parallelograms, but the figures Sy d x or, in the language of fluxions, fluent of y x A process similar to the preceding is employed by surveyors in measuring a field whose boundaries are curvilinear. [See SURVEYING, Offset.] The investigation of the area of a curve was formerly called the quadrature of the curve (quadratum, a square), because, before the application of arithmetic to geometry, the most convenient method of representing an area was by giving the square to which it is equal. For some practical purposes the following experimental method of finding the above area might suffice. Cut out the figure ABCD in pasteboard (heavy wood or metal would be better). Out of the same pasteboard cut a square inch or other unit; and weigh both the pieces thus cut out accurately. Then the weight of the first piece divided by that of the second will give the number of square units in the area required, if the pasteboard, or other material, be of moderately uniform thickness. A method similar to that of Archimedes (see his Life) might easily be devised. ARE'CA, a genus of palms containing two species, both remarkable for the purposes to which they are applied. Botanically, areca is distinguished by a double membranous sheath in which its bunches of flowers are contained, by its female corollas containing the rudiments of stamens, its calyx being divided into three parts or leaves, and its fruit [Areca catechu.] VOL. II.-2 Q |