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numeral calculation of the perimeters of the inscribed and circumscribed polygons: from which calculation it appears that the perimeter of the circumscribed regular polygon of 192 sides is to the diameter in a less ratio than that of 3} or 3% to 1 ; and that the perimeter of the inscribed polygon of 96 sides is to the diameter in a greater ratio than that of 3}} to 1; and consequently that the ratio of the circumference to the diameter lies between these two ratios. Now the first ratio, of 34 to 1, reduced to whole numbers, gives that of 22 to 7, for 3} : 1 :: 22 : 7; which therefore is nearly the ratio of the circumference to the diameter. From this ratio between the circumference and the diameter, Archimedes computed the approximate area of the circle, and he found that it is to the square of the diameter, as 11 is to 14. He determined also the relation between the circle and ellipse, with that of their similar parts. And it is probable that he likewise attempted the hyperbola; but it is not to be expected that he met with any success, since approximations to its area are all that can be given by the various methods that have since been invented. Beside these figures, he determined the measures of the spiral, described by a point moving uniformly along a right line, the line at the same time revolving with a uniform angular motion; determining the proportion of its area to that of the circumscribed circle, as also the proportion of their sectors. Throughout the whole works of this great man, we every where perceive the deepest design, and the finest invention. He seems to have been, with Euclid, exceedingly careful of admitting into his demonstrations nothing but principles perfectly geometrical and unexceptionable: and although his most general method of demonstrating the relations of curved figures to straight ones, be by inscribing polygons in them : yet to determine those relations, he does not increase the number, and diminish the magnitude, of the sides of the polygon ad infinitum; but from this plain fundamental principle, allowed in Euclid's Elements, (riz. that any quantity may be so often multiplied, or added to itself, as that the result shall exceed any proposed finite quantity of the same kind), he proves that to deny his figures to have the proposed relations would involve an absurdity. And when he demonstrated many geometrical properties, particularly in the parabola, by means of

certain progressions of numbers, whose terms are similar to the inscribed figures; this was still done without considering such series as continued ad infinitum, and then collecting or summing up the terms of such infinite series. There have been various editions of the existing writings of Archimedes. But the most complete of any is the magnificent edition, in folio, lately printed at the Clarendon press, Oxford, 1792. This edition was prepared ready for the press by the learned Joseph Torelli, of Verona, and in that state presented to the university of Oxford. The Latin translation is a new one. Torelli also wrote a preface, a commentary on some of the pieces, and notes on the whole. An account of the life and writings of Torelli is prefixed, by Clemens Sibiliati. And at the end a large appendix is added, in two parts ; the first being a Commentary on Archimedes's paper upon Bodies that float on Fluids, by the Rev. Adam Robertson of Christ Church College; and the latter is a large collection of various readings in the manuscript works of Archimedes, found in the library of the late King of France, and of another at Florence, as collated with the Basil edition abovementioned. ARCHITECTURE is the art of forming dwellings, or erecting buildings of any kind. Animals of acute feelings, exposed to disagreeable extremes of seasons, uncertainties of weather, and to the depredations and attacks of each other, must have a strong desire to shelter and secure themselves. Consequently, those favoured by nature either for digging in the earth or building would, under these pressing circumstances, soon form places of retirement for themselves; and other animals, without such powers, would endeavour to seek such places of shelter as are either furnished by nature itself, or formed by others. Thus birds and insects build themselves nests; many kinds of quadrupeds form subterraneous retreats; and in time of storms cattle flee, and endeavour to shelter themselves among rocks, trees, &c. There can be little doubt but building began first among the brutes; but their modes of working have been uniformly the same from time to time, without improvement. Man, with feelings much more acute than any other animal, and also superior, both from his reasoning powers, and the construction of his frame, in being adapted to lift, remove, shape, and place inanimated matter wherever his mind directs, and from his imitative disposition, would be compelled by pressing necessity to form some kind of habitation where he might breathe the temperate air, amid the summer's heat, or winter's cold, secure himself from the attacks of ferocious animals, and when nature calls he might rest and sleep in ease and security. It is probable that the original habitations of men were natural caverns in the earth, and hollows in the trunks of trees. Also, from the example of brutes, he might excavate the ground; but being disgusted with darkness and damps, taking example from the birds, he would begin to build huts of such materials as the situations would afford: the natural eaverns might suggest the idea of using earth or stones. That the first attempts at building must have been extremely rude there can be little doubt; men, without cutting instruments or tools, could not shape, smooth, break, and join timbers or stones, as they do in the present day; timbers could only be supported by balancing each other, or driving them fast into the ground, or piling stones or other materials around their lower ends, or interlacing them with slender twigs or boughs. It is reasonable to conjecture, that wherever wood is found, the primitive hut would be constructed of a conic figure, not only from its form being the most simple of all solids, but also from the ease with which this covering is made. The builder collecting a few boughs, and perhaps breaking them to determinate lengths, would support them by leaning them against each other at the top, and spreading them out at the bottom, so as to make the interior of sufficient capacity, leaving an aperture on one side for entrance: the interstices he would interweave with smaller branches, and to render it impervious to disagreeable changes, or excesses of the surrounding element, he would plaster the interstices with mud, slime, or clay; such are the wigwams of the North American Indians, and the kraals of the Hottentots and Caffrees, in the present day. It would not be long before the inhabitant saw the inconvenience of the simple conic form, on account of its inclined sides, in preventing him from standing erect at the extremities of the floor. His former dwelling would readily suggest the plan on which he was to build. He might perhaps begin to dispose the timbers upright, and fasten their bottom ends as above, or by setting them upon the ground only, and interweaving interstices in the manner of basket work; or

perhaps by combining both these methods together, so as to make his hut still more durable : in this manner the first walls might have been made, or by collecting the most portable and shapely stones, and rearing a rough wall to a sufficient height: the roof would be constructed of the conic or pyramidal figure, as formerly, and the whole plastered over with mud, or any other tenacious material. As mankind began to associate, they would improve each other by degrees; and having found the use of tools, trunks of trees, divested of their bark and branches, would be used as pillars, and beams or lintels, instead of ramified boughs. In this improved state of joining and cutting the timbers, the beams would no doubt suggest a rectilineal plan instead of the circular one, as beams of the circular form could not be so readily procured as those of the straight form, the triangle being the only figure that includes a space by the fewest sides, it may first have been employed for the plan; but finding this form of building inconvenient, on account of the acuteness of its angles, the rectangle would be adopted in its stead, the hut erected thereon would have the form of a rectangular prism, which figure has been generally retained to the present day, with little variation, by almost the whole inhabitants of the globe, and exactly by those who live in the mildest climates; but in countries liable to rain, pyramidal and wedge-formed roofs have been coustantly in use. From this state of the hut has civil architecture advanced progressively to the present state of improvement. Vetruvius, the most ancient writer of architecture, informs us nearly as above, in the following words: “Mankind began to make themselves coverings with the boughs of trees; some dug caves in the mountains; and others, in imitation of the nests of swallows, with sprigs and loam made shelters which they might lye under; and by observing each others work and turning their thoughts to discover something new, they by degrees improved and made better kinds of habitations; but men being of an imitative and docile nature, glorying in their daily inventions, and shewing one another the houses they had made, they by these endeavours and exertions of their faculties became in time more skilful. “At first for the walls they erected forked stakes, and disposing twigs between them, covered them with loam; others piled up dry clods of clay, binding them togetherwith wood, and to avoid rain and heat, they made a covering with reeds and boughs; but finding that this roof could not resist the winter rains, they made it sloping and pointing at the top, plastering it over with clay, and by that means discharged the rain water. That the origin of things was as above written may be concluded from observing, that to this day some foreign nations construct their dwellings of the same kind of materials, as in Gaul, Spain, Lusitania, and Aquitain, they use oak shingles or straw. The Colchians, in the kingdom of Pontus, where they abound in forests, fix trees in the earth, close together in ranks to the right and left, leaving as much space between them as the length of the trees will permit; upon the ends others are laid transversely, which circumclude the place of habitation in the middle; then at the top the four angles are braced together with alternate beams; and thus the walls, by fixing other trees perpendicularly on these below, may be raised to the height of towers. The interstices which, on account of the coarseness of the materials, remain, are stopped with chips and loam. The roof is also raised by beams laid across from the extreme angles, gradually converging, and rising from the four sides to the middle point at the top, and then covered with boughs and loam. In this manner the barbarians make the testudinal roofs of their towers. The Phrygians, who inhabit a champaign country, being destitute of timbers by reason of the want of forests, select little natural hills, excavate them in the middle, dig an entrance, and widen the space within as much as the nature of the place will permit: above they fix stakes in a pyramidal form, bind them together and cover them with reeds or straw, heaping thereon great piles of earth. This kind of covering renders them very warm in winter and cool in summer; some also cover the roofs of their huts with the weeds of lakes; and thus in all nations and countries the dwellings are formed upon similar principles. At Marseilles we may observe the roofs without tiles, and covered with earth and straw. At Athens the Areopagus is an example of the ancient roofs of loam; at the Capitol also the house of Romulus in the sacred citadel may remind us of the ancient manner of covering our roof with straw. By these examples therefore we may be assured, that the first inventions of building happened in the manner we have related; but at length mankind by daily practice improved, and by re

peatedly exercising their faculties and talents arrived at the full knowledge of the art, those who were most experienced professing themselves artificers. When therefore these things were thus far advanced, as nature had not only given to mankind sense in common with other animals, but had also furnished their minds with judgment and foresight, and had subjected other animals to their power, they from the art of building gradually proceeded to other arts and sciences, and from a savage and rustic way of life became humane and civilized. Then when their minds were thus enlightened, and they became more judicious by experience, and the advancement of the various arts and sciences, they no longer built huts, but founded houses with walls constructed with bricks, stones, or other materials, covering the roofs with tiles.”

history of Architecture.

The origin of architecture is, like that of most other arts, involved in great obscurity. We are informed by Moses that Cain built a city, and called it after the name of his son Enoch; but concerning the mode of constructing the houses, or the quality of the materials, he is quite silent. The same author also informs us that Jabal was the father of such as dwell in tents. In the days of Noah architecture must have arrived at great perfection: to construct the ark of sufficient strength to withstand the tempests raging over the surface of the watery element would require considerable skill in the art of carpentry. Ashur built the cities of Nineveh, Rehoboth Calah, and Resen. The city and tower of Babel were built of well-burnt brick, and slime for mortar. Brick-making must have been well understood then, and perhaps at a period much anterior. Moses does not say, what either the dimensions or figure of the tower was, but that it was the intention of the people to make its top reach unto heaven: this vain design being frustrated by the intervention of the Almighty,the building was left unfinished. Whether this city and tower be the same Babylon and tower as described by Herodotus and Strabo is uncertain; the forme says it was a square building, each side . which at the base was a furlong, consequently half a mile in circumference; from a winding stair, or rather an inclined plane, o went around the exterior, making eight révolutions, the building appeared as if eight stories had been placed one upon the other; each such story was 75 feet high, and consequently the whole height 600 feet: the inclined plane was so broad as to allow carriages to pass each other. From very remote antiquity the Egyptians have been celebrated for their cultivation of architecture among other arts; the ruins of their ancient structures astonish the traveller of the present day, as may be seen in their huge pyramids and proud tombs,which have long outlived the memory of the mighty kings whose ashes they contain: granite temples as extensive as towns, which inclose in their courts or support upon their roofs villages of the modern inhabitants; long avenues of sphinxes, colossal statues, and obelisks. Yet the art of building among them consisted of but few principles, for they did not seem to understand the use of the arch; all the apertures and intercolumns of their walls were linteled with solid stone; the roofs of the chambers of their temples were generally covered with massy slabs for lintels; the ceiling or roof of the passage within the great pyramid is formed of stones in horizontal courses, projecting equally over each other from the two opposite walls to the summit, like inverted flights of steps: the roofs of some of their tombs are indeed arch formed, but these are only excavations cut out of the solid rock. Their walls were built of stones of an enormous size, without cement. The removal and placing of these huge materials would, even at this day, almost bid defiance to the boldest and best constructed of our mechanical inventions, though conducted with all the science of modern times. The stones of their edifices are squared and jointed with the utmost accuracy; the hieroglyphic carvings with which their walls and ceilings are charged are all recessed, but projecting in relief from the bottoms or backs of the recesses. The forms of Egyptian temples and gates are generally truncated rectangular pyramids, crowned with a cove and fillet, or cavetto, as a cornice around the four angles of the sides, and under the cornice project tori from each face. The entrance front of the temples has generally a large rectangular opening, in which are placed columns for supporting the architrave and cornice: over the middle of the door, and upon the linteling architrave, is carved a winged globe: the height of the columns, according to Denon's representation, is from five to six diameters. The colunius have in general little or no diminution, and are frequently placed upon a plinth, from which they sometimes rise in a convexity, forming what is called by workmen a quirk above the

plinth. The shafts of the columns are generally divided into two or more compartments, and sometimes charged with hieroglyphics, as well as the walls and ceilings: the compartments are sometimes also ornamented with vertical reeds, representing a bundle of rods, and separated from each other by annular incisions, or beads, which seem as bandages for tying the rods together. The whole of the compartments are not always reeded: sometimes there are only one or two, and the rest carved with hieroglyphics. The capitals sometimes swell out at the bottom from the upper part of the shaft, and diminish to the top, which is covered with a square projecting abacus; sometimes capitals have vases like the Corinthian order, which rise with a small convexity from the shaft, and change into a large concavity upwards, which as it approaches the top has more and more curvature until it terminates: above the termination it recedes with a convexity to the abacus, which is also recessed within the face of the linteling architrave. Sometimes the capitals are formed by the head of Isis, with a temple in miniature placed over it, and then crowned with the square abacus recessed; the lower parts of the intervals between the columns are shut by a kind of parapet, reaching from three to three and a half diameters from the ground. This parapet is sometimes flush with the columns; but is not extended so as to hide their convexity on the front, which shews nearly a quarter of the circumference. Architecture has also been carried to a wonderful extent among the ancient inhabitants of India, who have not only rivalled the Egyptians, but have been supposed to be even anterior to them in the knowledge of the art; their exertions were, however, directed almost exclusively to excavation. The Assyrians have been much reputed for their knowledge in the art of building: the walls of Nineveh and Babylon were of wonderful magnitude. Those of the latter were double, and surrounded with a ditch; the outer wall was regularly fortified; it was 15 miles square, or 60 in circumference, 200 royal cubits high, and 50 thick: in the circumference were placed 100 massy gates of brass; and on the top, watchtowers, corresponding to each other. The materials used in the construction of these works, were square bricks, baked in a fur. nace, and heated bitumen, mixed with the tops of reeds; this composition was placed between every thirteen courses of bricks: from this circumstance it is probable, that

the method of reducing calcareous stones

into lime for mortar was unknown at this time. The walls of Babylon are described to be one of the seven wonders of the world; they were first built by queen Semiramis, in the time of her regency, during the minority of her son Ninias; and it would seem that they were afterwards improved by the great Nebuchadnezzar. Of these mighty works there are no remains, nor hardly any trace of the ancient city. In the ruins of Persepolis, though the columns are of a character somewhat different from those of Egypt; yet the Egyptian style of building may be retraced in various parts of these ruins. Diodorus Siculus says, that the famous palaces of Susa and Persepolis were not built till after the conquest of Egypt by Cambyses, and that they were both conducted by Egyptian architects; it therefore seems probable that the Persians received the art of building in unwrought stone from the Egyptians. The Phoenicians were also very celebrated for their arts of design, but few or mone of their works have reached the present time. In the vast structures of Asia and Africa, greatness of design, ponderosity of parts, and stones of immense magnitude, seem to have been more regarded than elegance or utility: in all those great works there is no trace of an arch, but what is excavated out

of the solid rock, or may be made of a

single stone. The Greeks profess to have derived the knowledge of architecture from the Egyptians, but the art of building has been so much improved by transplanting, that scarcely any trace of the original remains: their edifices were at first constructed of wood and clay, but they soon began to imitate the wooden posts and beams of the original hut in stone and marble: from this imitation arose the first order in architecture, which also gave birth to two others. This ingenious people, favoured by nature with marble and other building materials, and, like the Egyptians, being anxious to make their works durable, employed very weighty stones in the construction, which, although laid without cement, as was the practice of all ancient nations, yet they were jointed with the utmost accuracy, which is the reason of the perfect state of their edifices at this day. There is little doubt but that the Greeks were the inventors of the arch, though they

never considered it as an oruament; it is.

only to be found in the theatres and gymnasia; the apertures of walls and intercolumns being linteled. * Greece, though a mild climate, is sometimes liable to rain ; the architects of this: country, therefore, found it necessary to raise the roofs of their edifices, to a ridge in the middle, the section being that of a rectilineal isosceles triangle: the base being the span or distance between the opposite walls. This form of roof, called a pediment roof, was frequently covered with marble tiles. The Grecians surpassed all contemporary nations in the arts of design; the remains of their ancient structures are mo-' dels of imitation, and confessed standards of excellence. They were the inventors of three orders of architecture, of which we have already hinted, and which we' shall detail in a subsequent part of this article. The remains of their sculptures far exceed that of any other people, and are, even at this day, most perfect models. Modern artists have no means so certain, in attaining a just knowledge of their profession, as in the study of those exquisite master-pieces. The progress of Grecian architecture appears to have occupied a period of about three centuries, from the age of Solon to the death of Alexander; and in this period it advanced rapidly, particularly from the defeat of Xerxes, to the death of Pericles, at which time it attained its utmost degree of excellence, and continued to flourish till the time it became a Roman province. Prior to the Macedonian conquest, all the temples of Greece, and its colonies in Sicily and Italy appear to have been of the Doric order; and of one general form, though slightly varied in particular parts, as occasional circumstances might require : their plan was an oblong, having one column more on the flank than double the number of those in front. The ancient Etrurians have left many excellent monuments of taste, and to them is generally ascribed the method of building with small stone and mortar, made of calcareous stone ; and this seems probable, as the most ancient vestiges of cementitious buildings are to be found in the country which the present Tuscans inhabit. They were employed by the Romans in many public works; the walls of the city of Rome were made of hewn stone, the capitol ..and the cloaca maxima are of their construction; the last of these is esteemed a very extraordinary piece of architecture, as

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