Prop. XXXVIII. Theor. If a plane be perpendicular to another plane, and a straight line be drawn from a point in one of the planes perpendicular to the other plane, this straight line shall fall on the common section of the planes.

Prop. XXXIX. Theor. In a solid parallelepiped, if the sides of two of the opposite planes be divided each into two equal parts, the common section of the planes passing through the points of division, and the diameter of the solid parallelepiped cut each other into two equal parts.

Prop. XL. Theor. If there be two triangular prisms of the same altitude, the base of one of which is a parallelogram, and the base of the other a triangle; if the parallelogram be double of the triangle, the prisms shall be equal to one another. Book XII. Lemma 1-Which is the first proposition of the tenth book, and is necessary to some of the propositions of this book.

If from the greater of two unequal magnitudes, there be taken more than its half, and from the remainder more than its half; and so on: there shall at length remain a magnitude less than the least of the proposed magnitudes.

Prop. I. Theor. Similar polygons inscribed in circles, are to one another as the squares of their diameters.

Prop. II. Theor. Circles are to one another as the squares of their diameters.

Prob. III. Theor. Every pyramid having a triangular base, may be divided into two equal and similar pyramids having triangular bases, and which are similar to the whole pyramid; and intɔ two equal prisms which together are greater than half of the whole pyramid.

Prop. IV. Theor. If there be two pyramids of the same altitude, upon triangular bases, and each of them be divided into two equal pyramids similar to the whole pyramid, and also into two equal prisms; and if each of these pyramids be divided in the same manner as the first two, and so on: as the base of one of the first two pyramids is to the base of the other, so shall all the prisms in one of them be to all the prisms in the other, that are produced by the same number of divisions.

Prop. V. Theor. Pyramids of the same altitude which have triangular bases, are to one another as their bases,

Prop. VI. Theor. Pyramids of the same altitude which have polygons for their bases, are to one another as their bases.

Prop. VII. Theor. Every prism having a triangular base may be divided into three pyramids that have triangular bases, and are equal to one another.

Prop. VIII. Theor. Similar pyramids having tri ingular bases are one to another in the triplicate ratio of that of their homologous sides.

Prop. IX. Theor. The bases and altitudes of equal pyramids having triangular bases are reciprocally proportional: and triangular pyramids of which the bases and altitudes are reciprocally proportional, are equal to one another.

Prop. X. Theor. Every cone is the third part of a cylinder which has the same base, and is of an equal altitude with it.

Prop. XI. Theor. Cones and cylinders of the same altitude, are to one another as their bases. Prop. XII. Theor. Similar cones and cylinders have to one another the triplicate ratio of that which the diameters of their bases have.

Prop. XIII. Theor. If a cylinder be cut by a plane parallel to its opposite planes, or bases; it divides the cylinder into two cylinders, one of

which is to the other as the axis of the first to the axis of the other.

Prop. XIV. Theor. Cones and cylinders upon equal bases are to one another as their altitudes. Prop. XV. Theor. The bases and altitudes of equal cones and cylinders, are reciprocally proportional: and if the bases and altitudes be reciprocally proportional, the cones and cylinders are equal to one another.

Prop. XVI. Prob. To describe in the greater of two circles that have the same centre, a polygon of an even number of equal sides, that shall not meet the lesser circle.

Prop. XVII. Prob. To describe in the greater of two spheres which have the same centre, a solid polyhedron, the superficies of which shall not meet the lesser sphere.

Prop. XVIII. Theor. Spheres have to one another the triplicate ratio of that which their diameters have.

GEOMETRY (Descriptive), the name given to a species of geometry almost entirely new, and which we owe in great measure to M. Monge.

When any surface whatever penetrates another, there most frequently results from their intersection curves of double curvature, the determination of which is necessary in many arts, as in groined vault-work, cutting arch-stones, wood-cutting, for ornamental work, &c. the form of which is frequently very fantastical and complicated: it is in the solution of problems appertaining to these subjects that descriptive geometry is especially useful.

Some architects more versed in geometry than persons of that profession commonly are, have long ago thrown some light on the first principles of this kind of geometry. There is, for example, a work by a jesuit named Father Courcier, who examined and shewed how to describe the curves resulting from the mutual penetration of cylindrical, spherical, and conical surfaces: this work was published at Paris in 1663. P. Deraud, Matheurin Jousse, Frezier, &c. had likewise contributed a little towards the promotion of this branch of geometry. But Monge has given it very great extension, not only by proposing and resolving various problems both curious and difficult, but by the invention of several new and interesting theorems. We can only mention in this place one or two of the problems and theorems. Thus among the problems. 1st, Two right lines being given in space, and which are neither parallel nor in the same plane, to find in both of them the points of their least distance, and the position of the line joining these points. 2d, Three spheres being given in space to determine the position of the plane which touches them. There are also curious problems relative to lines of double curvature, and to surfaces resulting from the application of a right line that leans continually upon two or three others given in position in space. Among the theorems, the following may be mentioned: if a plane surface given in space is projected upon three planes, the one horizontal, and the two others vertical and perpendicular to each other, the square of that surface will be equal to the sum of the squares of the three surfaces of projection, This theorem is as interesting in the geometry of solids, as Pythagoras's theorem (Euc. 1. xlvii.) is in plane geometry. But for more on this subject we must refer to Monge's and Lacroix's ingenious works entitled Geometrie Descriptive.

GEOMETRY OF THE COMPASSES, a species of geometry more ingenious than profound, lately

proposed by the Abbe Mascheroni. Hitherto both the ruler and the compasses have been made use of in the solution of problems in plane geometry, and it had not been imagined that problems could be solved or constructed without the combined use of these two instruments. But the Abbé Mascheroni has struck out a greater number of problems, which are rendered very piquant and

amusing, by the new condition of employing only the compasses in their solution. Thus: two points terminating a right line being given, to find either between those two points or exteriorly any number of points which shall be in the same right line with the former, and which shall be with the interval between them in a given ratio: to draw to a given line perpendiculars, or parallels, or lines making with them given angles: to inscribe or circumscribe within or about a circle, the various polygons which are constructible by plane geometry: to determine the mean proportional between two given distances; or to find third and fourth proportionals: all the problems indeed of the Euclidean geometry are here resolved by the simple intersection of arcs of circles, without drawing a single right line. He also resolves by ingenious approximations many problems which lie beyond the limits of common geometry, as those relating to duplications, trisections, &c. still employing only the compasses. The inquisitive reader will be much entertained with the work entitled Geometria del Compassa, in which these curious particulars are explained with much perspicuity and elegance. A French translation of this work was published in 1798, in 1 vol. 8vo.

GEOPO'NICAL. a. [yn and mors.] Relating to agriculture. (Brown).

GEOPO'NICS. s. [yn and ovos.] The science of cultivating the ground, the doctrine of agriculture.

GEORGE I. king of England, was the eldest son of Ernest Augustus, elector of Brunswick Lunenburg, or Hanover, by the princess Sophia, daughter of Frederic, elector Palatine, and king of Bohemia, and of Elizabeth daughter of James I. He was born in 1660, created duke of Cambridge in 1706, and succeeded queen Anne in 1714. The next year a rebellion broke out in Scotland, in favour of the pretender, which, however, was shortly quelled, In his reign parliaments were made septennial, and the order of the Bath was revived. In 1720 happened the failure of the famous South sea scheme, by which many thousands were ruined. He died June 11, 1727, at Osnaburgh.

-

The following character of this monarch is extracted from Allen's History of England: "George the first was plain and simple in his person and address, grave and composed in his deportment, though easy, familiar, and facetious in his hours of relaxation. He is said to have been the only prince of his time who knew what it was to enjoy the sweets of private friendship; a pleasure from which sovereigns in general are, by the elevation of their rank, excluded. He had given a proof of his political abilities before his arrival in this island. Naturally inclined to justice and equity, though he was absolute and despotic in his hereditary dominions, he ruled them with all the lenity and moderation of a limited monarch. He

considered civil and religious liberty as the unalienable right of mankind; and therefore he granted it to a people who pleaded no other claim to it than the known goodness and humanity of his temper. Possessed of these noble and generous sentiments, though his accession to the British throne enlarged his sphere of action, it did not alter his plan of conduct; that was uniformly and invariably the same, both before and after his advancement to that high dignity. In a word, it may be affirmed, that no prince was ever better qualified to sway the sceptre over a free people, nor any who ever exercised the virtues of a great and good governor with more distinguished ability, or more uninterrupted success."

GEORGE II. king of England, succeeded his father in 1727. In 1737 he lost his queen Caroline, a woman of excellent qualities, and of a strong mind. In 1739 admiral Vernon was sent with a squadron to the West Indies, where he demolished Porto Bello, but failed in his attempt on Carthagena. In 1743 George put himself at the head of his army on the continent, and gained the battle of Dettingen, June 16th. In August 1745, the pretender's eldest son landed in the Highlands, and was joined by several clans. After obtaining several successes, the rebels were defeated by the Duke of Cumberland, at Culloden, in 1746. In 1748 peace was concluded at Aix-la-Chapelle. In March, 1750, died, universally lamented, Frederic prince of Wales, between whom and his father there had never been any cordiality. The next year an act passed for regulating the commencement of the year, by abolishing the old style. In 1753 another famous act passed for preventing clandestine marriages. In 1755 war broke out between the English and the French, which was at first very unpromising; Braddock was defeated and killed in North America, and Minorca was taken in the Mediterranean, for which admiral Byng, who was sent out to relieve it, was shot at Portsmouth. About this time Mr. Pitt became prime minister, and public affairs began at length to wear a new face under his active management. In 1768, a treaty was entered into between Eugland and Prussia, which proved very burthensome to the former nation; but the successes with which our fleets and armies were attended, made the people cheerful under ali the expences of the war. The British flag waved triumphant in every sea; and Geo. II. died suddenly amidst the blaze of glory, October 25th 1760; and was succeeded by his grandson, our present gracious sovereign.

"The memory of George II. (says Guthrie), is reprehensible on no head but his predilection for his electoral dominions. He never could form an idea that there was any difference between them and his regal dominions; and he was sometimes ill enough advised to declare so much in his speeches to parliament. We are, however, to remember, that his people gratified him in this partiality, and that he never acted by power or prerogative. He was just rather than generous; and in matters of economy,

either in his state or his household, he was willing to connive at abuses, if they had the sanction of law and custom. He was not very accessible to conversation, and therefore it was no wonder that having left Germany, after he had attained to man's estate, he still retained foreign notions both of men and things. In government he had no favourite; for he parted with sir Robert Walpole's administration with great indifference, and shewed very little concern at the subsequent revolutions among his servants. This quality may be deemed a virtue, as it contributed greatly to the internal quiet of his reign, and prevented the people from loading the king with the faults of his ministers. In his personal disposition he was passionate, but placable; fearless of danger, fond of military parade, and enjoyed the memory of the campaigns in which he served when young. His affections, either public or private, were never known to interfere with the ordinary course of justice; and though his reign was distracted by party, the courts of justice were never better filled than under him."

GEORGE (St.), or GEORGE of Cappadocia, a name whereby several orders, both military and religious, are denominated. It took its rise from a saint or hero famous throughout all the East, called by the Greeks Meyaλqua,lug, q. d. great martyr. On some medals of the emperors John and Manuel Comneni, we have the figure of St. George armed, holding a sword or javelin in one hand, and in the other a buckler, with this inscription; Р

an O, and therein a little A, and rErioc, O

making O ATIO TEOPTIOE, O holy George. He is generally represented on horseback, as being supposed to have frequently engaged in combats in that manner. He is highly venerated throughout Armenia, Muscovy, and all the countries which adhere to the Greek rite; from the Greek, his worship has long ago been received into the Latin church; and England and Portugal have both chosen him for their patron saint. Great difficulties have been raised about this saint or hero. His very existence has been called in question. Dr. Heylin, who wrote first and most about him, concluded with giving him entirely up, and supposing him only a symbolical device; and Dr. Pettingal has turned him into a mere Basilidian symbol of victory. Mr. Pegg, in a paper in vol. i. of the Archæologia, has attempted to restore him. And finally, Mr. Gibbon (Hist. yol. II. p. 404.) has sunk him into an Arian bishop in the reigns of Constantius and Julian. The bishop alluded to,

GEORGE the Cappadocian, was so surnamed, according to our author, from his parents or education; and was born at Epiphania, in Cilicia, in a fuller's shop." From this obscure and servile origin he raised himself by the talents of a parasite and the patrons whom he assiduously flattered procured for their worthless dependent a lucrative commission, or contract, to supply the army with bacon. His employ

ment was mean: he rendered it infamous. He accumulated wealth by the basest arts of fraud and corruption; but his malversations were so notorious, that George was compelled to escape from the pursuits of justice. After this disgrace, in which he appears to have saved his fortune at the expence of his honour, he embraced, with real or affected zeal, the profession of Arianism. From the love, or the ostentation, of learning, he collected a valuable library of history, rhetoric, philosophy, and theology; and the choice of the prevailing faction promoted George of Cappadocia to the throne of Athanasius. His conduct in this station is represented by our historian as polluted by cruelty and avarice, and his death considered as a just punishment for the enormities of his life, among which Mr. Gibbon seems to rank his "enmity to the Gods."

The

The immediate occasion of his death, however, as narrated by ecclesiastical writers, will not probably appear calculated to add any stain to his memory. "There was in the city of Alexandria a place in which the heathen priests had been used to offer human sacrifices. This place, as being of no use, Constantius gave to the church of Alexandria, and George the bishop gave orders for it to be cleared, in order to build a Christian church on the spot. In doing this they discovered an immense subterraneous cavern, in which the heathen mysteries had been performed, and in it were many human skulls. These, and other things which they found in the place, the Christians brought out and exposed to public ridicule. heathens, provoked at this exhibition, suddenly took arms, and rushing upon the Christians, killed many of them with swords, clubs, and stones: some also they strangled, and several they crucified. On this the Christians proceeded no farther in clearing the temple; but the heathens, pursuing their advantage, seized the bishop as he was in the church, and put him in prison. The next day they dispatched him; and then fastening the body to a camel, he was dragged about the streets all day, and in the evening they burned him and the camel together. This fate, Sozomen says, the bishop owed in part to his haughtiness while he was in favour with Constantius, and some say the friends of Athanasius were concerned in this massacre; but he ascribes it chiefly to the inveteracy of the heathens, whose superstitions he had been very active in abolishing.

St. George, the patron saint of England, and of the order of the Garter, is much revered in Russia, and his figure occurs in all the churches. He is represented as usual, riding on a horse, and piercing a dragon with his lance. This same device also forms part of the arms of the Russian sovereign, and is observed upon several of the coins. As most persons endeavour, if possible, to derive every custom and allusion from their own country, some of the English historians have conjectured, that Ivan Vassilievitch the Second, being presented with the Garter by Queen Elizabeth, assumed the George and the Dragon for his arms, and

ordered it to be stamped upon the current money. But this supposition is erroneous, as it by no means appears, that the Tzar was created a knight of the Garter; and it is certain that the sovereigns of Moscow bore this device before they had the least connection with England; for the arms of Moscovy are thus described by Chanceler, the first Englishman who discovered Russia, as being affixed to a dispatch sent in 1554, from Ivan Vassilevitch to Queen Mary:-"This letter was written in the Muscovian tongue, in letter much like to the Greeke letters, very faire written in paper, with a broade seale hanging at the same, scaled in paper upon waxe. This seale was much like the broad scale of England, having on the one side the image of a man on horseback in complete harnesse fighting with a dragon." Hackluyt, vol. 1. p. 255.

Many writers are of opinion that the history of St. George is an allegory; others maintain that the device is borrowed by the Christian church from the ancient story of Perseus and the sea-monster, of Bellerophon and the Chimera, or of Apollo and the serpent Pithon, or from some Egyptian hieroglyphics, or from charms and amulets worn by the Pagans; others have traced his legend from the history of George of Cappadocia, the Arian archbishop of Alexandria. Amongst the authors who have written upon this subject with the greatest success, must not be omitted Mr. Byrom (Miscellaneous Poems, vol. i. p. 100), who has composed a metrical rhapsody addressed to lord Willoughby, president of the Antiquarian Society, in which he endeavours to prove that St. George is a corruption of St. Gregory; and to use his own words:

Now, my lord, I would ask of the learn'd and laborious,

If Ge-orgius ben't a mistake for Gregorius? In names so like letter'd it would be no wonder,

If hasty transcribers had made such a blunder;

And mistake in the names by a slip of their

In opposition to these opinions, several authors have entered the lists; particularly Heylin, Selden, and Fuller; and lately, Mr. Pegge endeavours to rescue this saint from annihilation, and to prove, at the same time, that he was a different personage from St. Gregory. But unfortunately for the arguments of the learned antiquarian, Mr. Byrom's treatise, which he opposes, is written in verse with so much ease, humour, and whimsical rhyming, that the reader is almost divested of his judg ment, and with difficulty is prevented by the most weighty reasons from siding with the poet, who, instead of dull and dry disquisitions, treats us with much wit, couched in an easy flow of doggrel metre.

GEORGE. S. (Georgius, Latin.) 1. A figure of St. George on horseback worn by the knights of the garter (Shak.). 2. A brown loaf (Dry.). GEORGE (Fort St.). See MADRAS. GEORGE (St.), one of the Azores, inhabited wheat. Lat. 38. 39 N. Lon. 28. 0 W. by about 5000 persons, who cultivate much

GEORGE'S ISLANDS (King), are two islands in the South sea, lying in W. lon. 144. 56 S. lat. 14. 28. They were first discovered by commodore Byron in 1765, and have since been visited by captain Cook in 1774. Commodore Byron's people had an encounter with the inhabitants, which proved fatal to some of the natives; but captain Cook was more fortunate. A lieutenant and two boats well armed wers sent on shore by captain Cook, and landed without opposition. As soon as the gentlemen landed, the islanders embraced them by touching noses, a mode of civility used in New Zealand, which is 900 leagues distant, and the only place besides this where the custom has been observed to prevail. Notwithstanding this ceremony, however, very little real friendship seemed to take place on the part of the islanders.

GEORGE TOWN, the seat of justice in a dis

rict of the same name in S. Carolina. Lat. 33. 20 N. Lon. 79. 30 W.

GEORGIA, a country of Asia, called by the Persians Gurgistan, and by the Turks Gurtshi. It is one of the seven Caucasian nations, in the countries included between the Black sea and the Caspian, and comprehends the ancient Iberia and Colchis. It is bounded on the N. by Circassia, on the E. by Daghestan and Schirvan, on the S. by Armenia, and on the W. of the Cuban, or new Russian government of Caucasus. It is divided into nine provinces. Of these, five are subject to Heraclius, and form what is commonly called the kingdom of Georgia; and four, which are subject to David, form the kingdom or principality of Imeretia. This country is so extremely beautiful, that some fanciful travellers have imagined they had here found the situation of the original garden of Eden. The hills are covered with forests of oak, ash, beech, chesnuts, walnuts, and elms, enriched with vines, growing perfectly wild, but producing vast quantities of grapes. From these is annually made as much wine as is necessary for their yearly consumption; the remainder are left to rot on the vines. Cotton grows spontaneously, as well as the finest European fruit-trees. Rice, wheat, millet, hemp, and Hax, are raised on the plains almost without culture. The valleys afford the finest pasturage in the world; the rivers are full of fish; the mountains abound in minerals; and the climate is delicious; so that nature appears to have lavished on this favoured country every production that can contribute to the happiness of its inhabitants. On the other hand, the rivers of Georgia, being fed by mountain torrents, are always either too rapid or too shallow for the purposes of navigation; the Black sea, by which commerce and civilization might be introduced from Europe, has been till very lately in the exclusive possession of the Turks; the trade of Georgia by land is greatly obstructed by the high mountains of Caucasus; and this obstacle is still increased by the swarms of preda tory nations, by which those mountains are inhabited. The inhabitants are Christians of the Greek communion, and appear to have received their present name from their attachments to St. George, the tutelary saint of these countries. Their dress nearly resembles that of the Cossacs; but men of rank frequently wear the habit of Persia. They usually dye their hair, beards, and nails with red. The women employ the same colour to stain the palms of their hands.

GEORGIA, the most southern of the United States of N. America, bounded on the E. by the Atlantic ocean, on the S. by E. and W. Florida, on the W. by the river Mississippi, and on the N. by N. and S. Carolina, being divided from the latter by the river Savannah. It is about 600 miles long and 250 broad. It is divided into 11 counties, namely, Chatham, Effingham, Burke, Richmond, Wilkes, Liberty, Glynn, Camden, Washington, Greene, and Franklin. The capital is Augusta. The

principal rivers are the Savannah, Ogeechee, Alatamaha, Turtle River, Little Sitilla, Great Sitilla, Crooked River, St. Mary's, and Appalachikola. The winters in Georgia are very mild and pleasant. Snow is seldom or never seen. The soil and its fertility are various, according to situation and different improvements. By culture are produced rice, indigo, cotton, silk, India corn, potatoes, oranges, figs, pomegranates, &c. Rice at present is the staple commodity; but great attention begins to be paid to the raising of tobacco. The whole coast of Georgia is bordered with islands, the principal of which are Skidaway, Wassaw, Ossahaw, St. Catharine's, Sapelo, Frederica, Jekyl, Cumberland, and Amelia. For more on the subject of this state, see Cruttwell's Gazetteer.

GEORGIA (New), or SOUTH GEORGIA, an island in the South Atlantic ocean, about thirty leagues in length, and ten in breadth. It abounds in bays and harbours, but the prodigious quantity of ice on the coast renders it inaccessible during a great part of the year, and even at other times floating masses of ice render the anchorage dangerous. The appearance of the land is the same throughout; the lofty mountains towards the south are divided into numberless parts, and appear like flames of fire. The coasts are bounded with high perpendicular rocks of ice, large portions of which frequently break off and fall into the valleys or into the sca, where they are tossed about by the waves, and resemble small detached islands. The interior country is not less savage, the summits of the rocks are lost in the clouds, and the valleys are covered with eternal snow; there is neither tree nor shrub. The only vegetables discovered were a kind of coarse grass, a species of burnet, and a plant like moss. The rocks are composed of a kind of slate, of a blueish grey colour, disposed in horizontal beds: many shining fragments of which cover the strand, and appear to have no mineral in the composition. In all the coast there was found neither river nor fresh-water spring. Lat. 54. 30 S. Lon 37.0 W.

GEORGIC, a poetical composition upon the subject of husbandry, containing rules therein, put into a pleasing dress, and set off with all the beauties and embellishments of poetry. The word is borrowed from the Latin georgicus, and that of the Greek yewplixos, of yn, terra, earth, and pyaas, opero, I work, or labour, of yor, opus, work. Hesiod and Virgil are the two greatest masters in this kind of poetry. The moderns have produced nothing in this way, except Rapin's book of Gardening; and the celebrated poem intitled Cyder, by Mr. Philips, who, if he had enjoyed the advantage of Virgil's language, would have been second to Virgil in a much nearer degree.

GEORGINA, in botany, a genus of the class syngenesia, order polygamia superflua. Receptacle chaffy, downless; calyx double, the outer many-leaved; inner one-leaved; eightparted. Three species; all natives of Mexico.

GEORGIUM SIDÚS, the name given by