other, and 2 to express the position of their plane. The value of these quaternions may be made apparent by a simple case. Suppose 3 lines of equal or unequal length to radiate from a fixed point. Let the quaternion quotients obtained by dividing either of these lines by another be denoted by r, p, and q. As these express all the relations of the rays, of the 3 extremities, and of the angles made by them when of equal length, it is evident that a single equation between r, p, and q may express all the theorems both of plane and spherical trigonometry. Prof. Peirce of Cambridge, Mass., in 1858 proposed a general method of expressing curves, which promises to lead into unexplored fields. He defines a curve by an equation between angles which it makes with 2 known curves allowed to vary either in parameter or in some other mode so as to produce an intersection of each other at each point of the 3d curve. Thus, to take a familiar example, the conic sections may be defined by their making equal angles with straight lines radiating from 2 fixed points; and for another, the logarithmic spiral makes with straight lines radiating from a point, and circles concentric round that point, angles whose ratio is constant. -The relation of geometry to other sciences is, like that of the other mathematics, twofold, giving and receiving. To mechanics it gives the only possibility of understanding the laws of motion, and from mechanics it receives the conception of moving points, lines, and surfaces, and thus generating lines, surfaces, and solids. To chemistry it gives the only means of investigating crystallization, polarization, &c., and from chemistry receives new ideas concerning the symmetry of planes. To botany and zoology it gives the laws of form, and from them receives those highest problems of morphology which are as yet barely touched upon by the masters in either of these sciences. In the study of the human spirit, geometry proposes the question of the foundations of belief, by giving the first examples of demonstration; and from the psychological inquiries thus aroused, geometry has received from age to age the new conceptions which have been the base of many new methods of investigation and proof. To theology geometry gives definite conceptions of the order and the wisdom of the natural creation, and a priori demonstrations of the optimism of actually existing material laws; and from theology she has been stimulated to many fresh exertions in the investigation of these teleological questions.-The beginner in geometry will find many text books, from which he may select according to his taste. None is perhaps more popular than the "Elements of Geometry and Trigonometry," by Prof. Charles Davies, from the works of A. M. Legendre (New York, 1858). Much more_condensed and suggestive is an "Elementary Treatise on Plane and Solid Geometry," by Prof. Benjamin Peirce (Boston, 1858). An easier treatise than either of these, by Prof. G. R. Perkins, has been published in

New York. The true style of Greek geometry may be found in Playfair's "Euclid," of which there is a New York edition. For advanced studies the following list of works is recommended: "Modern Geometry," by Mulcahy (London, 1859), giving some idea of the new methods, but not employing analytical geometry; "Elementary Treatise on Plane and Spherical Trigonometry," and "Elementary Treatise on Curves, Functions, and Forces," by Benjamin Peirce (Boston, 1858), giving in its most condensed form the necessary introductory knowledge of the notation of trigonometry, analytical geometry, and the calculus; "Analytical Geometry," and "Differential and Integral Calculus," by Charles Davies (New York, 1855), giving a more popular expression of the same knowledge; a "Treatise on Conic Sections, containing an Account of some of the most important Modern Algebraic and Geometric Methods," by G. Salmon (London, 1855); a "Treatise on the Higher Plane Curves," by the same author; Traité de géométrie supérieure, by M. Chasles (Paris, 1852); Mémoire de géométrie sur les propriétés géométriques des coniques sphériques, by Chasles (Brussels, 1831; soon after translated into English); Aperçu historique sur l'origine et le développement des méthodes en géométrie, particulièrement de celles qui se rapportent à la géométrie moderne ; suivi d'un mémoire de géométrie sur deux principes généraux de la science, la dualité et l'homographie, by Chasles (Brussels, 1837; translated into German, Halle, 1839; a work which will richly repay a close study); Systematische Entwickelung der Abhängigkeit geometrischer Gestalten von einander, mit Berücksichtigung der Arbeiten alter und neuer Geometer über Porismen, projective Methoden, Geometrie der Lage, Transversalen, Dualität und Reciprocitat, by Steiner (Berlin, 1832); Traité des propriétés projectives des figures, by Poncelet (Paris, 1822); Des méthodes en géométrie, by Serret (Paris, 1855); Mémoire sur les lignes du second ordre, by Brianchon (Paris, 1817), and his Mémoire sur les courbes de raccordement (Paris, 1823); four books of Plücker of great merit, the Analytische geometrische Entwickelungen (2 vols. Essen, 1828-'31), System der analytischen Geometrie auf neue Betrachtungsweisen gegründet und insbesondre eine ausführliche Theorie der Curven dritter Ordnung enthaltend (Berlin, 1833), Theorie der algebraischen Curven gegründet auf eine neue Behandlungsweise der analytischen Geometrie (Bonn, 1839), and System der Geometrie des Raumes in neuer analytischer Behandlungsweise, insbesondre die Theorie der Flächen zweiter Ordnung und Classe enthaltend (2d ed. Düsseldorf, 1852); Géométrie de position, by Carnot (Paris, 1803), his De la correlation des figures de géométrie (Paris, 1801), and his Mémoire sur la relation qui existe entre les distances respectives de cinque points quelconques pris dans l'espace; suivi d'un essai sur la théorie des transversales (Paris, 1806, and 4to. 1815); Géométrie descriptive, by

Monge, and his Application de l'algèbre à la géométrie (Paris, 1829, 1838); Développements de géométrie, avec des applications à la stabilité des vaisseaux, aux déblais et remblais, au défilement, à l'optique, &c., pour faire suite à la géométrie descriptive de G. Monge, by Dupin (Paris, 1813); Sir Isaac Newton's Enumeratio Linearum Tertii Ordinis; Euler's Introductio in Analysin Infinitorum (London, 1797; translated into French, Paris, 1796); "Lectures on Quaternions, by Sir William Rowan Hamilton, containing a systematic statement of a new mathematical method, of which the principles were communicated in 1843 to the royal Irish academy, and which has since formed the subject of successive courses of lectures, delivered in 1848 and subsequent years, in the halls of Trinity college, Dublin; with numerous illustrative diagrams, and with some geometrical and physical applications" (Dublin, 1853).

GEORGE (LEWIS) I., king of Great Britain and Ireland, first sovereign of the Hanoverian line, born in Osnaburg, May 28, 1660, died near the same place, June 10, 1727. By the act of settlement, passed in 1701, the British crown was entailed on Sophia, electress dowager of Hanover, granddaughter of James I., and her posterity.Sophia died June 8, 1714, and so missed the throne to which she had been called by only 54 days, Queen Anne dying Aug. 1. George Lewis, her eldest son, was, like his father, the elector Ernest Augustus, partial to military pursuits, which he commenced at the age of 15. He served in Greece, Hungary, Germany, and Flanders, and was present at the great battle near Vienna in which John Sobieski defeated the Turks. He also served under William III., and took part in the battles of Steenkerke and Lauden. He succeeded his father in 1698. He sought the hand of Anne, daughter of the duke of York, in 1681, but failed; and in 1682 he married his first cousin, Sophia Dorothea, daughter of the duke of Zell, whose affections, if he ever possessed them, were soon alienated by his infidelities and brutality. Sophia Dorothea was suspected of an intrigue with a Swedish nobleman, Count Philip Christopher von Königsmark, and was arrested and imprisoned for 32 years in the castle of Ahlden, her supposed lover having previously been put to death. The elector was proclaimed king of Great Britain and Ireland immediately on the death of Anne, arrived in London Sept. 20, 1714, and was crowned Oct. 20. The majority of the English people were undoubtedly opposed to his accession, but the activity of the whigs was more than a match for the strength of the Jacobites and tories. The rebellion of 1715, in Scotland and the north of England, was put down; and though there were many conspiracies entered into against him, and even foreign potentates and ministers planned his downfall, George I. was never in any danger of losing the throne to which he had been called by parliament. He was not a popular monarch, and the grossness of his tastes (he is said to have

been fond of nothing but punch and fat women) was not calculated to make his vices pardonable. Both he and his attendants regarded England as a land to be plundered. His quarrel with his son originated in politics, and dated back to years before the king's accession; but it was aggravated by the prince's attachment to his mother, the imprisoned Sophia Dorothea. After the family had secured the British throne, the father and son behaved toward one another quite decorously for a time, the prince acting as regent during the first of the many visits which the king made to Germany. This state of things did not last, and the king soon came to hate his son almost as bitterly as he hated his wife. There were separate courts, the prince of Wales residing at Leicester house, and the king declared that he should regard all who visited his son as his enemies, to be excluded from the royal presence. The quarrel became one of the scandals of the time, and as it was followed by similar quarrels in the royal family, the triumph of constitutionalism was coupled with domestic dissensions. The king connected himself with the whigs, whose long ascendency in the 18th century dates from his accession. The ministry of 1714 contained such men as Townshend, Stanhope, Cowper, Halifax, Sunderland, and other eminent whigs. Robert Walpole, who was soon to be the leading minister, had at first no higher place than that of paymaster-general, and did not belong to the cabinet. His talent was so superior, however, that in 1715 he was made first lord of the treasury, which office did not then confer the premiership, though it was in his person that it finally became the first place in the ministry. breach that took place in the whig party in 1717 caused Walpole, Pulteney, and others to resign their places, and the Stanhope-Sunderland ministry was formed. Previous to this the septennial act was passed, by which the legal existence of each house of commons was extended to 7 years, a proceeding that worked well, though it was in opposition to old whig principles. The new ministry formed the quadruple alliance, by which England, France, the empire, and Holland united to preserve the peace of Europe. The short war that followed was very humiliating to Spain. Admiral Byng annihilated the Spanish fleet at Passaro. The government was very strong both at home and abroad, all the plots in favor of the Stuarts having failed, while the schemes of Alberoni and Görtz had ruined only those who had formed them. A sort of reconciliation had been effected between the king and the prince of Wales. Everything appeared prosperous, when the bursting of the South sea bubble wrought the ruin of the ministry, and it was even expected that the king would abdicate. The ministry were held responsible, not only for the South sea act, which was indeed their work, but for the mad speculations that had followed its passage. Stanhope was seized with a fit while replying in the house of lords to an unjust attack,

The

and died; Craggs, secretary of state, died of small pox, and Sunderland left the treasury. Walpole, who had not been in office when the act was passed, took the lead in the work of change, and was made Sunderland's successor. The latter soon dying, peace was restored, and Walpole's long turn of power began with a new house of commons, in which the now united whigs had a heavy majority. The hopes of the Jacobites had revived, and a plot was formed for an insurrection in 1722, which was detected, and Bishop Atterbury was banished for his part in it. Walpole's love of power, and resolution to monopolize it, led to dissensions in the ministry. Carteret was compelled to give up the place of secretary of state, and go to Ireland as lord lieutenant, where Swift had raised a storm against Wood's halfpence. Walpole was jealous of his relative Townshend, who had become conspicuous from the part he had in making the treaty of Hanover, by which an alliance was formed between England, France, and Prussia, in opposition to Austria, Russia, and Spain. The king did not like the measure, and Walpole disapproved of it because of the consequence it gave to Townshend; but the time had not come for an open breach between them. William Pulteney, so celebrated for his eloquence and his avarice, was removed from a petty office in the royal household, for his opposition to a motion for discharging the civil list debts. The union of Spain and the empire came near to bringing about a general war in the last year of George I., but Walpole, whose merit it was ever to prefer peace to war, averted the danger. At this time Bolingbroke, who had been allowed to return to England, though not to resume his former state, was engaged in strong efforts to overthrow Walpole. Having secured the interest of the duchess of Kendal, the king's mistress, he was confident of ultimate success with the king, who at first treated his representations of the evils that must follow from Walpole's continuance in power with contempt. But the reign of George I. was drawing to a close. He loved Hanover and disliked England, and often visited his German dominions, which gave rise to much animadversion from the Jacobites. On June 3, 1727, he left England for Hanover, accompanied by the duchess of Kendal and Lord Townshend. On the 10th, while in his carriage, he was seized with a fit, and died before the party could reach Osnaburg, though attempts were made to revive him after they arrived there, where his death is generally stated to have happened. He is said to have been fully impressed when he left England with the belief that he never should return, and among the gossip of the time was a story that he had been warned that he should not survive his wife a year; she died at Ahlden, Nov. 2, 1726. He was a man of moderate faculties, and but for the circumstance that his fortunes happened to be identified with those of the constitutional party in a great nation, he probably never would have been more heard of than a thousand other

German princes whose very names are unknown. He was a cruel husband, a bad father, and a vicious man, and yet by no means a bad sovereign; but this distinction was owing to the constitutional restraints that were imposed on him, and to the want of knowledge on his part of the nature of English politics and parties. The union of England and France, which was the chief event of the reign of George I., and which was brought about by the very party that had upheld the war policy in Anne's reign, shows a breadth of policy such as was not common in that age; yet it ought to be mentioned, that this alliance was dictated by the circumstances of both governments, it being for the interest of the Hanoverian dynasty and the whigs that the connection between France and the Stuarts should cease, and that the reliance of the tories on that country should be broken up; while the hope of the duke of Orleans, then regent, that he might come to the crown of France, when he would need foreign support, made it equally for his interest that both England and Holland should be engaged on his side.

GEORGE (AUGUSTUS) II., son of the preceding and of Sophia Dorothea, born in Hanover, Oct. 30, 1683, died at Kensington palace, Oct. 25, 1760. But little is known of his early history, except that he was neglected by his father, and that he was brought up by his grandmother, the electress Sophia. He visited Holland in 1699, and in 1705 was married to Wilhelmina Dorothea Caroline, daughter of the margrave of Brandenburg-Anspach, a woman of marked character and superior talent. The next year he was made a peer of England, his chief title being duke of Cambridge, with precedence over the peerage. He made the campaign of 1708 under the duke of Marlborough, and conducted himself with great bravery at the battle of Oudenarde, having his horse shot under him. In the opposite ranks, and showing equal valor, was the pretender, son of James II. He accompanied his father to England in 1714, and took the titles of prince of Wales, and earl of Chester and Flint, Sept. 27. The quarrel between father and son broke out soon, and they hated each other cordially. The prince had been preferred by the electress Sophia to her own son, and that son was attached to his mother, two causes that sufficed to increase his father's original dislike of him; but political differences were the groundwork of the quarrel. The king also hated the princess of Wales, and was jealous of her popularity. So vindictive was his feeling, that he entertained a proposition, made by the earl of Berkeley, to carry off the prince to America, there to be so disposed of as never to trouble his father again. The king sought to deprive his son of all control of his children, and 10 of the 12 judges decided in his favor. A sort of reconciliation was effected in 1720, through Walpole's influence, but the parties hated each other as bitterly as ever, their formal intercourse being a hypocritical tribute to

public opinion. When he ascended the throne, George II. endeavored to transfer power to the hands of Sir Spencer Compton, but his incapacity was so evident that Walpole retained his place, the more easily that he was supported by Queen Caroline, who saw that he was the man for the times and for the dynasty. The coronation took place Oct. 11, 1727. The pretender, on hearing of the death of George I., had made some movements, and the Jacobites had their hopes roused, but the new king was not in the least disturbed thereby. The history of the first 14 years of the reign of George II. is that of the struggle of Walpole and the opposition, the fiercest civil political contest, unstained by blood, that England has ever known. The hopes that had been entertained of Walpole's overthrow, as a consequence of the death of George I., had been disappointed, and that great minister's power was now fixed on a firm basis, from which it could never have been shaken but for his jealousy of all able men. The new parliament contained an overwhelming ministerial majority, and the king soon became strongly attached to the minister both on personal and political grounds. The royal avarice was gratified and the royal ease consulted by the minister, and hence the king supported the latter with all his influence; but the support he received from the queen, who governed her husband without his knowing it, was of greater importance. George II. was as fond of Hanover as his father had been, and visited it often, to the disgust of his English subjects. He hated his son Frederic, prince of Wales, as bitterly as he had himself been hated by his father, and the queen shared his feelings in that respect. Frederic was not allowed to visit England until 1728. The fault was originally on the side of the parents, and the prince long bore the king's harsh treatment without complaint, and perhaps would have done so to the last if it had not been for the interest of the opposition to make him acquainted with his political importance, and to stir him to resentment, because the king supported Walpole personally. The first great act of this reign was the treaty of Seville, concluded in 1729, between England, France, Spain, and Holland, which was very advantageous to England, and by which Spain silently acquiesced in the English possession of Gibraltar. Walpole quarrelled with his colleague Townshend in 1730, and the latter resigned office, and retired altogether from public life. Walpole was supreme in the cabinet, and appears to have been disposed to make some improvements in the laws, and to correct abuses; but the virulence with which all his measures were opposed in parliament compelled him to be cautious. The use of Latin in the courts of justice was discontinued, and English substituted. The sinking fund, which Walpole himself had aided to establish, was now so encroached upon that it soon ceased to be of any value. The great contest on the excise question now commenced, and was the most remarkable incident of the first half

of the reign. The mere report of the intention to introduce a scheme of general excise caused alarm, and the opposition, which had been reduced very low, immediately became vigorous. The battle was fought with ability and courage on both sides, and though in some of its stages the ministerial majorities were 60, they finally fell to 16, in a house of commons which had given Walpole almost 200 majority on other questions. The bill was then withdrawn, greatly to the satisfaction of the people. The king was as much beaten as the minister, and they revenged themselves by dismissing from office, or from sinecure places, a number of distinguished noblemen who had been prominent in opposing the measure, the chief of whom was Lord Chesterfield. The election of 1734 resulted in the return of a strong Walpolian majority. The opposition sought the repeal of the test act, and were beaten by 128 majority. The gin act, which sought to do something to lessen drunkenness, was passed in 1736. The prince of Wales was married in 1736, and the question of his income afforded the opposition means to annoy the ministry, and caused the quarrel between father and son to become bitterer than ever. The queen died in 1737, recommending her husband to Walpole with her last breath. The opposition endeavored to have the army reduced in 1738, but failed. They then assailed the ministry because of its indifference to the outrages perpetrated by the Spaniards in America on Englishmen there trading. An arrangement made with Spain was unpopular. The ministerial majority was greatly reduced, but the minister was saved by the folly of his enemies, a number of whom seceded. The troubles with Spain went on, and war was declared against that country, Oct. 19, 1739. The ministerial strength now diminished, and the hopes of the Jacobites revived. The war was by no means brilliantly conducted. Anson's cruise in the Pacific and Indian seas revived the recollections of the Elizabethan age, and Vernon took Portobello; but the English failed at Carthagena, and also at Santiago de Cuba. The war of the Austrian succession began in 1741, and England was drawn into it. Parliament was dissolved, and the new elections took place under circumstances unfavorable to Walpole. When parliament assembled, the opposition found themselves in a majority, and after a hard battle Walpole gave way, much to the grief of the king, who continued to take his advice to the last days of his life. Sir Robert was created earl of Orford, and the attempts made to prosecute him fell through. Lord Wilmington became premier, and Carteret secretary of state. Success was ruinous to the opposition, which showed it could not administer affairs, though so eloquent in fault-finding. The public was greatly disappointed, and the refusal of the victors to repeal the septennial act, which had been the object of their especial indignation when out of office, caused much disgust. England had now become actively engaged in the

Austrian war, and sent a large force to Flanders, which did nothing. Some success was achieved by the navy, and a British squadron compelled Naples to observe a neutrality. The king, who was fond of military life, and personally brave, was desirous of placing himself at the head of an army, and a large German force was taken into English pay. This added to the hatred of Hanover already felt in England. The king joined the allied army, June 19, 1743, and 7 days later was fought the battle of Dettingen, in which the French were beaten, the monarch showing much courage. The death of Lord Wilmington led to Henry Pelham's elevation to the premiership, the king acting under the advice of Walpole. Carteret continued to manage foreign affairs, and was much liked by the king. The Hanoverian policy was still vigorously opposed, but the resolute conduct of France, the fear of invasion, and the revival of the Jacobite party, caused some remission of party feeling, and the adoption of strong measures by government, the whigs of all views uniting in their support. The French government called Charles Edward Stuart to France, and extensive preparations were made to invade England, which failed through the occurrence of a severe storm, the fleet being destroyed or dispersed. Lord Carteret, now Earl Granville, was compelled to leave the ministry, and Pelham forced the king to admit Chesterfield and some others of the old opposition to office. The Hanoverian policy was kept up, despite these changes, and England entered on an extensive system of German subsidies. Great preparations for the campaign of 1745 were made, but with no gain to England. The allies were beaten by the French at Fontenoy, when under command of the duke of Cumberland. Charles Edward landed in Scotland, was joined by many Highlanders and others, and, after occupying much of the ancient kingdom of his race, marched into England as far as Derby, when his leading supporters compelled him to retrace his steps. He had won the victory of Gladsmuir, and it is now the current opinion that if he had pressed forward from Derby to London the capital would have fallen into his hands. George II., though very brave, and prepared to place himself at the head of his guards for a last fight, made preparations to fly. The rebels defeated the royal troops at Falkirk, but 3 months later their army was annihilated at Culloden. From that time dates the extinction of the Stuart party. The rest of the war was inglorious, and it was terminated by the treaty of Aix la Chapelle in 1748. For some years there was but little political discussion, and the opposition had dwindled into a small faction, headed by the prince of Wales, which became extinct soon after his death in 1751. Private bills excited more interest in parliament than those of a public character. An increase in the stringency of the mutiny bill, and the passage of a regency bill, caused some discussion. The reformation of the calendar was

effected in 1752, providing that the year should commence Jan. 1 instead of March 25. The bill for the naturalization of the Jews was passed in 1753, and, though one of the most creditable acts of English legislation, caused so much popular excitement that it was immediately repealed. The premier died March 6, 1754, and was succeeded by his brother, the duke of Newcastle, who found himself compelled to share power with others. His first ally was Henry Fox, afterward Lord Holland, and at a later day William Pitt. The 7 years' war began in 1756, and England was involved in a contest of the severest character with France, while at the same time she was the ally of Prussia, which was at war with France, Russia, the empire, and lesser powers. The contest extended over the world, and was marked by great actions in Europe, in North America, and in the East Indies. The early part of the war was inglorious to England, but shortly after the formation of the Pitt and Newcastle ministry in 1757, the genius of Pitt changed the fortune of the contest, and the English were triumphant in every quarter. Ample subsidies were furnished to Frederic of Prussia, so that he was enabled to make head against the coalition formed for his overthrow. An army of English and Germans defeated the French in Germany, at Crefeld, Minden, and elsewhere. North America was the scene of great operations, which ended in the expulsion of the French. An expedition to France, twice renewed, inflicted considerable damage on that country, destroying, among other things, the works at Cherbourg. The success of Clive laid the foundation of the British Indian empire. Senegal and Goree were conquered; and the victory of Admiral Hawke over Conflans, in the naval battle of Quiberon, established English supremacy on the ocean. Never had England stood so high as she stood in 1760. In the midst of these successes, forming so striking a contrast to most of his reign, George II. suddenly died, at the age of 77. He was a man of ordinary character, and never had been popular with his English subjects; but he had governed constitutionally, and in his reign the liberal polity was fairly established. The industrial system of England then began to display itself, and the world saw the first evidences of that power which was to receive a development so vast and unparalleled. Personally the monarch had little to do with these things, but he was the chief in the political system under which and through which they came to pass. As a despot he would probably have proved a bad ruler, for his tastes were arbitrary, and so were his acts when he was not restrained by law and circumstances; but as a constitutional sovereign he has a respectable place in history.

GEORGE (FREDERIO WILLIAM) III., grandson of the preceding, and son of Frederic, prince of Wales, and of Augusta of SaxeGotha, born June 4, 1738, died at Windsor castle, Jan. 29, 1820. He was not likely to be