Scotland.—In Scotland the courts of session and justiciary have, at common law, and exercise the power of punishing contempt committed during a judicial proceeding by censure, fine or imprisonmentproprio motuwithout formal proceedings or a summary complaint. The nature of the offence is there in substance the same as in England (see Petrie, 1889: 7 Rettie Justiciary 3; Smith, 1892: 20 Rettie Justiciary 52).Ireland.—In Ireland the law of contempt is on the same lines as in England, but conflicts have arisen between the bench and popular opinion, due to political and religious differences, which have led to proposals for making juries and not judges arbiters in cases of contempt.British Dominions beyond Seas.—The courts of most British possessions have acquired and freely exercise the power of the court of king’s bench to deal summarily with contempt of court; and, as already stated, it is not infrequently the duty of the privy council to restrain too exuberant a vindication of the offended dignity of a colonial court.
Scotland.—In Scotland the courts of session and justiciary have, at common law, and exercise the power of punishing contempt committed during a judicial proceeding by censure, fine or imprisonmentproprio motuwithout formal proceedings or a summary complaint. The nature of the offence is there in substance the same as in England (see Petrie, 1889: 7 Rettie Justiciary 3; Smith, 1892: 20 Rettie Justiciary 52).
Ireland.—In Ireland the law of contempt is on the same lines as in England, but conflicts have arisen between the bench and popular opinion, due to political and religious differences, which have led to proposals for making juries and not judges arbiters in cases of contempt.
British Dominions beyond Seas.—The courts of most British possessions have acquired and freely exercise the power of the court of king’s bench to deal summarily with contempt of court; and, as already stated, it is not infrequently the duty of the privy council to restrain too exuberant a vindication of the offended dignity of a colonial court.
(W. F. C.)
CONTI, PRINCES OF.The title of prince of Conti, assumed by a younger branch of the house of Condé, was taken from Conti-sur-Selles, a small town about 20 m. S.W. of Amiens, which came into the Condé family by the marriage of Louis of Bourbon, first prince of Condé, with Eleanor de Roye in 1551.
François(1558-1614), the third son of this marriage, was given the title of marquis de Conti, and between 1581 and 1597 was elevated to the rank of a prince. Conti, who belonged to the older faith, appears to have taken no part in the wars of religion until 1587, when his distrust of Henry, third duke of Guise, caused him to declare against the League, and to support Henry of Navarre, afterwards King Henry IV. of France. In 1589 after the murder of Henry III., king of France, he was one of the two princes of the blood who signed the declaration recognizing Henry IV. as king, and he continued to support Henry, although on the death of Charles cardinal de Bourbon in 1590 he himself was mentioned as a candidate for the throne. In 1605 Conti, whose first wife Jeanne de Cöeme, heiress of Bonnétable, had died in 1601, married the beautiful and witty Louise Marguerite (1574-1631), daughter of Henry duke of Guise and Catherine of Cleves, whom, but for the influence of his mistress Gabrielle d’Estrées, Henry IV. would have made his queen. Conti died in 1614. His only child Marie having predeceased him in 1610, the title lapsed. His widow followed the fortunes of Marie de’ Medici, from whom she received many marks of favour, and was secretly married to François deBassompierre(q.v.), who joined her in conspiring against Cardinal Richelieu. Upon the exposure of the plot the cardinal exiled her to her estate at Eu, near Amiens, where she died. The princess wroteAventures de la cour de Perse, in which, under the veil of fictitious scenes and names, she tells the history of her own time.
In 1629 the title of prince de Conti was revived in favour ofArmand de Bourbon(1629-1666), second son of Henry II. ofBourbon, prince of Condé, and brother of Louis, the great Condé. He was destined for the church and studied theology at the university of Bourges, but although he received several benefices he did not take orders. He played a conspicuous part in the intrigues and fighting of the Fronde, became in 1648 commander-in-chief of the rebel army, and in 1650 was with his brother Condé imprisoned at Vincennes. Released when Mazarin went into exile, he wished to marry Mademoiselle de Chevreuse (1627-1652), daughter of the famous confidante of Anne of Austria, but was prevented by his brother, who was now supreme in the state. He was concerned in the Fronde of 1651, but soon afterwards became reconciled with Mazarin, and in 1654 married the cardinal’s niece, Anne Marie Martinozzi (1639-1672), and secured the government of Guienne. He took command of the army which in 1654 invaded Catalonia, where he captured three towns from the Spaniards. He afterwards led the French forces in Italy, but after his defeat before Alessandria in 1657 retired to Languedoc, where he devoted himself to study and mysticism until his death. At Clermont Conti had been a fellow student of Molière’s for whom he secured an introduction to the court of Louis XIV., but afterwards, when writing a treatise against the stage entitledTraité de la comédie et des spectacles selon les traditions de l’Église(Paris, 1667), he charged the dramatist with keeping a school of atheism. Conti also wroteLettres sur la grâce, andDu devoir des grands et des devoirs des gouverneurs de province.
Louis Armand de Bourbon, prince de Conti (1661-1685), eldest son of the preceding, succeeded his father in 1666, and in 1680 married Marie Anne, a daughter of Louis XIV. and Louise de la Vallière. He served with distinction in Flanders in 1683, and against the wish of the king went to Hungary, where he assisted the Imperialists to defeat the Turks at Gran in 1683. After a dissolute life he died at Fontainebleau from smallpox.
François Louis de Bourbon, prince de Conti (1664-1709), younger brother of the preceding, was known until 1685 as prince de la Roche-sur-Yon. Naturally of great ability, he received an excellent education and was distinguished both for the independence of his mind and the popularity of his manners. On this account he was not received with favour by Louis XIV.; so in 1683 he assisted the Imperialists in Hungary, and while there he wrote some letters in which he referred to Louis asle roi an théâtre, for which on his return to France he was temporarily banished to Chantilly. Conti was a favourite of his uncle the great Condé, whose grand-daughter Marie Thérese de Bourbon (1666-1732) he married in 1688. In 1689 he accompanied his intimate friend Marshal Luxembourg to the Netherlands, and shared in the French victories at Fleurus, Steinkirk and Neerwinden. On the death of his cousin, Jean Louis Charles, duc de Longueville (1646-1694), Conti in accordance with his cousin’s will, claimed the principality of Neuchâtel against Marie, duchesse de Nemours (1625-1707), a sister of the duke. He failed to obtain military assistance from the Swiss, and by the king’s command yielded the disputed territory to Marie, although the courts of law had decided in his favour. In 1697 Louis XIV. offered him the Polish crown, and by means of bribes the abbé de Polignac secured his election. Conti started rather unwillingly for his new kingdom, probably, as St Simon remarks, owing to his affection for Françoise, wife of Philip II., duke of Orleans, and daughter of Louis XIV. and Madame de Montespan. When he reached Danzig and found his rival Augustus II., elector of Saxony, already in possession of the Polish crown, he returned to France, where he was graciously received by Louis, although St Simon says the king was vexed to see him again. But the misfortunes of the French armies during the earlier years of the war of the Spanish Succession compelled Louis to appoint Conti, whose military renown stood very high, to command the troops in Italy. He fell ill before he could take the field, and died on the 9th of February 1709, his death calling forth exceptional signs of mourning from all classes.
Louis Armand de Bourbon, prince de Conti (1606-1727), eldest son of the preceding, was treated with great liberality by Louis XIV., and also by the regent, Philip duke of Orleans. He served under Marshal Villars in the War of the Spanish Succession, but he lacked the soldierly qualities of his father. In 1713 he married Louise Elisabeth (1693-1775), daughter of Louis Henri de Bourbon, prince de Condé, and grand-daughter of Louis XIV. He was a prominent supporter of the financial schemes of John Law, by which he made large sums of money.
Louis François de Bourbon, prince de Conti (1717-1776), only son of the preceding, adopted a military career, and when the war of the Austrian Succession broke out in 1741 accompanied Charles Louis, duc de Belle-Isle, to Bohemia. His services there led to his appointment to command the army in Italy, where he distinguished himself by forcing the pass of Villafranca and winning the battle of Coni in 1744. In 1745 he was sent to check the Imperialists in Germany, and in 1746 was transferred to the Netherlands, where some jealousy between Marshal Saxe and himself led to his retirement in 1747. In this year a faction among the Polish nobles offered Conti the crown of that country, where owing to the feeble health of King Augustus III. a vacancy was expected. He won the personal support of Louis XV. for his candidature, although the policy of the French ministers was to establish the house of Saxony in Poland, as the dauphiness was a daughter of Augustus. Louis therefore began secret personal relations with his ambassadors in eastern Europe, who were thus receiving contradictory instructions; a policy known later as thesecret du roi. Although Conti did not secure the Polish throne he remained in the confidence of Louis until 1755, when his influence was destroyed by the intrigues of Madame de Pompadour; so that when the Seven Years’ War broke out in 1756 he was refused the command of the army of the Rhine, and began the opposition to the administration which caused Louis to refer to him as “my cousin the advocate.” In 1771 he was prominent in opposition to the chancellor Maupeou. He supported the parlements against the ministry, was especially active in his hostility to Turgot, and was suspected of aiding a rising which took place at Dijon in 1775. Conti, who died on the 2nd of August 1776, inherited literary tastes from his father, was a brave and skilful general, and a diligent student of military history. His house, over which the comtesse de Boufflers presided, was the resort of many men of letters, and he was a patron of Jean Jacques Rousseau.
Louis François Joseph, prince de Conti (1734-1814), son of the preceding, possessed considerable talent as a soldier, and distinguished himself during the Seven Years’ War. He took the side of Maupeou in the struggle between the chancellor and the parlements, and in 1788 declared that the integrity of the constitution must be maintained. He emigrated owing to the weakness of Louis XVI., but refused to share in the plans for the invasion of France, and returned to his native country in 1790. Arrested by order of the National Convention in 1793, he was acquitted, but was reduced to poverty by the confiscation of his possessions. He afterwards received a pension, but the Directory banished him from France, and as he refused to share in the plots of the royalists he lived at Barcelona till his death in 1814, when the house of Conti became extinct.
See F. de Bassompierre,Mémoires(Paris, 1877); G. Tallemant des Reaux,Historiettes(Paris, 1854-1860); L. de R. duc de Saint Simon,Mémoires(Paris, 1873); C. E. duchesse d’Orleans,Mémoires(Paris, 1880); R. L. Marquis d’Argenson,Journal et mémoires(Paris, 1859-1865); F. J. de P. cardinal de Bérnis,Mémoires et lettres(Paris, 1878); J. V. A. duc de Broglie,Le Secret du roi(Paris, 1878); P. A. Cheruel,Histoire de la minorité de Louis XIV et du ministère de Mazarin(Paris, 1879); E. Boutaric,Correspondence secrète de Louis XV sur la politique étrangère(Paris, 1866); P. Foncin,Essai sur le ministère de Turgot(Paris, 1877); E. BourgeoisNeuchâtel et la politique prussienne en Franche-Comté(Paris, 1877).
See F. de Bassompierre,Mémoires(Paris, 1877); G. Tallemant des Reaux,Historiettes(Paris, 1854-1860); L. de R. duc de Saint Simon,Mémoires(Paris, 1873); C. E. duchesse d’Orleans,Mémoires(Paris, 1880); R. L. Marquis d’Argenson,Journal et mémoires(Paris, 1859-1865); F. J. de P. cardinal de Bérnis,Mémoires et lettres(Paris, 1878); J. V. A. duc de Broglie,Le Secret du roi(Paris, 1878); P. A. Cheruel,Histoire de la minorité de Louis XIV et du ministère de Mazarin(Paris, 1879); E. Boutaric,Correspondence secrète de Louis XV sur la politique étrangère(Paris, 1866); P. Foncin,Essai sur le ministère de Turgot(Paris, 1877); E. BourgeoisNeuchâtel et la politique prussienne en Franche-Comté(Paris, 1877).
CONTI, NICOLO DE’(fl. 1419-1444), Venetian explorer and writer, was a merchant of noble family, who left Venice about 1419, on what proved an absence of 25 years. We next find him in Damascus, whence he made his way over the north Arabian desert, the Euphrates, and southern Mesopotamia, to Bagdad. Here he took ship and sailed down the Tigris to Basra and the head of the Persian Gulf; he next descended the gulf to Ormuz, coasted along the Indian Ocean shore ofPersia (at one port of which he remained some time, and entered into a business partnership with some Persian merchants), and so reached the gulf and city of Cambay, where he began his Indian life and observations. He next dropped down the west coast of India to Ely, and struck inland to Vijayanagar, the capital of the principal Hindu state of the Deccan, destroyed in 1555. Of this city Conti gives an elaborate description, one of the most interesting portions of his narrative. From Vijayanagar and the Tungabudhra he travelled to Maliapur near Madras, the traditional resting-place of the body of St Thomas, and the holiest shrine of the native Nestorian Christians, then “scattered over all India,” the Venetian declares, “as the Jews are among us.” The narrative next refers to Ceylon, and gives a very accurate account of the Cingalese cinnamon tree; but, if Conti visited the island at all, it was probably on the return journey. His outward route now took him to Sumatra, where he stayed a year, and of whose cruel, brutal, cannibal natives he gained a pretty full knowledge, as of the camphor, pepper and gold of this “Taprobana.” From Sumatra a stormy voyage of sixteen days brought him to Tenasserim, near the head of the Malay Peninsula. We then find him at the mouth of the Ganges, and trace him ascending and descending that river (a journey of several months), visiting Burdwan and Aracan, penetrating into Burma, and navigating the Irawadi to Ava. He appears to have spent some time in Pegu, from which he again plunged into the Malay Archipelago, and visited Java, his farthest point. Here he remained nine months, and then began his return by way ofCiampa(usually Cochin-China in later medieval European literature, but here perhaps some more westerly portion of Indo-China); a month’s voyage from Ciampa brought him toColoen, doubtless Kulam or Quilon, in the extreme south-west of India. Thence he continued his homeward route, touching at Cochin, Calicut and Cambay, to Sokotra, which he describes as still mainly inhabited by Nestorian Christians; to the “rich city” of Aden, “remarkable for its buildings”; toGiddaor Jidda, the port of Mecca; over the desert toCarrasor Cairo; and so to Venice, where he arrived in 1444.
As a penance for his (compulsory) renunciation of the Christian faith during his wanderings, Eugenius IV. ordered him to relate his history to Poggio Bracciolini, the papal secretary. The narrative closes with Conti’s elaborate replies to Poggio’s question on Indian life, social classes, religion, fashions, manners, customs and peculiarities of various kinds. Following a prevalent fashion, the Venetian divides his Indies into three parts, the first extending from Persia to the Indus; the second from the Indus to the Ganges; the third including all beyond the Ganges; this last he considered to excel the others in wealth, culture and magnificence, and to be abreast of Italy in civilization. We may note, moreover, Conti’s account of the bamboo in the Ganges valley; of the catching, taming and rearing of elephants in Burma and other regions; of Indian tattooing and the use of leaves for writing; of various Indian fruits, especially the jack and mango; of the polyandry of Malabar; of the cockfighting of Java; of what is apparently the bird of Paradise; of Indian funeral ceremonies, and especiallysuttee; of the self-mutilation and immolation of Indian fanatics; and of Indian magic, navigation (“they are not acquainted with the compass”), justice, &c. Several venerable legends are reproduced; and Conti’s name-forms, partly through Poggio’s vicious classicism, are often absolutely unrecognizable; but on the whole this is the best account of southern Asia by any European of the 15th century; while the traveller’s visit to Sokotra is an almost though not quite unique performance for a Latin Christian of the middle ages.
The original Latin is in Poggio’sDe varietate Fortunae, book iv.; see the edition of the Abbé Oliva (Paris, 1723). The Italian version, printed in Ramusio’sNavigationi et viaggi, vol. i., is only from a Portuguese translation made in Lisbon. An English translation with short notes was made by J. Winter Jones for the Hakluyt Society in the vol. entitledIndia in the Fifteenth Century(London, 1857); an introductory account of the traveller and his work by R. H. Major precedes.(C. R. B.)
The original Latin is in Poggio’sDe varietate Fortunae, book iv.; see the edition of the Abbé Oliva (Paris, 1723). The Italian version, printed in Ramusio’sNavigationi et viaggi, vol. i., is only from a Portuguese translation made in Lisbon. An English translation with short notes was made by J. Winter Jones for the Hakluyt Society in the vol. entitledIndia in the Fifteenth Century(London, 1857); an introductory account of the traveller and his work by R. H. Major precedes.
(C. R. B.)
CONTINENT(from Lat.continere, “to hold together”; hence “connected,” “continuous”), a word used in physical geography of the larger continuous masses of land in contrast to the great oceans, and as distinct from the submerged tracts where only the higher parts appear above the sea, and from islands generally.
On looking at a map of the world, continents appear generally as wedge-shaped tracts pointing southward, while the oceans have a polygonal shape. Eurasia is in some sense an exception, but all the southern terminations of the continents advance into the sea in the form of a wedge—South America, South Africa, Arabia, India, Malaysia and Australia connected by a submarine platform with Tasmania. It is difficult not to believe that these remarkable characters have some relation to the structure of the great globe-mass, and according to T. C. Chamberlin and R. D. Salisbury, in theirGeology(1906), “the true conception is perhaps that the ocean basins and continental platforms are but the surface forms of great segments of the lithosphere, all of which crowd towards the centre, the stronger and heavier—the ocean basins—taking precedence and squeezing the weaker and lighter ones—the continents—between them.” “The area of the most depressed, or master segments, is almost exactly twice that of the protruding or squeezed ones. This estimate includes in the latter about 10,000,000 sq. m. now covered with shallow water. The volume of the hydrosphere is a little too great for the true basins, and it runs over, covering the borders of the continents” (seeContinental Shelf). Several theories have been advanced to account for the roughly triangular shape of the continents, but that presenting the least difficulty is the one expressed above, “since in a spherical surface divided into larger and smaller segments the major part should be polygonal, while the minor residual segments are more likely to be triangular.”
As bearing on this geological idea, it is interesting to notice in this connexion that the areas of volcanic activity are mostly where continent and ocean meet; and that around the continents there is an almost continuous “deep” from 100 to 300 m. broad, of which the Challenger Deep (11,400 ft.) and the great Tuscarora Deep are fragments. If on a map of the world a broad inked brush be swept seawards round Africa, passing into the Mediterranean, round North and South America, round India, then continuously south of Java and round Australia south of Tasmania and northward to the tropic, this broad band will represent the encircling ribbon-like “deep,” which gives strength to the suggestion that the continents in their main features are permanent forms and that their structural connexion with the oceans is not temporary and accidental. The great protruding or “squeezed” segments are the Eurasian (with an area roughly of twenty-four, reckoning in millions of square miles), strongly ridged on the south and east, and relatively flat on the north-west; the African (twelve), rather strongly ridged on the east, less abruptly on the west and north; the North American (ten), strongly ridged on the west, more gently on the east, and relatively flat on the north and in the interior; the South American (nine), strongly ridged on the west and somewhat on the north-east and south-east, leaving ten for the smaller blocks. The sum of these will represent one-third of the earth’s surface, while the remaining two-thirds is covered by the ocean. The foundation structure of the continents is everywhere similar. Their resulting rocks and soils are due to differential minor movements in the past, by which deposits of varying character were produced. These movements, taking place periodically and followed by long periods of rest, produce continued stability for the development and migration of forms of life, the grading of rivers, the development of varied characteristic land forms, the migration and settlement of human beings, the facility or difficulty of intelligent intercourse between races and communities, with finally the commercial interchange of those commodities produced by varying climatic conditions upon different parts of the continental surface; in short, for those geographical factors which form the chief product of past and present human history. (SeeGeography.)
CONTINENTAL SHELF,the term in physical geography for the submerged platform upon which a continent or island stands in relief. If a coin or medal be partly sunk under water the image and superscription will stand above water and represent a continent with adjacent islands; the sunken part just submerged will represent the continental shelf and the edge of the coin the boundary between it and the surrounding deep, called by Professor H. K. H. Wagner the continental slope. If the lithosphere surface be divided into three parts, namely, the continent heights, the ocean depths, and the transitional area separating them, it will be found that this transitional area is almost bisected by the coast-line, that nearly one-half of it (10,000,000 sq. m.) lies under water less than 100 fathoms deep, and the remainder 12,000,000 sq. m. is under 600 ft. in elevation. There are thus two continuous plain systems, one above water and one under water, and the second of these is called the continental shelf. It represents the area which would be added to the land surface if the sea fell 600 ft. This shelf varies in width. Round Africa—except to the south—and off the western coasts of America it scarcely exists. It is wide under the British Islands and extends as a continuous platform under the North Sea, down the English Channel to the south of France; it unites Australia to New Guinea on the north and to Tasmania on the south, connects the Malay Archipelago along the broad shelf east of China with Japan, unites north-western America with Asia, sweeps in a symmetrical curve outwards from north-eastern America towards Greenland, curving downwards outside Newfoundland and holding Hudson Bay in the centre of a shallow dish. In many places it represents the land planed down by wave action to a plain of marine denudation, where the waves have battered down the cliffs and dragged the material under water. If there were no compensating action in the differential movement of land and sea in the transitional area, the whole of the land would be gradually planed down to a submarine platform, and all the globe would be covered with water. There are, however, periodical warpings of this transitional area by which fresh areas of land are raised above sea-level, and fresh continental coast-lines produced, while the sea tends to sink more deeply into the great ocean basins, so that the continents slowly increase in size. “In many cases it is possible that the continental shelf is the end of a low plain submerged by subsidence; in others a low plain may be an upheaved continental shelf, and probably wave action is only one of the factors at work” (H. R. Mill,Realm of Nature, 1897).
CONTINUED FRACTIONS.In mathematics, an expression of the form
where a1, a2, a3, ... and b2, b3, b4, ... are any quantities whatever, positive or negative, is called a “continued fraction.” The quantities a1..., b2... may follow any law whatsoever. If the continued fraction terminates, it is said to be a terminating continued fraction; if the number of the quantities a1..., b2... is infinite it is said to be anon-terminatingorinfinitecontinued fraction. If b2/a2, b3/a3..., thecomponent fractions, as they are called, recur, either from the commencement or from some fixed term, the continued fraction is said to berecurringorperiodic. It is obvious that every terminating continued fraction reduces to a commensurable number.
The notation employed by English writers for the general continued fraction is
Continental writers frequently use the notation
The terminating continued fractions
reduced to the forms
are called the successive convergents to the general continued fraction.
Their numerators are denoted by p1, p2, p3, p4...; their denominators by q1, q2, q3, q4....
We have the relations
pn= anpn-1+ bnpn-2, qn= anqn-1+ bnqn-2.
In the case of the fraction
we have the relations pn= anpn-1- bnpn-2, qn= anqn-1- bnqn-2.
Taking the quantities a1..., b2... to be all positive, a continued fraction of the form
is called acontinued fraction of the first class; a continued fraction of the form
is called acontinued fraction of the second class.
A continued fraction of the form
where a1, a2, a3, a4... are allpositive integers, is called asimple continued fraction. In the case of this fraction a1, a2, a3, a4... are called the successivepartial quotients. It is evident that, in this case,
p1, p2, p3..., q1, q2, q3...,
are two series of positive integers increasing without limit if the fraction does not terminate.
The general continued fraction
is evidently equal, convergent by convergent, to the continued fraction
where λ2, λ3, λ4, ... are any quantities whatever, so that by choosing λ2b2= 1, λ2λ3b3= 1, &c., it can be reduced to any equivalent continued fraction of the form
Simple Continued Fractions.
1. The simple continued fraction is both the most interesting and important kind of continued fraction.
Any quantity, commensurable or incommensurable, can be expressed uniquely as a simple continued fraction, terminating in the case of a commensurable quantity, non-terminating in the case of an incommensurable quantity. A non-terminating simple continued fraction must be incommensurable.
In the case of a terminating simple continued fraction the number of partial quotients may be odd or even as we please by writing the last partial quotient, anas
The numerators and denominators of the successive convergents obey the law pnqn-1- pn-1qn= (-1)n, from which it follows at once that every convergent is in its lowest terms. The other principal properties of the convergents are:—
The odd convergents form an increasing series of rational fractions continually approaching to the value of the whole continued fraction; the even convergents form a decreasing series having the same property.
Every even convergent is greater than every odd convergent; every odd convergent is less than, and every even convergent greater than, any following convergent.
Every convergent is nearer to the value of the whole fraction than any preceding convergent.
Every convergent is a nearer approximation to the value of the whole fraction than any fraction whose denominator is less than that of the convergent.
The difference between the continued fraction and the nthconvergent is
These limits may be replaced by the following, which, though not so close, are simpler, viz.
Every simple continued fraction must converge to a definite limit; for its value lies between that of the first and second convergents and, since
so that its value cannot oscillate.
The chief practical use of the simple continued fraction is that by means of it we can obtain rational fractions which approximate to any quantity, and we can also estimate the error of ourapproximation. Thus a continued fraction equivalent to π (the ratio of the circumference to the diameter of a circle) is
of which the successive convergents are
the fourth of which is accurate to the sixth decimal place, since the error lies between 1/{q4q5} or .0000002673 and a6/{q4q6} or .0000002665.
Similarly the continued fraction given by Euler as equivalent to ½(e - 1) (e being the base of Napierian logarithms), viz.
may be used to approximate very rapidly to the value of e.
For the application of continued fractions to the problem “To find the fraction, whose denominator does not exceed a given integer D, which shall most closely approximate (by excess or defect, as may be assigned) to a given number commensurable or incommensurable,” the reader is referred to G. Chrystal’sAlgebra, where also may be found details of the application of continued fractions to such interesting and important problems as the recurrence of eclipses and the rectification of thecalendar(q.v.).
Lagrange used simple continued fractions to approximate to the solutions of numerical equations; thus, if an equation has a root between two integers a and a + 1, put x = a + 1/y and form the equation in y; if the equation in y has a root between b and b + 1, put y = b + 1/z, and so on. Such a method is, however, too tedious, compared with such a method as Homer’s, to be of any practical value.
The solution in integers of the indeterminate equation ax + by = c may be effected by means of continued fractions. If we suppose a/b to be converted into a continued fraction and p/q to be the penultimate convergent, we have aq - bp = +1 or -1, according as the number of convergents is even or odd, which we can take them to be as we please. If we take aq-bp = +1 we have a general solution in integers of ax + by = c, viz. x = cq - bt, y = at - cp; if we take aq - bp = -1, we have x = bt - cq, y = cp - at.
An interesting application of continued fractions to establish a unique correspondence between the elements of an aggregate of m dimensions and an aggregate of n dimensions is given by G. Cantor in vol. 2 of theActa Mathematica.
Applications of simple continued fractions to the theory of numbers, as, for example, to prove the theorem that a divisor of the sum of two squares is itself the sum of two squares, may be found in J. A. Serret’sCours d’Algèbre Supérieure.
2.Recurring Simple Continued Fractions.—The infinite continued fraction
where, after the nthpartial quotient, the cycle of partial quotients b1, b2, ..., bnrecur in the same order, is the type of a recurring simple continued fraction.
The value of such a fraction is the positive root of a quadratic equation whose coefficients are real and of which one root is negative. Since the fraction is infinite it cannot be commensurable and therefore its value is a quadratic surd number. Conversely every positive quadratic surd number, when expressed as a simple continued fraction, will give rise to a recurring fraction. Thus
The second case illustrates a feature of the recurring continued fraction which represents a complete quadratic surd. There is only one non-recurring partial quotient a1. If b1, b2, ..., bnis the cycle of recurring quotients, then bn= 2a1, b1= bn-1, b2= bn-2, b3= bn-3, &c.
In the case of a recurring continued fraction which represents √N, where N is an integer, if n is the number of partial quotients in the recurring cycle, and pnr/qnrthe nrthconvergent, then p2nr-Nq2nr= (-1)nr, whence, if n is odd, integral solutions of the indeterminate equation x² - Ny² = ±1 (the so-called Pellian equation) can be found. If n is even, solutions of the equation x² -Ny² = +1 can be found.
The theory and development of the simple recurring continued fraction is due to Lagrange. For proofs of the theorems here stated and for applications to the more general indeterminate equation x² -Ny² = H the reader may consult Chrystal’sAlgebraor Serret’sCours d’Algèbre Supérieure; he may also profitably consult a tract by T. Muir,The Expression of a Quadratic Surd as a Continued Fraction(Glasgow, 1874).
The General Continued Fraction.
1.The Evaluation of Continued Fractions.—The numerators and denominators of the convergents to the general continued fraction both satisfy the difference equation un= anun-1+ bnun-2. When we can solve this equation we have an expression for the nthconvergent to the fraction, generally in the form of the quotient of two series, each ofnterms. As an example, take the fraction (known as Brouncker’s fraction, after Lord Brouncker)
Here we have
un+1= 2un+ (2n-1)²un-1,
whence
un+1- (2n + 1)un= -(2n - 1){un- (2n - 1)un-1},
and we readily find that
whence the value of the fraction taken to infinity is ¼π.
It is always possible to find the value of the nthconvergent to a recurring continued fraction. If r be the number of quotients in the recurring cycle, we can by writing down the relations connecting the successive p’s and q’s obtain a linear relation connecting
pnr+m, p(n-1)r+m, p(n-2)r+m
in which the coefficients are all constants. Or we may proceed as follows. (We need not consider a fraction with a non-recurring part). Let the fraction be
leading to an equation of the form Aunun-1+ Bun+ Cun-1+ D = 0, where A, B, C, D are independent of n, which is readily solved.
2.The Convergence of Infinite Continued Fractions.—We have seen that the simple infinite continued fraction converges. The infinite general continued fraction of the first class cannot diverge for its value lies between that of its first two convergents. It may, however, oscillate. We have the relation pnqn-1- pn-1qn= (-1)nb2b3...bn, from which
and the limit of the right-hand side is not necessarily zero.
The tests for convergency are as follows:
Let the continued fraction of the first class be reduced to the form
then it is convergent if at least one of the series d3+ d5+ d7+ ..., d2+ d4+ d6+ ... diverges, and oscillates if both these series converge.
For the convergence of the continued fraction of the second class there is no complete criterion. The following theorem covers a large number of important cases.
“If in the infinite continued fraction of the second class an≥ bn+ 1 for all values of n, it converges to a finite limit not greater than unity.”
3.The Incommensurability of Infinite Continued Fractions.—There is no general test for the incommensurability of the general infinite continued fraction.
Two cases have been given by Legendre as follows:—
If a2, a3, ..., an, b2, b3, ..., bnare all positive integers, then
I. The infinite continued fraction
converges to an incommensurable limit if after some finite value of n the condition an≠ bnis always satisfied.
II. The infinite continued fraction
converges to an incommensurable limit if after some finite value of n the condition an≥ bn+ 1 is always satisfied, where the sign > need not always occur but must occurinfinitely often.