Chapter VI: Act 1888: , s. 162), and may also attack persons who having means refuse (1)

to comply with an order to pay money, or refuse to comply with an order to deliver up a specific chattel or disobey an injunction. A court of quarter sessions has at common law a like power as to contempts _in facie curiae_ and is said to have power to punish its officials for contempt in non-attendance or neglect of duty.

Punishment.

Contempt of court is a misdemeanour and is punishable by fine and imprisonment or either at discretion. The offence may be tried summarily, or may be prosecuted on information or on indictment as was done in the case of the _Weekly Dispatch_ already mentioned. The prerogative of pardon extends to all contempts of court which are dealt with by a sentence of clearly punitive character; but it is doubtful whether it extends to committals for disobedience to orders made in aid of the execution of a civil judgment.

Contempt is usually dealt with summarily by the court contemned in the case of contempt _in facie curiae_. The offender may be instantly apprehended and without further proof or examination fined or sent to prison. In the case of other contempts the High Court not only can deal with contempts affecting itself, but can also intervene summarily to protect inferior courts from contempts. This jurisdiction was asserted and exercised in the Moat Farm case (1903) and the _South Wales Post_ case (1905) already mentioned.

Except in cases of contempt _in facie curiae_ evidence on oath as to the alleged contempt must be laid before the court, and application made for the "committal" or "attachment" of the offender. The differences between the two modes are technical rather than substantial.

The procedure for dealing with contempt of court varies somewhat according as the contempt consists in disobeying an order of the High Court made in a civil cause, or consists in interference with the course of justice by persons not present in court nor parties to the cause. In the first class of cases the court proceeds by order of committal or giving leave to issue writ of attachment. In either case the person said to be in contempt must have full notice of the proposed motion and of the grounds on which he is said to be in contempt; and the rules regulating such proceedings must be strictly complied with (_R._ v. _Tuck_, 1906, 2 Ch. 692). In proceedings on the crown side of the king's bench division it is still usual to apply in the first place for a rule nisi for leave to attach the alleged offender who is given an opportunity of explaining, excusing or justifying the incriminated acts. It is essential that before punishment the alleged offender should have had full notice as to the specific offence charged and opportunity of answering to it. The king's bench procedure is that generally used for interference with the due course of criminal justice or disobedience to prerogative writs such as _mandamus_.

An order of committal is an order in execution specifying the nature of the detention to be suffered, or the penalty to be paid. The process of attachment merely brings the accused into court; he is then required to answer on oath interrogatories administered to him, so that the court may be better informed of the circumstances of the contempt. If he can clear himself on oath he is discharged; if he confesses the court will punish him by fine or imprisonment, or both, at its discretion. But in very many cases on proper apology and submission, and undertaking not to repeat the contempt, and payment of costs, the court allows the proceedings to drop without proceeding to fine or imprison.

From time to time proposals have been made to deprive the superior courts of the power to deal summarily with contempts not committed _in facie curiae_, and to require proceedings on other charges for contempt to go before a jury. This distinction has already been made in some British colonies, e.g. British Guiana, by an ordinance of 1900 (No. 31). Recent decisions in England have so fully defined the limits of the offence and declared the practice of the courts that it would probably only result in undue licence of the press if the power now carefully and judicially exercised of dealing summarily with journalistic interference with the ordinary course of justice were taken away and the delay involved in submitting the case to a jury were made inevitable. The courts now only act in clear cases, and in cases of doubt can always send the question to a jury. The experience of other countries makes it undesirable to part with the summary remedy so long as it is in the hands of a trusted judicature.

_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 imprisonment
_proprio motu_ without 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 Conde, was taken from Conti-sur-Selles, a small town about 20 m. S.W. of Amiens, which came into the Conde family by the marriage of Louis of Bourbon, first prince of Conde, with Eleanor de Roye in 1551.

FRANCOIS (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 Coeeme, heiress of Bonnetable, 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'Estrees, 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 Francois de Bassompierre (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 wrote _Aventures 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 of ARMAND DE BOURBON (1629-1666), second son of Henry II. of Bourbon, prince of Conde, and brother of Louis, the great Conde. 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 Conde 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 Moliere's for whom he secured an introduction to the court of Louis XIV., but afterwards, when writing a treatise against the stage entitled _Traite de la comedie et des spectacles selon les traditions de l'Eglise_ (Paris, 1667), he charged the dramatist with keeping a school of atheism. Conti also wrote _Lettres sur la grace_, and _Du 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 Valliere. 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.

FRANCOIS 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 as _le roi an theatre_, for which on his return to France he was temporarily banished to Chantilly. Conti was a favourite of his uncle the great Conde, whose grand-daughter Marie Therese 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 Neuchatel 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 abbe de Polignac secured his election. Conti started rather unwillingly for his new kingdom, probably, as St Simon remarks, owing to his affection for Francoise, 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 Conde, 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 FRANCOIS 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 the _secret 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 FRANCOIS 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, _Memoires_ (Paris, 1877); G. Tallemant des
Reaux, _Historiettes_ (Paris, 1854-1860); L. de R. duc de Saint
Simon, _Memoires_ (Paris, 1873); C. E. duchesse d'Orleans, _Memoires_
(Paris, 1880); R. L. Marquis d'Argenson, _Journal et memoires_
(Paris, 1859-1865); F. J. de P. cardinal de Bernis, _Memoires et
lettres_ (Paris, 1878); J. V. A. duc de Broglie, _Le Secret du roi_
(Paris, 1878); P. A. Cheruel, _Histoire de la minorite de Louis XIV
et du ministere de Mazarin_ (Paris, 1879); E. Boutaric,
_Correspondence secrete de Louis XV sur la politique etrangere_
(Paris, 1866); P. Foncin, _Essai sur le ministere de Turgot_ (Paris,
1877); E. Bourgeois _Neuchatel et la politique prussienne en
Franche-Comte_ (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 of Persia (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 of _Ciampa_ (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 to _Coloen_, 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"; to _Gidda_ or Jidda, the port of Mecca; over the desert to _Carras_ or 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 especially _suttee_; 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's _De varietate Fortunae_, book iv.;
see the edition of the Abbe Oliva (Paris, 1723). The Italian version,
printed in Ramusio's _Navigationi 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. entitled _India 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 their _Geology_ (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" (see Continental 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. (See Geography.)

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

b2
a1 +- -----------
b3
a2 +- -----------
b4
a3 +- ----------
b5
a4 +- ----
a5 +- ...,

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 a _non-terminating_ or _infinite_ continued fraction. If b2/a2, b3/a3 ..., the _component fractions_, as they are called, recur, either from the commencement or from some fixed term, the continued fraction is said to be _recurring_ or _periodic_. 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

b2 b3 b4
a1 +- -- -- -- ...
a2 +- a3 +- a4 +-

Continental writers frequently use the notation

b2 b3 b4 b2 | b3 | b4 |
a1 +- -- +- -- +- -- +- ..., or a1 +- |----| +- |----| +- |----| +- ...
a2 a3 a4 | a2 | a3 | a4

The terminating continued fractions

b2 b2 b3 b2 b3 b4
a1, a1 + --, a1 + -- --, a1 + -- -- --, ...
a2 a2 + a3 a2 + a3 + a4

reduced to the forms

a1 a1a2 + b2 a1a2a3 + b2a3 + b2a1
--, ---------, --------------------,
1 a2 a2a3 + b3

a1a2a3a4 + b2a3a4 + b3a1a4 + b4a1a2 + b2b4
------------------------------------------, ...
a2a3a4 + a4b3 + a2b4

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

p_n = a_{n}p_{n-1} + b_{n}p_{n-2}, q_n = a_{n}q_{n-1} + b_{n}q_{n-2}.

b2 b3 b4
In the case of the fraction a1 - -- -- -- ..., we have the
a2 - a3 - a4 -

relations

p_n = a_{n}p_{n-1} - b_{n}p_{n-2}, q_n= a_{n}q_{n-1} - b_{n}q_{n-2}.

Taking the quantities a1 ..., b2 ... to be all positive, a continued

b2 b3
fraction of the form a1 + -- -- ... is called a _continued fraction
a2 + a3 +

b2 b3 b4
of the first class_; a continued fraction of the form -- -- -- ...
a2 - a3 - a4 -

called a _continued fraction of the second class_.

1 1 1
A continued fraction of the form a1 + -- -- -- ..., where
a2 + a3 + a4 +

a1, a2, a3, a4 ... are all _positive integers_, is called a _simple continued fraction_. In the case of this fraction a1, a2, a3, a4 ... are called the successive _partial 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.

b2 b3 b4
The general continued fraction a1 + -- -- -- ... is evidently
a2 + a3 + a4 +

equal, convergent by convergent, to the continued fraction

[lambda]2b2 [lambda]2[lambda]3b3 [lambda]3[lambda]4b4
a1 + ----------- -------------------- -------------------- ...,
[lambda]2a2 + [lambda]3a3 + [lambda]4a4 +

where [lambda]2, [lambda]3, [lambda]4, ... are any quantities whatever, so that by choosing [lambda]2b2 = 1, [lambda]2[lambda]3b3 = 1, &c., it can be reduced to any equivalent continued fraction of the form

1 1 1
a1 + -- -- -- ...
d2 + d3 + d4 +

_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

1
partial quotient, a_n as a_n - 1 + --.
1

The numerators and denominators of the successive convergents obey the law p_{n}q_{n-1} - p_{n-1}q_n = (-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 n^{th} convergent

1 a_{n+2}
is less than ------------, and greater than ------------. These limits
q_{n}q_{n+1} q_{n}q_{n+2}

may be replaced by the following, which, though not so close, are

1 1
simpler, viz. ------- and ------------------ .
q^{2}_n q_n(q_n + q_{n+1})

Every simple continued fraction must converge to a definite limit; for its value lies between that of the first and second convergents and, since

p_n p_{n-1} 1 p_n p_{n-1}
--- ~ ------- = ------------, Lt. ----- = Lt. -------,
q_n q_{n-1} q_{n}q_{n-1} q_n q_{n-1}

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 our approximation. Thus a continued fraction equivalent to [pi] (the ratio of the circumference to the diameter of a circle) is

1 1 1 1 1 1
3 + - -- -- --- -- --
7 + 15 + 1 + 292 + 1 + 1 + ...

of which the successive convergents are

3 22 333 355 103993
--, --, ---, ---, ------, &c.,
1 7 106 113 33102

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 1/2(e -1) (e being the base of Napierian logarithms), viz.

1 1 1 1 1
-- -- -- -- --
1 + 6 + 10 + 14 + 18 + ...,

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's _Algebra_, 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 the calendar (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 the _Acta 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's _Cours d'Algebre Superieure_.

2. _Recurring Simple Continued Fractions._--The infinite continued fraction

1 1 1 1 1 1 1 1 1 1
a1 + -- -- --- -- -- --- -- -- --- --
a2 + a3 ... + a_n + b1 + b2 ... + b_n + b1 + b2 ... + b_n + b1 + ...,

where, after the n^{th} partial quotient, the cycle of partial quotients b1, b2, ..., b_n recur 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

__ 1 1 1 1 1
2 - \/ 3 = -- -- -- -- --
3 + 1 + 2 + 1 + 2 + ...,

___ 1 1 1 1 1 1 1 1
\/ 28 = 5 + -- -- -- -- -- -- -- --
3 + 2 + 3 + 10 + 3 + 2 + 3 + 10 + ...

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, ..., b_n is the cycle of recurring quotients, then b_n = 2a1, b1 = b_{n-1}, b2 = b_{n-2}, b3 = b_{n-3}, &c.

In the case of a recurring continued fraction which represents [sqr]N, where N is an integer, if n is the number of partial quotients in the recurring cycle, and p_{nr}/q_{nr} the nr^{th} convergent, then p^2_{nr} -Nq^2_{nr} = (-1)^{nr}, whence, if n is odd, integral solutions of the indeterminate equation x squared - Ny squared = +-1 (the so-called Pellian equation) can be found. If n is even, solutions of the equation x squared -Ny squared = +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 x squared -Ny squared = H the reader may consult Chrystal's _Algebra_ or Serret's _Cours d'Algebre Superieure_; 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 u_n = a_{n}u_{n-1} + b_{n}u_{n-2}. When we can solve this equation we have an expression for the n^{th} convergent to the fraction, generally in the form of the quotient of two series, each of n terms. As an example, take the fraction (known as Brouncker's fraction, after Lord Brouncker)

1 1 squared 3 squared 5 squared 7 squared
-- -- -- -- --
1 + 2 + 2 + 2 + 2 + ...

Here we have

u_{n+1} = 2u_n + (2n-1) squaredu_{n-1},

whence

u_{n+1} - (2n + 1)u_n = -(2n - 1){u_n - (2n - 1)u_{n-1}},

and we readily find that

p_n 1 1 1 1
----- = 1 - -- + -- - -- + ... +- ------,
q_n 3 5 7 2n + 1

whence the value of the fraction taken to infinity is 1/4[pi].

It is always possible to find the value of the n^{th} convergent 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

p_{nr+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

a1 a2 a_r a1
-- -- --- --
b1 + b2 + ... + b_r + b1 + ...

p_{nr+m} a1 a2 a_r
Let u_n = --------; then u_n = -- -- ------------, leading
q_{nr+m} b1 + b2 + ... + b_r + u_{n1}

to an equation of the form Au_{n}u_{n-1} + Bu_n + Cu_{n-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 p_{n}q_{n-1} - p_{n-1}q_n = (-1)^{n}b2b3...b_n,

p_n p_{n-1} b2b3 ... b_n
from which --- - ------- = (-1)^n ------------, and the limit of the
q_n q_{n-1} q_{n}q_{n-1}

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

1 1 1
d1 + -- -- -- , then it is convergent if at least one of the
d2 + d3 + d4 + ...

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 a_n [>=] b_n + 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, ..., a_n, b2, b3, ...,b_n are all positive integers, then

b2 b3 b_{n}
I. The infinite continued fraction -- -- ----- converges
a2 + a3 + ... + a_{n} + ...

to an incommensurable limit if after some finite value of n the condition a_{n} [not <] b_{n} is always satisfied.

b2 b3 b_{n}
II. The infinite continued fraction -- -- -----
a2 - a3 - ... - a_{n} - ...

converges to an incommensurable limit if after some finite value of n the condition a_{n} [>=] b_{n} + 1 is always satisfied, where the sign > need not always occur but must occur _infinitely often_.

_Continuants._

The functions p_{n} and q_{n}, regarded as functions of a1, ..., a_{n}, b2, ..., b_{n} determined by the relations

p_{n} = a_{n}p_{n-1} + b_{n}p_{n-2},
q_{n} = a_{n}q_{n-1} + b_{n}q_{n-2},

with the conditions p1 = a1, p0 = 1; q2 = a2, q1 = 1, q0 = 0, have been studied under the name of _continuants_. The notation adopted is

/ b2,...,b_{n}\
p_{n} = K ( ),
\a1, a2,...,a_{n}/

and it is evident that we have

/ b3,...,b_{n}\
q_{n} = K ( ).
\a2, a3,...,a_{n}/

The theory of continuants is due in the first place to Euler. The reader will find the theory completely treated in Chrystal's _Algebra_, where will be found the exhibition of a prime number of the form 4p + 1 as the actual sum of two squares by means of continuants, a result given by H. J. S. Smith.

The continuant

/ b2, b3, ..., b_{n}\
K ( ) is also equal to the determinant
\a1, a2, a3, ..., a_{n}/

is also equal to the determinant

| a1 b2 0 0 . . . 0 |
| -1 a2 b3 0 . . . 0 |
| 0 -1 a3 b4 . . . 0 |
| 0 0 -1 a4 b5 . . -- |
| |
| u -1 a_{n-1} b_{n} |
| 0 0 -- -- 0 0 -1 a_{n} |,

from which point of view continuants have been treated by W. Spottiswoode, J. J. Sylvester and T. Muir. Most of the theorems concerning continued fractions can be thus proved simply from the properties of determinants (see T. Muir's _Theory of Determinants_, chap. iii.).

Perhaps the earliest appearance in analysis of a continuant in its determinant form occurs in Lagrange's investigation of the vibrations of a stretched string (see Lord Rayleigh, _Theory of Sound_, vol. i. chap. iv.).

_The Conversion of Series and Products into Continued Fractions._

1. A continued fraction may always be found whose n^{th} convergent shall be equal to the sum to n terms of a given series or the product to n factors of a given continued product. In fact, a continued fraction

b1 b2 b_{n}
-- -- ----- can be constructed having for the
a1 + a2 + ... + a_{n} + ...

numerators of its successive convergents any assigned quantities p1, p2, p3, ..., p_{n}, and for their denominators any assigned quantities q1, q2, q3, ..., q_{n} ...

The partial fraction b_{n}/a_{n} corresponding to the n^{th} convergent can be found from the relations

p_n = a_{n}p_{n-1} + b_{n}p_{n-2}, q_n = a_{n}q_{n-1} + b_{n}q_{n-2};

and the first two partial quotients are given by

b1 = p1, a1 = q1, b1a2 = p2, a1a2 + b2 = q2.

If we form then the continued fraction in which p1, p2, p3, ..., p_{n} are u1, u1 + u2, u1 + u2 + u3, ..., u1 + u2 + ..., u_{n}, and q1, q2, q3, ..., q_{n} are all unity, we find the series u1 + u2 + ..., u_{n} equivalent to the continued fraction

u1 u2/u1 u3/u2 u_n/u_{n-1}
-- ------ ------ ----------
1 - u2 u3 u_{n}
1 + -- - 1 + -- - ... - 1 + -------
u1 u2 u_{n-1}

which we can transform into

u1 u2 u1u3 u2u4 u_{n-2}u_{n}
-- ------- ------- ------- ---------------,
1 - u1 + u2 - u2 + u3 - u3 + u4 - ... - u_{n-1} + u_{n}

a result given by Euler.

2. In this case the sum to n terms of the series is equal to the n^{th} convergent of the fraction. There is, however, a different way in which a Series may be represented by a continued fraction. We may require to represent the infinite convergent power series a0 + a1x + a2x squared + ... by an infinite continued fraction of the form

[beta]0 [beta]1 x [beta]2 x [beta]3 x
------- --------- --------- ---------
1 - 1 - 1 - 1 - ...

Here the fraction converges to the sum to infinity of the series. Its n^{th} convergent is not equal to the sum to n terms of the series. Expressions for [beta]0, [beta]1, [beta]2, ... by means of determinants have been given by T. Muir (_Edinburgh Transactions_, vol. xxvii.).

A method was given by J. H. Lambert for expressing as a continued fraction of the preceding type the quotient of two convergent power series. It is practically identical with that of finding the greatest common measure of two polynomials. As an instance leading to results of some importance consider the series

x x squared
F(n,x) = 1 + --------------- + -------------------------------- + ...
([gamma] + n)1! ([gamma] + n)([gamma] + n + 1)2!

We have

x
F(n + 1,x) - F(n,x) = - ------------------------------ F(n + 2,x),
([gamma] + n)([gamma] + n + 1)

whence we obtain

F(1,x) 1 x/[gamma]([gamma] + 1) x/([gamma] + 1)([gamma] + 2)
------ = -- ---------------------- ----------------------------
F(0,x) 1 + 1 + 1 + ...,

which may also be written

[gamma] x x
------- ----------- -----------
[gamma] + [gamma] + 1 + [gamma] + 2 + ...

By putting +- x squared/4 for x in F(0,x) and F(1,x), and putting at the same time [gamma] = 1/2, we obtain

x x squared x squared x squared x x squared x squared x squared
tan x = -- -- -- -- tanh x = -- -- -- --
1 - 3 - 5 - 7 - ... 1 + 3 + 5 + 7 + ...

These results were given by Lambert, and used by him to prove that [pi] and [pi] squared incommensurable, and also any commensurable power of e.

Gauss in his famous memoir on the hypergeometric series

F([alpha], [beta], [gamma], x) =

[alpha].[beta] [alpha]([alpha] + 1)[beta]([beta] + 1)
--------------x + -------------------------------------- x squared + ...
1.[gamma] 1.2.[gamma].([gamma] + 1)

gave the expression for F([alpha], [beta] + 1, [gamma] + 1, x) / F([alpha], [beta], [gamma], x) as a continued fraction, from which if we put [beta] = 0 and write [gamma] - 1 for [gamma], we get the transformation

[alpha] [alpha]([alpha] + 1)
1 + -------x + --------------------x squared +
[gamma] [gamma]([gamma] + 1)

[alpha]([alpha] + 1)([alpha] + 2)
---------------------------------x cubed + ... =
[gamma]([gamma] + 1)([gamma] + 2)

1 [beta]1 x [beta]2 x
-- --------- --------- where
1 - 1 - 1 - ...

[alpha] ([alpha] + 1)[gamma]
[beta]1 = -------, [beta]3 = --------------------------, ...,
[gamma] ([gamma] + 1)([gamma] + 2)

([alpha] + n - 1)([gamma] + n - 2)
[beta]_{2n-1} = ------------------------------------,
([gamma] + 2n - 3)([gamma] + 2n - 2)

[gamma] - [alpha] 2([gamma] + 1 - [alpha])
[beta]2 = --------------------, [beta]4 = --------------------------,
[gamma]([gamma] + 1) ([gamma] + 2)([gamma] + 3)

n([gamma] + n - 1 - [alpha])
..., [beta]_{2n} = ------------------------------------.
([gamma] + 2n - 2)([gamma] + 2n - 1)

From this we may express several of the elementary series as continued fractions; thus taking [alpha] = 1, [gamma] = 2, and putting x for -x,

x 1 squaredx 1 squaredx 2 squaredx 2 squaredx 3 squaredx 3 squaredx
we have log(1 + x) = -- --- --- --- --- --- ---
1 + 2 + 3 + 4 + 5 + 6 + 7 + ...

Taking [gamma] = 1, writing x/[alpha] for x and increasing [alpha] indefinitely, we have

1 x x x x x
e^x = -- -- -- -- -- --
1 - 1 + 2 - 3 + 2 - 5 + ...

For some recent developments in this direction the reader may consult a paper by L. J. Rogers in the _Proceedings of the London Mathematical Society_ (series 2, vol. 4).

_Ascending Continued Fractions._

There is another type of continued fraction called the ascending continued fraction, the type so far discussed being called the descending continued fraction. It is of no interest or importance, though both Lambert and Lagrange devoted some attention to it. The notation for this type of fraction is

b5 +
b4 + ----
a5
b3 + ---------
a4
b2 + --------------
a3
a1 + -------------------
a2

It is obviously equal to the series

b2 b3 b4 b5
a1 + -- + ---- + ------ + -------- + ...
a2 a2a3 a2a3a4 a2a3a4a5

_Historical Note._

The invention of continued fractions is ascribed generally to Pietro Antonia Cataldi, an Italian mathematician who died in 1626. He used them to represent square roots, but only for particular numerical examples, and appears to have had no theory on the subject. A previous writer, Rafaello Bombelli, had used them in his treatise on Algebra (about 1579), and it is quite possible that Cataldi may have got his ideas from him. His chief advance on Bombelli was in his notation. They next appear to have been used by Daniel Schwenter (1585-1636) in a _Geometrica Practica_ published in 1618. He uses them for approximations. The theory, however, starts with the publication in 1655 by Lord Brouncker of the continued fraction

1 1 squared 3 squared 5 squared
-- -- -- -- as an equivalent of [pi]/4. This he is supposed
1 + 2 + 2 + 2 + ...

to have deduced, no one knows how, from Wallis' formula for

3 . 3 . 5 . 5 . 7 . 7 ...
4/[pi], viz. -------------------------
2 . 4 . 4 . 6 . 6 . 8 ...

John Wallis, discussing this fraction in his _Arithmetica Infinitorum_ (1656), gives many of the elementary properties of the convergents to the general continued fraction, including the rule for their formation. Huygens (_Descriptio automati planetarii_, 1703) uses the simple continued fraction for the purpose of approximation when designing the toothed wheels of his _Planetarium_. Nicol Saunderson (1682-1739), Euler and Lambert helped in developing the theory, and much was done by Lagrange in his additions to the French edition of Euler's _Algebra_ (1795). Moritz A. Stern wrote at length on the subject in _Crelle's Journal_ (x., 1833; xi., 1834; xviii., 1838). The theory of the convergence of continued fractions is due to Oscar Schloemilch, P. F. Arndt, P. L. Seidel and Stern. O. Stolz, A. Pringsheim and E. B. van Vleck have written on the convergence of infinite continued fractions with complex elements.