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himself and changed the name to Erismatura. Mr. Gould gave the name of Undina to the genus, and figures the European species under the name of Undina leucocephala.

It should be remembered that the subgeneric term Macropus has been long applied as a generic name for the Kangaroos.

Micropterus is the genus containing the well-known Race-Horse of Cook (Micropterus brachypterus, Anas brachyptera of authors). Captain Phillip Parker King, R.N., who has added a second species (Micropterus Patachonicus), gives these short-winged but rapidly progressing Sea Ducks the familiar name of Steamer Ducks or Steamers.

At a meeting of the Zoological Society, in December 1837, Mr. T. C. Eyton made some observations on the Anatidae, which family he regarded as connected with the Grallato rial Birds by means of the Flamingo on the one side and the semipalmated Goose on the other, with the Divers of the family Alcade by the Mergansers, and also with the Cormorants through the Erismaturince. Mr. Eyton divides the Anatidae into the subfamilies Plectropterinæ, Anserinæ, Anatina, Fuligulina, Erismeurince, and Mergine. The Anatine, according to Mr. Eyton, contain the following genera: Tadorna, Lach; Casarka, Bonaparte; Dendrocygna, Swainson; Leptotarsis, Gould (L. Eytoni); Dafila, Leach; Marec, Stephens; Aia, Boié (Anas sponsa, Linn.); Paciloneta, Eyton (Anas marmorata, Temm.); Querquedula, Auct.; Cyanopterus (Anas Rafflesii, King); Rhynchasps, Leach; Malacorhynchus, Swainson; Chauliodus,* Swainson; Anas, Auct.; Carina, Fleming.

Mr. Eyton's Fuliguline consist of the genera-Micropter us, King; Melanitta, Boié; Somateria, Leach; PolysLicta, Eyton (Anas dispar, Gmel.); Kamptorhynchus, Eyton (Anas Labradora, Wilson); Callicher, Brehm; Fuligula, Ray; Nyroca, Fleming; Harelda, Leach; and Clangula, Leach.

Mr. Eyton stated, that characters of the genera and species would be given in his forthcoming monograph on the

Anatidae.

FULLER, THOMAS, was the son of the Rev. Thomas Fuller, rector of Aldwincle, in Northamptonshire, where he was born in 1608. He was educated under his father, and was sent at the early age of twelve years to Queen's College, Cambridge. He became B.A. in 1625, and M.A. in 1628, but afterwards removed to Sidney College, where he obtained a fellowship in 1631, and nearly at the same time the prebend of Netherby, in the church of Salisbury. In this year also he issued his first publication, a poem, now little known, entitled 'David's Hainous Sin, Hearti Repentence, Heavie Punishment,' in 12mo. He was soon after ordained priest, and presented to the rectory of Broad Windsor, in Dorsetshire; but growing weary of a country parish, and uneasy at the unsettled state of public affairs, he removed to London, and distinguished himself so much in the pulpits there, that he was invited by the master and brethren of the Savoy to be their lecturer. In 1639 he published his History of the Holy War' it was printed at Cambridge, in folio, and so favourably received that a third edition appeared in 1647. On April 13, 1640, a parliament was called, and a convocation also began at Westminster, in Henry VIIth's chapel, having licence granted to make new canons for the better government of the church: of this convocation he was a member, and has detailed its proceedings in his Church History. During the commencement of the Rebellion, and when the king left London, in 1641, to raise an army, Mr. Fuller continued at the Savoy, to the great satisfaction of his congregation and the neighbouring nobility and gentry, labouring all the while in private and in public to serve the king. On the anniversary of his inauguration, March 37, 1642, he preached at Westminster Abbey on this text, 2 Sam. xix. 30, Yea, let him take all, so that my lord the king return in peace,' which, being printed, gave great offence to those who were engaged in the opposition, and exposed the preacher to a good deal of danger.

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before him at St. Mary's Church. He had before preached and published a sermon in London, upon the newmoulding church-reformation, which caused him to be censured as too hot a royalist; and now, from his sermon at Oxford, he was thought to be too luke-warm, which can only be ascribed to his moderation, which he would sincerely have inculcated upon each party as the only means of reconciling both. During his stay here, his residence was in Lincoln College, but he was not long after sequestered, and lost all his books and manuscripts. This loss, the heaviest he could sustain, was made up to him partly by Henry Lord Beauchamp, and partly by Lionel Cranfield, earl of Middlesex, who gave him the remains of his father's library. That, however, he might not lie under the suspicion ef want of zeal or courage in the royal cause, he determined to join the army, and therefore, being well recommended to Sir Ralph Hopton in 1643, he was admitted by him in quality of chaplain. For this employment he was at liberty, being deprived of all other preferment. Though he attended the army from place to place, and constantly exercised his duty as chaplain, he yet found proper intervals for his favorite studies, which he employed chiefly in making historical collections, and especially in gathering materials for his Worthies of England,' which he did, not only by an extensive correspondence, but by personal inquiries in every place which the army had occasion to pass through.

After the battle at Cheriton-Down, March 29, 1644, Lord Hopton drew on his army to Basing-House, and Fuller, being left there by him, animated the garrison to so vigorous a defence of that place, that Sir William Waller was obliged to raise the siege with considerable loss. But the war coming to an end, and part of the king's army being driven into Cornwall under Lord Hopton, Fuller, with the permission of that nobleman, took refuge at Exeter, where he resumed his studies, and preached constantly to the citizens. During his residence here he was appointed chaplain to the infant princess, Henrietta Maria, who was born at Exeter in June, 1643. He continued his attendance on the princess till the surrender of Exeter to the parliament, in April, 1646. He is said to have written his Good Thoughts in Bad Times' at Exeter, where the book was published in 1645, 16mo. On the garrison being forced to surrender, he came to London, where he found his lectureship at the Savoy filled by another. It was not long however before he was chosen lecturer of St. Clement's, near Lombard Street, and shortly afterwards removed to St. Bride's, Fleet Street. In 1647 he published, in 4to., 'a Sermon of Assurance, fourteen years ago preached at Cambridge, since in other places, now by the importunity of his friends exposed to public view.' He dedicated it to Sir John Danvers, who had been a royalist, was then an Oliverian, and next year one of the king's judges; and in the dedication he says, that it had been the pleasure of the present authority to make him mute, forbidding him, till further order, the exercise of his public preaching.' Notwithstanding his being thus silenced, he was, about 1648, presented to the rectory of Waltham Abbey, in Essex, by the earl of Carlisle. In 1648 he published his 'Holy State,' folio, Cambr.

His Pisgah-sight of Palestine and the Confines thereof, with the History of the Old and New Testament, acted thereon,' was published, fol. Lond. 1650, and reprinted in 1662. At this period he was still employed upon his Worthies.' In 1651 he published Abel Redivivus, or the Dead yet Speaking; the Lives and Deaths of the Modern Divines.' Lond. 4to. In the two or three following years he printed several sermons and tracts upon religious subjects: The Infant's Advocate,' 8vo. Lond. 1653; Perfection and Peace, a Sermon,' 4to. Lond. 1653 A Comment on Ruth, with two Sermons,' 8vo. Lond. 1654; A Triple Reconciler,' 8vo. Lond. 1654. About this last year he took as a second wife a sister of the Viscount Baltinglasse. In 1655, notwithstanding Cromwell's prohibition of all persons from preaching or teaching school who had been adherents to the late king, he continued preaching and exerting his charitable disposition towards those ministers who were ejected, as well as towards others. In 1655 he published in folio The Church History of Britain, from the birth of Jesus Christ until the year MDCXLVIII.,' to which he subjoined "The History of the University of Cambridge since the Conquest,' and 'The History of Waltham Abbey, in Essex, founded by King Dr. Peter Heylyn in his Examen Historicum,' to which Harold. The Church History was animadverted upon by

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12} = One equiv. cyanogen

14.

Fuller replied in his Appeal of Injured Innocence,' fol. | Two equivs. of carbon. 121
Lond. 1659. It is said that Lord Berkeley, in 1658 or 1659, One do. azote
took him over to the Hague, and introduced him to Charles
II. It is certain however that a short time before the Res-
toration he was re-admitted to his lecture in the Savoy, and
on that event restored to his prebend of Salisbury. He
was chosen chaplain extraordinary to the king; and created
D.D., at Cambridge, by a mandamus dated Aug. 2, 1660.
Upon his return from Salisbury, in August, 1661, he was
attacked by a fever, of which he died on the 15th of that
month. His funeral was attended by at least two hundred
of his brethren of the ministry. He was buried in his
church of Cranford, on the north wall of the chancel of
which his monument is still remaining. His History of
the Worthies of England' was not published till after his
death, fol. Lond. 1662: reprinted in two volumes, 4to. Lond.
1811, with explanatory notes by John Nichols. The Uni-
versity of Oxford intend publishing another reprint of this
work.

Besides the works already mentioned, Fuller was the
author of several others of a smaller kind. 1. Andronicus,
or the Unfortunate Politician,' 12mo. Lond. 1646. 2. Good
Thoughts in Worse Times,' 16mo. Lond. 1647, reprinted
with his Good Thoughts in Bad Times,' 1652, and again,
12mo. Lond. 1669, and both since reprinted at Oxford;
3. Mixt Contemplations in Better Times, 12mo. Lond.
1660; 4. The Speech of Birds, also of Flowers, partly
moral, partly mystical,' 8vo. 1660. In 1651 he published
Dr. Holdsworth's Valley of Vision,' with a preface. A
specimen of his Latin composition, in what is called 'An
Eccho,' occurs in the first book of Ayres and Dialogues,
for one, two, and three Voyces,' by Henry Lawes, fol.
Lond. 1653. Fuller was a man of great wit, and of powers
of memory almost incredible. (Life of Dr. Thomas Fuller,
12mo. Lond. 1661; Biogr. Britan., vol. iii. 2049-2069;
Chalmers's Biogr. Dict., vol. xv. p. 168-176.)

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FULLERS' EARTH, a mineral product, formerly much used in the fulling of cloth, whence it derives its name. It occurs massive, and is usually of a greenish brown or dull grey colour; sometimes it is nearly of a slate colour. It is opaque, dull, and its specific gravity is 18, 22. Greasy and soft, yielding to, and polished by, the nail. Fracture uneven, earthy; in water it breaks down into a soft pulpy mass. Before the blow-pipe it fuses into a white blobby glass. It is found at Nutfield, near Reigate, in Surrey, and occurs in regular beds near the summit of a hill, between beds of sand or sandstone, containing fossil wood, cornua ammonis, &c. There are two distinct beds of fullers' earth; the upper has a greenish colour, is five feet in thickness, and rests upon the other, which has a bluish tint, and is eleven feet thick; in these beds, but especially in the latter, there are found considerable masses of sulphate of barytes, frequently in regular crystals. Fullers' earth is also found in Kent, Bedfordshire, Bath, Nottinghamshire, and Sussex. It is met with also in Styria, Saxony, and some other places. According to Dr. Thomson's analysis, this substance con

sists of

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Dr. Thomson observes that, allowing the lime, magnesia, and protoxide of iron, to be in the state of silicates, and as mere accidental constituents, fullers' earth is a hydrous bisilicate of alumina, consisting of 2 equivalents of silica, 1 equivalent of alumina, and two equivalents of water.

FULLING. [WOOLLEN MANUFACTURES.] FULMINATING POWDERS. [DETONATION.] FULMINIC ACID. An acid which appears to be isomeric with cyanic acid [CYANIC ACID], that is, composed of the same elements in the same proportions, and they appear to have similar saturating powers. Fulminic acid is that which exists in the detonating mercury and silver discovered by Mr. Howard. These fulminates, as shown under the respective metals, are prepared by the simultaneous action of nitric acid and alcohol upon them in this operation the metals are oxidized, and such portions of the carbon, of the alcohol, and azote and oxygen of the decomposed nitric acid combine, as to form the fulminic acid, which may be regarded as composed of

One do. oxygen

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Equivalent 34

The subject of the perfect identity of the cyanic acid and fulminic acid is, however, obscure; and as their combining weight is similar, we have not the resource available in the case of the isomeric carburetted hydrogens, of supposing that they consist of the same proportions, but of a different number of equivalents of the same elements. Fulminic acid may be separated from the oxides of silver and of mercury, and combined with other bases, as with potash, and it still retains its power of forming detonating compounds.

FULTON, ROBERT, distinguished as having been the first to establish steam-navigation on the American seas and rivers, was born in 1765, in Little Britain, Pennsyl vania. His parents were emigrants from Ireland. He received a common English education at a village school. Besides a fondness for mechanical pursuits, he early displayed a taste for drawing, and in his eighteenth year went to Philadelphia, and began to paint portraits and landscapes as a means of subsistence.

In November, 1786, he embarked for England, and on his arrival in London was received as an inmate in the house of West, the historical painter, with whom he continued to reside for some years, and who also gave him instructions in his profession.

After leaving West, painting was for some time his chief employment. But with Fulton the fine arts were destined to give place to the mechanical. He spent about two years in Devonshire, where he became acquainted with the Duke of Bridgewater, and projects for the improvement of canals then began to occupy the chief share of his attention. In 1794 he took out a patent for an inclined plane, which was intended to set aside the use of locks; he invented a machine to facilitate excavation, and wrote a work on canals, in which he first styled himself a civil engineer. He also invented a mill for sawing marble, and took out patents for spinning flax and making ropes.

He seems however to have had little success; and at the latter end of 1796 went to Paris, on the invitation of Joel Barlow, then resident minister from the United States, in whose house he resided during seven years. While at Paris two projects appear to have occupied a large portion of his time and attention; one, a carcass, or box filled with combustibles, which was to be propelled under water, and made to explode beneath the bottom of a vessel; the other, a submarine boat, to be used for a similar destructive purpose. The first was a failure; but of his submarine boat he made many trials and exhibitions, some of them at the expense of the French government, with occasional failures and partial success, on the Seine, at Havre, and at Rouen. But for all practical purposes this was as much a failure as the other. He appears however to have clung to it with great perseverance, and not long before his death exhibited its power by blowing up an old vessel in the neighbourhood of New York.

But while at Paris he had other and better pursuits. He made himself acquainted with the higher branches o. science, and with the modern European languages; he projected the first panorama exhibited at Paris, and in conjunction with Mr. R. Livingston, the American ambassador, began to make experiments on the Seine with small steamboats: a larger one was built, which broke asunder, but a second, completed in 1803, was successful.

Soon after this time he was invited to England by the English ministry, at the suggestion of Earl Stanhope, with whom Fulton had become acquainted about the time of his introduction to the Duke of Bridgewater. The object of the English ministry appears to have been to employ him in the construction of his submarine implements of war. After some trials on the Thames, the negotiation failed, and Fulton resolved to embark for America.

In 1806 Fulton arrived at New York, and soon after, with funds supplied by Mr. Livingston, commenced the construction of a steam-vessel of considerable size, which began to navigate the Hudson in 1807. He afterwards built others of large dimensions, one of them a frigate, which bore his name. His reputation became established, and his fortune was rapidly increasing, when his patent for steam-vessels, which he had taken out in conjunction with

Mr. Livingston, was disputed, and his opponents were, in a considerable degree, successful. His constitution had been impaired by his numerous labours, and a severe cold which he caught by incautious exposure in giving directions to his workmen, together with the anxiety and fretfulness occasioned by the law-suits about his patent rights, brought his life to a premature termination on the 24th of February, 1815, in his forty-ninth year. His death occasioned extraordinary demonstrations of national mourning in the United States.

In person he was tall, and though slender, well formed. He appears to have been an amiable, social, and liberal man. (Encyclopædia Americana; Dictionnaire de la

Conversation.)

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paper, which constitute the second kind of fumigation, does not require any particular notice; their operation can hardly be regarded as any other than that of substituting one smell for another. In the last kind of fumigation three substances have been chiefly employed, and in the gaseous state: first, the vapour of burning sulphur, or sulphurous acid gas, muriatic acid gas, nitric acid gas, and chlorine gas; all but the last of these, or at any rate the first and second named, appear to have been first used and recommended by Dr. James Johnstone of Worcester, about the year 1758; in 1773 Guyton de Morveau also mentioned the application of muriatic and nitric acid gases, and in 1802 their use was still further extended by Dr. J. C. Smith, who received a public remuneration as the discoverer, which he certainly was not.

We shall give an abstract of the mode adopted by Mr. Faraday in fumigating the Penitentiary at Milbank in 1825. (Quarterly Journal, vol. xviii., p. 92.)

FUMARIA CEÆ, a small natural order of Exogenous plants, consisting of slender-stemmed, herbaceous plants, Chlorine gas, which is undoubtedly preferable to any dismany of which scramble up others by aid of their twist-infectant, was first recommended by Dr. Rollo, who pubing leafstalks. They are rather succulent in texture, with lished a work on diabetes in 1797; he liberated the gas by watery juice. Their leaves, which have no stipules, are the usual method of mixing sulphuric acid, binoxide of repeatedly divided till the terminal lobes become small manganese, and common salt. When it is desirable to proovate leaflets; their flowers, which are extremely irregular, duce a great effect in a short time, this is still unquestionconsist of two membranous, minute, ragged sepals, two ex- ably the best mode of proceeding. terior distinct linear petals, and two others, which hold firmly together at the points; there are six stamens united into two parcels, and the ovary is a one-celled case with one or many seeds, whose placentation is parietal; finally, the seeds consist principally of albumen, in which there ripens a very small embryo. Fumaria officinalis is one of the commonest of weeds; many are objects of cultivation by the gardener for the sake of their showy flowers; all are reputed diaphoretics. They only inhabit the cooler parts of the world, alike avoiding extremes of heat or cold. It is probable that notwithstanding the diversity of their appear ance they are only a low irregular form of Papaveraceæ.

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2,000,000 cubic feet; and the surface of the walls, floors, The space requiring fumigation amounted to nearly ceilings, &c., was about 1,200,000 square feet. This surface was principally stone and brick, most of which had been lime-washed. A quantity of salt reduced to powder was mixed with an equal weight of binoxide of manganese, and upon this mixture were poured two parts of sulphuric acid, previously diluted with one part of water, and cold. The acid and water were mixed in a wooden tub, the water being first put in, and it being more convenient to measure than to weigh the water and acid, ten measures of water and nine of acid were used; half the acid was first used, and when the mixture had cooled the remainder was added. Into common red earthen pans, each capable of holding about a gallon, were put 3 lbs. of the mixed salt and manganese, and there was then added such a measure of the diluted acid as weighed 4 lbs.; the mixture was well stirred and then left to itself, and all apertures were well stopped. The action did not commence immediately, so that there was sufficient time for the operator to go from pan to pan without inconvenience. On entering a gallery 150 feet in length, a few minutes after the mixture had been made, the general diffusion of chlorine was sufficiently evident; in half an hour it was often almost impossible to enter, and frequently on looking along the gallery the yellow tint of the atmosphere could easily be perceived. Up to the fifth day the colour of the chlorine could generally be observed in the building; after the sixth day the pans were removed, though sometimes with difficulty, and the gallery thus fumigated had its windows and doors thrown open. The charge contained in each pan was estimated to yield about 5 cubic feet of chlorine gas; in fumigating a space of 2,000,000 cubic feet, about 700 lbs. of common salt and the same of binoxide of manganese were employed: and it will appear by a slight calculation, that about 1710 cubic feet of chlorine were employed to disinfect this space. In common cases, Mr. Faraday conceives that about onehalf to one-fourth of this quantity of chlorine would be sufficient.

When any cause is continually recurring, and in some cases almost imperceptibly so, the chloride of lime or soda, and especially of the former, has been within a few years successfully employed by M. Labarraque; the exact nature of these compounds is still under discussion, but the chloride of lime is a substance well known and extensively employed under the name of bleaching-powder.

FUMIGATION is the application of the vapour or fumes from metallic or other preparations to the body, with the intention of healing either generally, or particular parts. The vapours of hot vinegar, burning sulphur, and of aromatic vegetable matters, have been long used to counteract unpleasant or unwholesome smells: this is effected chiefly by the formation of such as are stronger. The most im- We shall relate a few experiments performed by M. Gualportant kind of fumigation is that which consists in the tier de Claubry, illustrative of the mode in which these subemployment of such vapours or gases as do not merely de- stances produce their effects. A solution of chloride of stroy unhealthy odours by exciting such as are more power-lime exposed to the air for about two months, ceased to ful, but which by their chemical action convert dangerous act upon litmus, contained no chlorine, but a precipitate was miasmata into innocuous matter. formed in it which consisted entirely of carbonate of lime, without any admixture of chlorine; it was therefore evident that the carbonic acid of the atmosphere had decomposed the chloride of lime, evolved the chlorine, and precipitated the lime. That this was the case was proved by passing atmospheric air through a solution of potash, before it was made to traverse one of chloride of lime; in this case the

The fumigation of the first kind, that which is intended to produce a healing effect, is now much less employed than formerly; still, however, the bisulphuret of mercury is occasionally used in vapour, as what is termed a mercurial fumigation, in certain forms of syphilis. The use of vinegar, of aromatic pastilles, and even the smoke of burning brown

potash separated the carbonic acid, so that no chlorine was evolved from the solution of chloride of lime, nor was any precipitate formed in it; in fact no change whatever occurred. That it was the carbonic acid which produced this effect, was further proved by passing a current of this gas into a solution of chloride of lime; by this it lost its bleaching power, the whole of the chlorine was expelled, and all the lime converted into carbonate.

In order to show the manner in which these compounds of chlorine and lime, and of chlorine and soda, act on putrid miasmata floating in the air, some further experiments were made in the following manner :-Air was passed through blood which had been left to putrefy for eight days; being then passed through a solution of the chloride of lime, carbonate of lime was deposited, and the air was rendered inodorous and completely purified. In a second similar experiment the fetid air was passed through a saturated solution of potash before it arrived at the solution of chloride of lime; the latter had then no effect upon it, and the air retained its insupportable odour; this happened evidently because the carbonic acid, which would otherwise have evolved chlorine to have acted upon the putrid matter, was absorbed by the potash. Another experiment was made with air left for twenty-four hours over putrescent blood; the portion of it which was passed directly through the chloride was perfectly purified, but when previously freed from carbonic acid the chloride had no effect upon it.

These experiments sufficiently prove that the carbonic acid in the air, arising from the various sources of respiration, combustion, and the decomposition of animal and vegetable matter, liberates the chlorine from its combination with lime or soda; and as this action is slow, the chlorine, though scarcely susceptible of affecting the animal economy, readily decomposes putrid miasmata. It is therefore true fumigation by chlorine, only it is less violent than that effected by the rapid evolution of the gas, and it continues for a longer time.

It is to be observed that chloride of lime is used in solution, and is obtained by dissolving one part of bleaching powder in about 100 times its weight of water, and allowing the solution to become clear. This is to be exposed to infected air, or in rooms which have any unpleasant odour, in flat vessels, in order that a sufficient surface may be acted upon. If it should be required, the operation may be quickened by the addition of a little vinegar, or of muriatic acid largely diluted. In some cases, where the disagreeable smell is extremely strong, and where it would be difficult to expose a solution to slow action, it may be thrown into the place, or the powder may be used, the action of which would be more gradual and effectual. Chloride of soda is prepared only in solution; the process is given in the last edition of the London Pharmacopoeia: it is however less easily obtained than the chloride of lime, is more expensive, and not in any respect preferable; the solution is then called liquor soda chlorinatæ.

FUNCHAL. [MADEIRA.]

FUNCTIONS, CALCULUS OF. By the term function of a quantity is meant any algebraical expression, or other quantity expressed algebraically or not, which depends for its value upon the first. Thus the circumference of a circle is a function of the radius; the expression (a2 —x2) (b2 + y2) is a function of a, b, x, and y. For the distinctive names of functions, see TRANSCENDENTAL and ALGEBRAICAL.

All algebra is, in one sense, a calculus of functions; but the name is peculiarly appropriate, and always given, to that branch of investigation in which the form of a function is the thing sought, and not its value in any particular case, nor the conditions under which it may have a particular value. [EQUATIONS, FUNCTIONAL.] For instance, What is that function of x which, being multiplied by the same function of y, shall give the same function of x+y?'-is a question of the calculus of Functions.

and the artic.e, Calculus of Functions,' in the 'Encyclo-
pædia Metropolitana,' are the only formal treatises on the
subject, of which we know.

A function of x is denoted by x, x, xx, fx, Fx, ox,
&c., &c., the first letter being a symbol of an operation to
be performed. Thus, Ffx denotes that when the operation
signified by fhas been performed upon x, that signified by
F is performed upon the result. When the same operation
is repeated, the results may be denoted by fx, ffx, ƒƒƒx,
&c., which may be abbreviated into fx, ƒax, ƒ3x, &c. For
different points of interest connected with the relations of
functional forms, see PERIODIC; INVERSE.

FUNCTIONS, THEORY OF, a name given by Lagrange to a view of the principles of the Differential Calculus, of which we have expressed our opinion in the article DIFFERENTIAL CALCULUS. The works of Lagrange, in which its details are to be found, are Théorie des Fonctions Analytiques,' first edition, 1797; second edition, 1813; and Leçons sur le Calcul des Fonctions,' of which the first edition is volume 10 of the Leçons de l'Ecole Normale (1801), and the second was published in 1806.

Taking Lagrange's intention to have been the proof that algebra, as it existed in his time, was sufficient to demonstrate the principles of the Differential Calculus without the introduction of limits, we have only to remark that the end is completely attained. [DIFFERENTIAL CALCULUS.] It is plain to any one acquainted with that calculus, that a demonstration of Taylor's Theorem being once attained, all the rest follows. We now proceed to look at the proof of this theorem given by Lagrange, with reference to absolute correctness or incorrectness.

Lagrange first attempts to prove that every function pr has this property, that (x + h) can be expanded in a series of the form

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$ (x + h) 4x + Ah + Bh2 + Ch3 +
He says, firstly, that no negative powers of h can enter the
expansion, for if such were the case (x + 0), instead of
being pr, would be infinite. This is true as to any finite
number of negative powers of h, but does not exclude an
infinite series of negative powers. For instance,

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when h 0, all the terms become infinite, but the first side
of the equation is not infinite. Secondly, he assumes that
there cannot be fractional powers of h, for if such were the
case, there must be fractional powers in the original
function px, and if pæ had m different values, and if
Kh" were one of the terms of the development, the n values
of this latter, combined with the m values of px, would give
mn different values to (x + h), instead of m.
to this it may be asked how is it known, à priori, that
there must be a series of powers of h, every value of which
is an expansion of 4 (x + h)? May it not possibly be true
that there is an expression of the form

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In answer

4(x + h) = x + Aha + Вht +
which is true under certain conditions, determining which
of the values of the several terms are to be taken? Thirdly,
he assumes that (having thus obtained a series, in which
only whole powers of h are found) the supposition h
must reduce it to its first term; an assumption which can
only be admitted of such a series as M + Ah + Bh2 + ..........
when it can be made convergent by giving sufficiently
small values to h.

1 0

Having once proved or assumed that (x + h) can be expanded in a series of the form pa +Ah + Bh2 + ... the proof of Taylor's Theorem, given by Lagrange, does not differ from the common one. He calls A the derived function of 4.x, and denotes it by p': generally, if changing

Various isolated questions connected with this calculus
have been treated, from the time of Newton downwards,
particularly by Lagrange, Laplace, Monge, and Euler. But
the direct solution of functional equations, or at least the
first attempt to form general methods in the case of func-
tions of a single variable, appears to have been made by
Mr. Babbage and Sir J. Herschel (1810-1813). To the
treatise entitled Examples of the Calculus of Differences, and by changing ʼn into h + k, ¢ (x + h) becomes

into x + h change Pinto P+ Ph+...., P' is the de-
rived function of P. The derived function of p'r, denoted
by "x, is called the second derived function of or, and so
on. By changing x into x + k, p (x + h), or øx + Ah +
Bh2 + becomes
....

by the latter, the former appended another, containing ex-
amples of the solutions of functional equations. This last,

(px + p'x.k + ..) + (A + A'k + . .) h + (B + B'k + . .) h3

+

....

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px + A (h + k) + B (h + k)* + ....

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These must be the same, since both represent (x+h+k): | other, preserving the east-north-eastern direction, is named and by equating the terms which contain the first powers of Chignecto Bay. The whole length of the Bay of Fundy is k, we find about 180 miles.

4'x + A'h + B'h2 +*... = A + 2Bh +.... Both shores of the large bay are rocky and bold, but especially so on the side of Nova Scotia, where a chain of whence Apx, 2B = A' = "x, and so on. The reader hills, probably not less than 500 feet above the sea, will recognise in this process the proof frequently given by rises at a short distance from the coast. The entrances, means of the preliminary lemma, that if

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The works of Lagrange on this subject, though defective in their fundamental positions, except upon the explanation given in DIFFERENTIAL CALCULUS, yet abound in new and useful details, given with all the elegance for which his writings are distinguished: and the student will find them well worth his attention.

both of the Bay of Minas and of Chignecto, are likewise rocky; but in the interior the shores are low, sandy, and flat.

The navigation of the bay is both difficult and dangerous, on account of the great strength of the tide and the prevailing fogs. The tide rises to a great height, someat the entrance about three miles an hour, increasing times seventy feet, and flows with great rapidity, running as it advances to more than seven, and at length rushing with great impetuosity into the bays of Minas and Chignecto. Fogs cover the bay when the wind blows from the east and south-east, or from the Atlantic; and during their prevalence many vessels are cast on the rocky shores by the violence of the tides.

FUNDAMENTAL BASE, in music, is the lowest note of the Perfect Chord, or Triad, as the Germans call it, and of the chord of the 7th: hence it is the root of all real chords;-for chords not derived from either the perfect chord or that of the 7th, are considered as suspensions or The Bay of Minas has been united with Halifax Harretardations; or, to speak in unaffected language, the dis-bour, which is situated on the Atlantic side of Nova Scotia, cordant notes of which they are composed are simply by a canal fifty-four miles long, and capable of receiving appogiaturas. [CHORD.] vessels which draw only eight feet of water. It is called the Shubenacadie Canal. Another canal was projected a few years ago, which was to connect the most northern corner of Chignecto Bay, called Cumberland Basin, with Northumberland Strait. This strait separates Prince Edward Island from New Brunswick and Nova Scotia, and one of its bays, called Bay Verte, is separated from Cumberland Basin only by an isthmus eleven miles across. The advantages of such a canal are obvious; but we are not aware that it has been executed. (M'Gregor's British America; Bouchette's British Dominions in North America.)

The following will show the two Fundamental Chords, and their inversions, with the continued [CONTINUED], ordinary base, and the Fundamental Base.

000

6 4

7

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65

000

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or

The

This term is not the best that might have been chosen;
the same meaning is much better conveyed by the word
rudicul, introduced, we believe, by Dr. Callcott.
system of the Fundamental Base, founded on harmonics,
and a continual addition of thirds to the triad, is indebted
for its origin to Rameau, the celebrated French composer
[RAMEAU], and was once almost universally received.
D'Alembert wrote a book to explain and eulogise it, and
Marpurg, a most distinguished theorist, adopted it in his
Handbuch bey dem Generalbasse. But though it may be
rendered in some degree serviceable in the analysis of
chords, it is in more than one respect erroneous, and
the rules drawn from it by its author would cruelly fetter
genius, were they allowed to exert any influence on the
composition of music. Rameau's once vaunted system is
now therefore entirely laid aside, even in the country
that gave it birth.

FUNDS and FUNDING SYSTEM. [NATIONAL
DEBT.]

FUNDY, BAY OF, is the most extensive gulf on the eastern coast of North America, between Cape Florida and the mouth of the St. Laurence river. It separates Nova Scotia from New Brunswick, and lies between 44° and 46° N. lat. and 63° and 67° W. long. Its direction is from cast-north-east to west-south-west; its entrance is at the west-south-western extremity.

This entrance is formed by Brier Island, on the side of Nova Scotia and Quoddy Head, on the mainland; a straight line between these two points passes through the island of Grand Manan, which lies about 8 or 9 miles from Quoddy Point, and 35 miles from Brier Island. In this part the bay is about 50 miles wide; but it narrows by degrees to about 30 miles and less, after which it again attains a width of between 30 and 35 miles, which breadth it preserves for the greatest part of its extent, the shores of Nova Scotia and New Brunswick running nearly parallel. Towards its inner extremity it is divided, by a bold headland terminating with Cape Chignecto, into two smaller bays, of which one extends due east, and is called the Bay of Minas; the

FÜNEN, or FUHNEN (in Danish Fyen), a stift' or province of Denmark, consisting of the islands of Fünen, Langeland, Taasing, and several islets. It has an area of 1286 square miles, and a population of about 160,000 (in 1801, 121,378), and is divided into the two circles or bailiwicks of Odense and Svendborg, which contain 3 earldoms, 4 baronies, 9 towns, and 201 parishes. It is a bishop's see, and is subdivided into 15 minor circles or herreder, in which there are 180 seignorial estates. The soil is a layer of rich loam on a substratum of clay or sand: it has some hills, but no streams deserving the name of rivers. The produce is grain, vegetables, flax, &c., and great numbers of horses and cattle are reared. The whole of the towns are in the island of Fünen, with the exception of Rudkiöbing, in the island of Langeland, a place of much trade, with about 250 houses and 1500 inhabitants.

FÜNEN, or FYEN, an island situated in the Baltic, between the eastern coast of the duchy of Schleswig and of Jutland, and the western shores of the island of Seeland, from which parts it is separated by the Great and Little Belts, between 55° 2' and 55° 47′ N. lat. and 9° 46′ and 10° 51' E. long. Its area is about 1176 square miles, and its population, which was 91,333 in 1769, is at present about 144,000. The surface is a level, varied by hills in the southern districts, but they never rise above 500 feet. The north-east of the island is deeply indented with bays of the Kattegat, particularly the Odense fiord,' and is more uniform and less wooded than the south. The soil is in general rich and productive. Fünen abounds in small streams, here called Aas, and lakes: the most considerable lakes are those of Arreskov, Brendegards, and Juulbye. The canal of Odense, which commences at Odense and terminates at Skibhusene, on the Odense fiord, is about two miles and a half in length, ten feet deep, and fifty feet in breadth at the surface. The climate is damp and variable, but milder than that of Seeland. About 610,000 acres are arable and meadow land. The principal crops are barley, oats, and buckwheat, and the quantity of grain annually exported amounts to about 100,000 quarters. Much flax and hemp are raised, and the growth of hops exceeds 2300 cwts. yearly. With the exception of potatoes, the cultivation of vegetables is limited, but the orchards are numerous, and an inferior kind of cider is made. About 78,600 acres are occupied by woods and forests, which, with the peat-moors, supply fuel. The Fünen breed of horses is much sought after, and the stock of the island, including that of Langeland, is upwards of 42,000: that of horned cattle is about

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