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 a 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.

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 funetions 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, by the latter, the former appended another, containing examples of the solutions of functional equations. This last,

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

A function of x is denoted by px, 4x, xx, fx, Fx, x, &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 fr, f2x, 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

$ (x + h) = px + Ah + Bh2 + Ch3 + ... He says, firstly, that no negative powers of h can enter the expansion, for if such were the case (+0), instead of being ox, 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,

Р

=

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 x 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 4 (x + h), instead of m. In answer 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 p (x + h)? May it not possibly be true that there is an expression of the form

(x + h) = x + Aha+ Bh' + 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 = 0 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.

Having once proved or assumed that p (x + h) can be expanded in a series of the form x + 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 px, and denotes it by o'r generally, if changing

into +h change Pinto P+Ph+...., P' is the derived function of P. The derived function of p'a, denoted by "x, is called the second derived function of pr, and so on. By changing x into x + k, 4 (x + h), or 4x + Ah + Bh2 + becomes ....

'x.k + ..) + (A + A'k + . .) h + (B + B'1⁄2 + . .) h3

+

....

and by changing ħ into h + k, ¢ (x + h) becomes px + A (h + k) + B (h + k)2 + ....

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.

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 retardations; or, to speak in unaffected language, the discordant notes of which they are composed are simply appogiaturas. [CHORD.]

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

This term is not the best that might have been chosen ; the same meaning is much better conveyed by the word radical, introduced, we believe, by Dr. Callcott. The 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 east-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

about 180 miles.

Both shores of the large bay are rocky and bold, but especially so on the side of Nova Scotia, where a chain of rises at a short distance from the coast. The entrances, hills, probably not less than 500 feet above the sea, both of the Bay of Minas and of Chignecto, are likewise rocky; but in the interior the shores are low, sandy, and flat.

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

The Bay of Minas has been united with Halifax Harbour, which is situated on the Atlantic side of Nova Scotia, by a canal fifty-four miles long, and capable of receiving the Shubenacadie Canal. Another canal was projected a few vessels which draw only eight feet of water. It is called 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.)

FÜNEN, or FUHNEN (in Danish Fyen), a stift' or province of Denmark, consisting of the islands of Füren, 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 earidoms, 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 bing, in the island of Langeland, a place of much trade, with are in the island of Fünen, with the exception of Rudkiöabout 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 Jütland, 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

The exports of Fünen consist of corn, pease, brandy, apples, horses, oxen, butter, salted meat, tallow, hides, hops, linen, honey, and wax. Odense, which by its canal has a direct access to the sea, is the great trading mart of the island. There is a good road from Middelfahrt to this town; but the roads are in general very bad. The people of Fünen are, like their neighbours, somewhat indolent and shy of work, as well as phlegmatic: they are however an honest, sound-hearted race. Their religion is the Lutheran.

The principal towns in Fünen are Odense, the capital and episcopal residence, pleasantly situated, and reputed to be the most antient town in Denmark; in 55° 25' N. lat., and 10° 22′ E. long. It has about 1100 houses, and 8600 inhabitants. Here are a royal palace, built by Frederick IV., a townhall, four churches (of which that of St. Canute is a noble Gothic pile, erected eight centuries ago, and containing the mausolea of St. Canute, Erichslaf, John, and Christian III., kings of Denmark and Norway), a chapter | seminary, gymnasium, theatre, two public libraries, hospital, house of correction, &c. Assens, on the western coast, at the entrance into the Little Belt, another old town, has an indifferent harbour, a townhall, one church, about 350 houses, and 2330 inhabitants. Bogense, on the north coast, the smallest town in the province, has one church, about 250 houses, and 1000 inhabitants. Kierteminde, beautifully situated on a bight of the Great Belt, which is crossed by a large wooden bridge, has one church, a school, two hospitals, about 260 houses, and 1500 inhabitants. Middelfahrt, on the Little Belt, has a townhall, church, hospital, school, about 240 houses, and 1300 inhabitants, and a ferry about a mile across to Snoghoi on the Jutland coast. Svendborg, the chief town of the bailiwick of this name, is at the south-eastern extremity of Fünen, on an arm of the Baltic which separates that island from Taasing; in 55° 5 N. lat. and 10° 38' E. long. It has two churches, a townhall, three schools, about 350 houses, and 3400 inhabitants, and exports much grain, &c. Nyeborg. a fortified town on the eastern coast, contains the remains of the palace in which the kings of Denmark held their courts and national diets, with a church, townhall, several schools, a hospital and an infirmary, about 300 houses, and 2900 inhabitants. The Swedes were totally defeated by the Danes under its walls in 1659. And lastly, Faaborg, in the south-west, is a small town with about 260 houses and 1500 inhabitants, a handsome church, &c., and a good harbour on an arm of the Little Belt, protected at its entrance by the three islands of Lyöe, Avernaröe, and Biömöe.

FUNERAL, the performance of the rites of sepulture or burial; generally supposed to be derived from the Latin funis, a torch, because, at least in the Roman times, funerals were sometimes performed by torch-light. Others derive the word from phónos (póvoc), 'slaughter,' as designating death.

The Egyptians are among the earliest people of whose religious ceremonies we have authentic accounts, more particularly in what related to their dead. Upon this occasion the parents and friends of the deceased put on mourning habits, and abstained from gaiety and entertainments. The mourning lasted from forty to seventy days, during which time the body was embalmed; and, when the process was completed, placed in a sort of chest, which was afterwards preserved either in their houses or in the sepulchres of their ancestors. Before the dead were allowed to be deposited in a tomb, they underwent a solemn judgment, upon an unfavourable issue of which they were deprived of the rite of burial.

The mourning customs of the antient Jews can only be collected from an examination of the Prophets and other parts P. C., No. 660.

The funeral rites of the Greeks and Romans have been collected with great research by Guichard in his Funérailles, et diverses Manières d'ensevelir des Romains, Grecs, et autres Nations,' 4to., Lyon, 1581; by Meursius, in his treatise De Funere Græcorum et Romanorum,' 12mo., Hag. Com. 1604; by Gutherius, De Jure Manium, seu de Ritu, More, et Legibus prisci Funeris,' 12mo., Par., 1613, reprinted in 4to., 1615, and again in 8vo., Lips., 1671; and by Kirchman, De Funeribus Romanorum Libri IV.,' 12mo., Hamb., 1605, and Lugd. Bat., 1672. See also the Ceremonies Funèbres de toutes Nations,' par le Sr. Maret, 12mo., Par., 1677.

In the religious creed both of the Greeks and Romans, sepulture was peculiarly an act of piety toward the dead, without which it was supposed the departed spirit could not reach a place of rest. To be deprived of the proper rites was considered the greatest misfortune. The funeral rites of the Greeks and Romans were in many respects similar, and among both nations the practice prevailed of burning the dead and collecting the ashes in urns. In the case of public funerals, according to Servius's Commentary on Virgil, the deceased was kept seven or eight days, and every day washed with hot water, or sometimes with oil, that in case he were only in a slumber he might be waked; and at stated intervals his friends meeting made a shout with the same view: this was called conclamatio. On the seventh day, if no signs of life appeared, he was dressed and placed on a couch in the vestibule, with the feet outwards, as if about to take his departure. In the course of these seven days, an altar was raised near the bed-side, called acerra, on which the friends offered incense. The scene here described is frequently represented in antient bas-reliefs. (See the Townley Marbles, vol. ii., pp. 167, 228, &c.) On the seventh day the last 'conclamatio' ended, when the couch and body were carried to the rostra, where the nearest of kin pronounced the funeral oration, and afterwards to the funeral pile. The body having been consumed, the ashes were gathered, inclosed in an urn, and finally laid in the sepulchre or tomb. An apotheosis or canonization was frequently part of the funeral ceremony of the emperor.

The Magi among the Medes and Persians neither burned nor buried their dead, but left them to birds of prey or dogs. (Herod. i., 140; Strabo, 735, 746.) Chardin, in his Travels,' vol. ii., p. 186, has given a full description of a modern Persian cemetery; and Niebuhr describes the Parsees near Bombay as still exposing their dead after the antient fashion mentioned in Herodotus. (Niebuhr, Reisebeschreibung, ii., 50.) Tacitus, in his treatise 'De Moribus Germanorum,' (c. 27) notices the simplicity of the funerals among the antient Germans. Like the Romans, they burned their dead. The things which a German valued most were his arms and his horse: these were added to the funeral pile, with a persuasion that the deceased would have the same pursuits in his new state of existence.

In the tomb of Childeric, king of the Franks, his spear, his sword, with his other warlike weapons, and even his horse's head, were found. (See Montfaucon, Monumens de la Monarchie Françoise, tom. i., p. 10.)

Lafitau, Charlevoix, and other travellers describe the same notions of a future state and the same funeral ceremonies as prevalent among the savages of America. Dr. Robertson (Hist. of Amer., vol. ii., b. 4) says, as they imagine that departed spirits begin their career anew in the world whither they are gone, they bury together with the bodies of the dead, their bow, their arrows, and other weapons used in hunting or war; they deposit in their tomb the skins or stuffs of which they make garments, Indian corn, venison, domestic utensils, and whatever is reckoned among the necessaries in their simple mode of life.

·

For the funeral rites of the early Christians, the reader may consult Gretser De Funere Christiano,' 4to., Ingolst., 1611; and he may learn the customs of a later period from Durand, who wrote his 'Rationale Divinorum Officiorum' in the twelfth century.

Brand, in his Popular Antiquities,' vol. ii., p. 139 to 212, has much upon the English ceremonials, beginning with VOL. XI.-D