demand for various objects in metal was now so greatly increased that it was evident, first from the consumption of timber for the foundries, and next from the destruction of forests in consequence of the agricultural improvements which were taking place in various parts of the country, that this provision must soon prove insufficient to meet the wants of the trade. It was then that the attempt was made to bring coal into use; and it is curious at this time to know how much difficulty, first from ignorance of its real value in the operations of metallurgy, and then from prejudice against such an innovation upon the old habit of burning wood only (to which must also be added the interference with the interests of wood monopolists), was thrown in the way of its introduction. One of the earliest and most zealous advocates for its use, Dudley, had all his works destroyed, and was nearly ruined by the violence of his opponents; but at length the employment of coal was fully established; and from that time the rapid advance of the iron and other metal works of this country may be dated, We have stated the chief reason for having recourse to the coalmines for fuel to have been the apprehension that the supply of wood fuel would fail. It is important to bear in mind, however, that, but for this well-grounded fear, wood or charcoal would always have been preferred for many of the operations of metallurgy; from its being less objectionable, as regards its chemical composition, than coal. The iron that is smelted in Sweden by wood fuel is considered a superior article, and is much sought after; and the smelters in this country find it necessary to char or coke the pit and sea-coal which they use, in order to adapt it to the purposes required. [COKE.] We shall now glance rapidly at some of the operations in the reduction of the ores of the four metals which are produced and worked in the greatest abundance in this country-iron, tin, copper, and lead: referring the reader to the articles under the names of those metals for further details. Iron is obtained from a very abundant ore in this country, namely, the common ironstone of our coal-measures. For the reduction of the ore to a metallic state it is necessary to add a certain quantity of lime, which acts as a flux; and it is worthy of remark that, while the ore itself from which the metal is produced, and the coal for smelting it, are found together, the limestone by which its reduction is facilitated usually abounds in the lower regions of the carboniferous strata. Sometimes, as in the great coal basin of South Wales, a bed of millstone grit capable of enduring the fire, and used in constructing the furnaces, is also found in connection or alternating with the iron ore and limestone. The first operation the ore undergoes is roasting. This is done in various ways, both in this and in other countries. Sometimes it is conducted in kilns, sometimes on the ground in the open air. The first method is by heaping the iron ore on a mass of ignited coal. In the other, a thick layer of ironstone, broken in pieces, is placed upon a bed of coal, wood, or charcoal (on the continent wood or charcoal is always used), 6 or 8 inches thick, and covering an area of several yards; upon this another layer of fuel is placed, and then another pile of ore, which diminishes both in area and thickness towards the top. The whole is then covered with small coal or charcoal dust till it reaches some feet above the ground. The lower stratum of fuel is then lighted, and by degrees ignites the whole mass. In the course of a few days the ironstone becomes cool, and the sulphur, arsenic, water, and inflammable matter being driven off, it is fit for smelting. It is then placed in a furnace, with fuel and limestone in deterinined proportions. At Dudley, in Staffordshire, for 2 tons of roasted ore, which affords a ton of cast metal, 19 cwt. of transition limestone are employed as flux. In the course of a few hours the whole runs down, and the iron is melted, and in that state is allowed to flow into furrows made in sand, where it forms what is termed pig-iron; or it is poured into moulds where it forms the various articles of cast-iron ware. There are various sorts of cast-iron, but it is usually divided into three classes relatively to its colour and qualities, which are in this country called numbers one, two, three; sometimes more descriptive names are given to the different qualities, as smooth-faced, gray, white, forge pigs, ballast-iron, &c., &c. Cast-iron is converted into bar-iron by smelting it by means of charcoal, when it is welded and hammered; of this there are also varieties, of which the toughest, called stub-iron, is used in forming fowling-piece barrels. It is made by inclosing old horse-shoe nails tightly in a broad iron ring, generally made of Swedish iron; a welding heat is then applied, and the whole mass is hammered till by degrees the nails and ring become completely united: it is then drawn into bars, which make an iron of peculiar closeness, toughness, and malleability. These matters are further treated under IRON MANUFACTURE. The best ore of tin is found in Cornwall. It is commonly blasted by gunpowder, and is procured in pieces of considerable size, which are stamped to powder by beams shod with iron: it is then well washed till the earthy particles are carried off, and the tin is fit for the smelting house. After being roasted in a reverberatory furnace, and again washed, it is a second time subjected to the furnace, being now mixed with small coal, and in some cases, with a small quantity of lime. The melted tin thus produced is at last placed in a small furnace and exposed to a very gentle heat, when the purest portion melts first and is drawn off. This is called common grain tin; and the inferior, which still contains a small proportion of copper and arsenic, is then cast into pigs called block tin. The finest grain tin is procured from the stream works of Cornwall. Good stream tin affords from 65 to 75 per cent. of the best grain. For the details of operation, see TIN. The The reduction of copper ore is made by several consecutive processes. The first is by calcining it, and when the ore is sufficiently roasted to oxidate the iron which it contains, it is melted. melted metal is after a time suffered to flow into a pit filled with water, by which it becomes granulated. It then undergoes further heating, and what is called technically its slag (or scoria) is taken off, and it is again allowed to run off into water. After other nearly similar processes the copper is cast in sand, when it becomes solid, and in this state is called blistered copper. It is now fit for what is termed the refinery, and undergoes an operation called refining or toughening. This is an operation of delicacy, requiring great skill and care in the workmen. The refining is conducted in a furnace similar to the melting furnace; the object is to thoroughly purify the metal from any portions of oxygen, which is performed by adding charcoal to the copper while it is in fusion, and stirring it occasionally till it is judged to be pure. The chemical relations of this process are noticed under COPPER. When tin is united with copper, it forms the compound called bronze. [BRONZE] a The greater part of the lead met with in England is procured from substance called galena, in which it is found combined with sulphur. There are, however, other ores of lead. The galena, being freed by hammering it and by the hand from whatever impurities can be separated from it by those means, is broken up into small pieces, and after repeated washings is placed in a reverberatory furnace; but only sufficiently heated to drive off certain ingredients without melting the lead itself. The roasting being finished, charcoal is added till the reduction is completed. The lead, after the slag has been removed from it, is suffered to run out of the furnace into a pan, and being first skimmed is ladled out into moulds and left to cool. There are various methods adopted in different places and under different circumstances for procuring the metal from the ores: these will be found succinctly noticed under LEAD. The furnaces that are used in founding are chiefly of two kinds, and though strictly speaking both are air furnaces, yet they are distinguished as air or wind furnaces and blast furnaces. The first acts by a draught through a chimney; whereas in the other the air is forced into the body of the furnace by means of bellows. The forms and relative proportions of the different parts of the furnace, and particularly the size, elevation, and direction of the chimnies, and the dimensions and space of the flues when these are required, are of great importance; the volume and intensity of heat and consequent certainty of the operations depending in a great measure on the knowledge and science displayed in adapting the parts to each other. The chief points of difference between wind, blast, and reverberatory furnaces are described under FURNACE. This Founding is practised either by melting or casting any quantity of metal in the solid, or with a core (by means of which the metal is preserved of a determined thickness or substance), or in plain casting. Before any object can be cast in metal it is necessary that a model of it be prepared. The models may be made of various substances; clay or wax, or sand with clay, are those usually employed, but they may also be of wood, stone, or any other material. Upon those models moulds must be made; these are commonly composed of plaster of Paris mixed with brickdust, sometimes sand, or sand with a mixture of cow-hair. For moulds for iron and brass work a yellowish sharp sand is preferred, which is prepared by mixing it with water and then rolling it on a flat board till it is well kneaded and fit for use. process is called, in technical language, tewing. If the model is cylindrical, or of a form that admits of such a process, it is moulded or cast in two pieces; these two parts are then carefully joined together, and the edges or seams trimmed. This doubling is an easy and cheap mode of execution, and only requires care to be successful. For the smaller class of works, instead of melting and running the metal at once from a large furnace, earthen crucibles are used, into which the metal is thrown in small pieces: the crucible is placed in a strong heat in a close stove, and as the metal is melted and sinks more is added till the vessel is full. It is then lifted out by means of iron instruments adapted to the purpose, and the metal is poured from it into the moulds, in which channels or ducts for receiving it have been previously made. There is one great advantage in using crucibles; namely, that the metal may be carried in them to any part of the foundry, whereas in general it is essential to have the moulds and the furnace close together. It is obvious however that melting metal in crucibles can only be practised where the casting is on a comparatively small scale. In noticing the different ways of casting, mention has been made of one in which a core is used, and which may require some explanation. The core, as its name denotes, is a part or portion situated within the body of the cast; and its purpose in founding is to form a centre to the work by which the thickness or substance of the metal may be regulated. In coring, the mould must first be made complete; into this, clay or wax, or any other fit substance or material, is then squeezed or pressed in a layer of uniform thickness; in large works it is usually from half an inch to an inch thick. This layer represents the metal. The mould, if in parts, is then to be put together, the above mentioned layer being left within it; and into the open space in the centre a composition (usually of plaster of Paris with other substances mixed with it) is introduced, and made to adhere to, or rather is filled up to the clay or wax. This is the core, and it is often made to occupy the whole interior of the mould. When this is set, or dry, the mould is taken to pieces and the material which has been made to represent the metal removed. The mould is then again put carefully together round its core or nucleus, the two portions being secured from contact by stops and keys properly arranged for that purpose. It is now obvious that when the mould is placed, with its channels and ducts, to receive the metal, this latter can only enter the interstice or space between the outer mould and the inner core; and thus, by an ingenious and simple contrivance, the cast is insured of sufficient substance to answer every object required, with, at the same time, a great saving of metal and reduction of weight. pinion work; by this agency the box is raised, lowered, and moved In all these operations it is essential that the mould and the cores be perfectly free from moisture; seeing that the sudden and violent expansion of air that is at all damp, upon the heated metal flowing into the mould, would cause it to burst, to the destruction of the work and the great danger of the workmen. In order to guard against this, the moulds and core are usually placed in an iron closet or dryingstove, in which large fires are constantly kept up, and from whence they are not removed till it is ascertained that they are perfectly dry, and just before they are required for the casting. The moulds and cores of works of large dimensions are usually strengthened with bars and hoops of iron, to prevent them from springing or changing their form during the drying, and during the necessary moving and shifting about in the foundry. All that is now necessary before casting is to cut the channels or ducts for the metal to penetrate easily and quickly into the mould; and to place the mould conveniently with respect to distance and inclination from the furnace. The first operation is easily performed; the founder takes care to distribute the channels, both in number and in their size (or width), according to the parts of his work into which he requires a greater or less volume of metal to flew, and also, if the object be of great extent or complicated form, that the different parts of the mould may as nearly as possible be filled simultaneously; it being most desirable that the whole getto, or cast, should be made before the metal in any of the parts has time to settle or lose its fluidity. Other channels are also made for allowing the air to escape as the melted metal enters the mould; these are called vents, and are very necessary where the works are on a considerable scale. With respect to placing the mould, it is only important to secure a sufficient inclination of plane from the mouth of the furnace to the mould that the metal may run easily and uninterruptedly, and not have time to grow cool and therefore sluggish. The usual method in great bronze works is to bury the mould in a pit a little below the level of the furnace, and by ramming sand firmly round it to insure its not being affected by any sudden or violent shock, or by the weight of metal running into it. When everything is ready, and the metal found to be in a state fit for running, the orifice or mouth of the furnace (which is usually plugged with clay and sand) is opened, when the metal descends, and in a few minutes the mould is filled. The metal is allowed to run till it overflows the mouths of the channels into the mould. The work is then left to cool, after which the mould is scraped or knocked off and the cast undergoes the necessary pro-lessness, mismanagement, or want of sufficient funds for the adminis cesses (such as cleaning, chasing, &c.) to render it fit for the purpose designed. The variations which the processes undergo, in treating different metals and manufacturing different articles, will be found noticed under Bell, Brass, Bronze, Cannon, Iron Manufacture, &c. Among the most remarkable modern specimens of founding, for their vastness of size and weight, are the cylinders and cranks of the Great Eastern ship; the iron tables for some of the plate-glass manufactories; the bed-plates for the larger kinds of machinery; some of the mortars on which such large sums of money have lately been unprofitably expended; and the two unfortunate bells for the Westminster palace. As was said under BRONZE, the largest castings for fine arts purposes are frequently not made in a single piece; thus, the largest bronze statue of modern times, the colossal figure of Bavaria,' 614 feet in height, placed in front of the Rühmeshalle, near Munich (modelled by Schwanthaler and cast by Fras. Miller at the royal foundry), was made in many pieces; and the largest of those did not nearly equal in weight the masses of iron above adverted to. We may briefly notice here a method introduced in 1859 by Mr. Jobson, for casting shells and other articles. The pattern is made with arms carrying pins or projections, entering sockets in the plate on which the casting-box is placed. When the pattern has been correctly adjusted upon the plate, the pins and the ends of the sockets are filed, so as to make the ends of the pins and those of the sockets coincide; this affords a test for the workman to judge of the accurate adjustment for each casting. To facilitate the introduction and removal of the moulding sand, apparatus is arranged whereby the sand is shovelled upon sieves, which are worked mechanically; the sand is damped and sifted, and is carried by a sort of Archimedean screw to the mixer, where fresh sand and other matters are added, and whence the sand is conveyed through another trough by a screw into a position to be used by the moulder. In instances where heavy casting-boxes are employed, a peculiar form of crane is used; it consists of a quadrant, a chain or band passing over the quadrant, and a train of rack-and In France, the philanthropist Vincent de Paul, the founder of the Society of the Missions in the first half of the 17th century, exerted himself to found an asylum for infants, which were at that time frequently left to perish in the streets of Paris. It was at first supported by private subscriptions, but afterwards was made a national establishment-"Hôpital des Enfans trouvés." Similar institutions were founded in other great French cities. Mortality appears to be very great in most foundling hospitals of the continent, owing to care tration of those institutions. The infants are given out to cheap nurses in the country where a great number of them die. At the same time, it is remarkable that the number of illegitimate births has greatly increased over all Europe. The principal objection that has been raised against foundling hospitals is, that they tend to increase the number of illegitimate offspring. In London, the institution of the Foundling Hospital, though it seemed to have prevented the exposure or murder of infants for a time, was found to produce such ill effects, that the object of the foundation was materially changed, and it is now a receptacle for illegitimate children on the application of the mother, who must prove the abandonment of it by the father, and whose character otherwise must be a good one. One distinction ought to be made, namely, that in countries where there is no legal provision for the poor, foundling hospitals appear to be more necessary, or at least less objectionable than in those where the mothers of illegitimate children, if unable to support them, have, like other destitute persons, the resource of the parish poor-house. It must also be observed that mothers of illegitimate children often neglect their unfortunate offspring, and are ill calculated by their habits to rear them up so as to make of them useful and honest members of society. FOUNTAIN, a jet or jets of water, flowing either naturally out of the earth, or from structures formed by art. Artificial fountains consist of water flowing from vases, statues, sculptured ornaments, or architectural buildings combined with sculptured figures and other ornamental decorations. Many ancient Greek cities were decorated with fountains. Pausanias informs us that Corinth was adorned with several fountains; and he mentions one in particular which stood near the statue of Diana, representing Pegasus, with the water flowing through his feet (ii. 3, 5). He describes another as consisting of a bronze Neptune seated on a dolphin, from the mouth of which the water issued (ii. 2, 8). Frontinus, who lived in the reigns of Nerva and Trajan, was superintendent of the fountains at Rome, and wrote a work, 'De Aquæductibus Urbis Romæ Commentarius,' in which he treats, among other things, of the distribution of the waters of fountains. The public fountains of Pompeii, some of which are almost perfect, evince the knowledge which the ancients possessed of the property of water to rise to its level, and their practical application of the principle. b b Section of Fountain, from Pompeii, showing the ascending pipe, a, a, a. Not only were the streets, but even the private houses of the Pompeians, decorated with fountains; and it appears that the ancients were acquainted with that law by which fluids may be made to ascend in a vertical jet to a height proportionate to the pressure which acts upon them. artistic power. The fountains of the Saracenic architects, as shown in Spain, Cairo, and Constantinople, exhibited in their best day great fancy, lightness, and brilliancy of design, and a style singularly appropriate to the purpose and to the character of the surrounding buildings. The city of Paris is well supplied with fountains, many of which are elegantly designed. The fountains of Versailles and St. Cloud in France, and the fountains at Wilhelmshöhe near Cassel, were the largest in Europe prior to the construction of those at the Crystal Palace, Sydenham. Now the entire "system of fountains," as it is called, at this latter place may take rank, both as regards extent and brilliancy of effect, with anything previously produced. London, though well supplied with water, has few fountains which make much pretension as works of art. During the past year, many small mural fountains, and a few standard fountains have however been erected in London and its vicinity, chiefly through the exertions of a society called the Metropolitan Free Drinking-Fountains Association. The primary object of this society is to erect as large a number of free drinking-fountains as possible, and with little loss of time. Eventually it is hoped that 400 such fountains may be erected in and around the metropolis. Utility rather than beauty, and economy of cost, are consequently leading principles in the preparation of the fountains of the association. In order, however, to combine a measure of ornament with utility, several designs have been selected, each of which serves as a pattern for a number of fountains; but few if any of the designs yet produced exhibit originality or fancy, and from the mode of reproduction the fountains show little individuality of character. Drinking fountains, similar in principle to these, were first introduced at Liverpool a few years back by Mr. Melly, a merchant in that city, who at his own cost erected several in the busiest localities. The example is being followed throughout the country, and already several hundred drinking-fountains have been erected. The majority of these are comparatively humble structures, but perhaps some of the most successful in an artistic point of view have been erected in provincial towns. FOURTH, an interval in music, and to be enumerated among the discords; though it seems to have puzzled many writers on music, some of whom are much inclined to view it as a concord. [CONCORD.] Its ratio is 4: 3. Of fourths there are three kinds: the Diminished Fourth, the Perfect Fourth, and the Extreme Sharp, or Superfluous Fourth (called also the Tritonus, from being composed of three whole tones). The first (c#, F) is composed of a whole tone and two semitones; the second (c, F) of two whole notes and a semitone; and the third (c, F#) of three whole tones. Example: Fountain, from the paintings of Pompeii. One of the domestic fountains of the Pompeians is encrusted with coloured glass and shells. The fountain of water flowed from a large mask set on steps, placed within a large niche. At Rome, the proper distribution of the rivers which flowed through her aqueducts was a matter of great importance, entrusted to the care of an officer of very high rank. It appears from Frontinus, who filled that office under the emperor Nerva, that the letting out of the public waters to private persons was a source of revenue. The right to a supply of water was strictly personal, not attached to houses, so that the supply was cut off at every change of ownership. The waters which had once been granted were sold by the superintendents, as they fell in, to the highest bidders. Those whose means or interest were insufficient to obtain a private pipe, were obliged to fetch water from the public fountains. (Pompeii,' vol. ii. pp. 73, 74.) The number of leaden pipes found in Pompeii leads us to conclude that they were universally employed in fitting up the fountains of that city. Some fountains flowed through bronze figures, of which several are preserved in the museum at Naples. Specimens of the domestic fountains of the Romans (of marble) may be seen in the Græco Roman Basement Room of the British Museum. Some of the cities of Italy and the East are adorned with fountains, which are no less agreeable to the eye than useful to the inhabitants. Of all places, modern Rome is perhaps most abundantly furnished with this agreeable convenience, though this profusion is probably only a tithe of the luxury with which the ancient city was supplied. Many of the fountains of Rome are highly decorated, of great magnitude, and very varied in their mode of ejecting the waters with which they are supplied from the existing aqueducts. The fountains of Trevi, and the Pauline fountain at San Pietro in Montorio, are immense piles of architecture, the former highly decorated with sculpture. In Italy, almost every species of design which the imagination can form has been adopted by their ingenious artists in the construction of fountains. Many of those produced during the Medicean period are works of great FOWLING, the act or art of taking birds with nets, by shooting, snares, the use of bird-lime, or other devices. It is also sometimes used for the taking of birds with hawks and falcons, more properly called falconry. In Latin this sport is termed Aucupium. See Bargæus de Aucupio,' liber i., 'ad Franciscum Medicem Florent. et Senens. Principem,' 4to. Flor. 1566. Olina's Uccelliera,' 4to. Rom. 1684, is another work on fowling, the plates of which, representing the different modes of following the sport, are extremely curious. English we have Blore's Gentleman's Recreation,' fol. Lond. 1686 and 1716; and The Experienced Fowler, or the Gentleman's Recrea tion,' 16mo. Lond. 1704. In FRACTIONS, COMMON AND DECIMAL. By a fraction is meant, in the first instance, a part of any magnitude. Thus, "three and a fraction" means three units and a part of a fourth. The next meaning of the term confines fractions, in an arithmetical point of view, to the aliquot parts or submultiples of the unit; which unit must therefore be divided into a number of equal parts, of which parts a certain number is to be taken. Under the heads ADDITION, &c., will be found the various rules by confine ourselves to fundamental points connected with the theory. which operations containing fractions are conducted. We shall here a b A fraction is thus denoted: means the quantity obtained by dividing a unit into b equal parts and taking a of those parts. If a be greater than b, it will obviously be necessary to divide more units than one, each into b equal parts, until enough have been subdivided to furnish the a parts required. It was usual, in English works on arithmetic, to call fractions in which a is less than b, proper fractions; and all others improper fractions; this absurd distinction is now beginning to be abolished. In the preceding fraction a is called the numerator, and b the denominator. The first term is correct, for a is the number of parts of a certain kind which are to be taken; the second is not quite so correct, for the denomination of which the number a is to be taken, is not b, but; ; the bth part of a unit (not b units) is to be repeated a times. On the whole, the terms numerator and denominator are very appropriate; but they are very long. Should they ever come into vulgar use, they will be shortened into numer and denomer; and it would be well if arithmeticians would do this for themselves. The preceding fraction may be considered in several different ways. It is 1st, the bth part of a unit repeated a times; or, in common language, a-bths of a unit; 2nd, the number of times, or parts of a time, or both, which a contains b; 3rd, the proportion which a is of b; 4th, the expression which ought to be written for a, on the supposition of that which was b units being made the unit. Thus expresses twofifths of a unit, the part of a time which 2 contains 5, the proportion which 2 is of 5, and the expression which must be written for what is now 2, when that which is now 5 is made the unit. All these meanings, except the first, are perfectly intelligible when we write a fraction in which the terms are both fractional. Thus may be thus explained. We can readily imagine the part of a time which 14 is of 3, the proportion which the first is of the second, and the expression which must be substituted for 14 when a larger unit is used, amounting to 34 of the present unit. But though we see clearly what is meant by dividing 1 into 3 equal parts and into 4 equal parts, what idea are we to attach to the division of 1 into 3 equal parts? The generality of mathematical conceptions is frequently destroyed by the peculiar idiom of a language. The science of arithmetic requires the abolition of all those distinctions which depend on singular and plural, noun and pronoun, &c. Thus, when we speak of the answer to a problem being a number of feet (unknown), it is better to allow the word to imply a part of a foot, a foot itself, or a number of feet together with a part of a foot, than to repeat all those possible cases every time a number is to be mentioned. Again, when one particular phrase seems absurd, but another which is synonymous appears clear, we must either reject the former altogether, or attribute to it the meaning of the latter, and the second course is generally the more We now observe that the direction to "divide one into convenient. 10 equal parts" is the same as "find a part such, that ten of them shall make a unit." Now there is no absurdity in requiring to "find a part such that 3 of them shall make a unit," though it is inconsistent with our idiom to speak of "dividing 1 into 34 equal parts." The meaning of the phrase which is intelligible should then be extended to that which is not, or " to divide 1 into 3 equal parts" should mean that the part is to be found which repeated 3 times and of a time shall give the unit. And this must be extended even to the case in which the number or fraction thus obtained is greater than a unit. Thus in the fourth of the preceding fractions such a number or fraction must be found, that 4th of it shall be a unit; that is,- and this must be repeated 3 times. The preceding considerations show that fractions with fractional denominators may be explained (without reference to any rule of reduction) by an extension of the definition which applies to integer denominators. The use of such an extension is as follows:-at present, algebraical students learn results which are perfectly intelligible with regard to whole numbers, or to fractions with integer terms, but of which they do not see the meaning when fractional or mixed terms are employed. In the latter case they trust to what they see in the former that their results will remain true; but they can have no distinct perception on this point until they have learnt to include every possible form of under one definition. The fundamental property of fractions on which all others depend is this that no fraction is changed in value by multiplying or dividing both its terms by the same number or fraction, that is, | preceding unit. Place a point on the unit's place (to mark its posi tion), and let the same method of valuation be carried further. Then in 111111111, the first 1 after the point should stand for one-tenth of the preceding, or one-tenth of a unit; the second for one-tenth of a tenth, or one-hundredth, and so on. The fundamental theorem of decimal fractions, in this view of the subject, is that which shows, for example, that 12-2345 (defined to mean 1 ten, 2 units, 2 tenths, 3 hundredths, 4 thousandths, and 5 ten-thousandths) is the same as 122345 ten thousandths; or that all the number, such as it would be if the units' column were on the right, may be taken as a numerator, and the denomination of the right hand figure as a denominator. Thus4 8 3 + 10 100 1000 + 65:483 or 60 + 5 + by only 41 of the hundred thousandth of a unit, or by less than the hundred thousandth part. It is from such a transformation that the common rule is derived. It is common to say that a result is true to a certain number of places of decimals when any alteration of any place would make it further from the truth. Thus, the diameter of a circle being unity, the circumference lies between 3.1415 and 31416, but nearer to the latter; whence the same circumference, true to four places of decimals, is 3.1416. Similarly, 62:13299, taken true to two places, is 62:13; to three, 62133; to four, 62:1330. Again, 625, taken true to two places, might be either 62, or 63; but the latter is generally taken. When a decimal fraction cannot be found exactly equal to a given common fraction, the division by which the numerator is found, leads to what is called a CIRCULATING DECIMAL. For subjects closely connected with the theory of fractions, see RATIO; PROPORTION; INCOMMENSURABLE. FRACTIONS, CONTINUED. A continued fraction is one which has a fraction in its denominator, which again has a fraction in its denominator, and so on: such as 1 2 + 7 +6 1 + A more convenient way of writing such fractions is desirable; in the present article we shall adopt the following:: 1 3 6 2 2 +7 +1+3 is written d + e } 3 3 x 10 5 5 × 10 the third is greater than the first, but less than the second. In practice it is convenient to employ fractions having either the same denominators, or which may easily be reduced to others of equal value having the same denominators. The numbers 10, 100, 1000, &c., suggest themselves for this purpose: indeed it may immediately be seen that the ordinary system of decimal numeration may be extended so as to allow of a representation of such fractions. If we consider the number 11111, we see that for every step which we make to the right, we find a unit which is only the tenth part of the a The use of continued fractions is as follows: by converting a common fraction with a large numerator and denominator, into a continued fraction, we are able to find a succession of more simple fractions which are alternately greater and less than the given fraction, and approach to it with great rapidity. Let be the given fraction, a being less than b; proceed as in the rule for finding the greatest common measure of a and b, and let q, r, s, t, &c., be the quotients obtained in the process; then 5, 1, 1, 7, 3, 1, 1, 2, 1, 3, 1, 2; the end of the process, observe that the numerator of the difference of any two succeeding fractions is unity. Thus 1 x 6 exceeds 1 x 5 by 1 1 x 11 falls short of 2 x 6 by 1 2 x 83 exceeds 15 x 11 by 1 15 × 260 falls short of 83 x 47 by 1 &c. No fraction, having a less denominator than one of the approximate fractions, can come so near to the original fraction as the one which 603 28319 which are to be used as follows in forming the succession of approxi- possible fraction which has an integer numerator, and an integer denois obtained by the process. Thus, 10 is nearer to 5 than any mate fractions. The first and second are always 280* 109 2 3206 343 1549* minator less than 603. FRACTIONS, DECOMPOSITION OF, a method of much use in the integral calculus for reducing products of the form x(x—a)—= (x-b)-n.. in which X is rational and integral, to the sum of terms of the form K(x-a)-, in which K is independent of x. But if x be of a higher dimension than m+n+... there is also a quotient. This quotient may be easily found by the short rule for division by x-a. To divide pa +qxm−1 + ... by x-a, take the first coefficient, multiply it by a and add the next; repeat this process to the end, taking care to use 0 for the coefficient of any missing term. The results, beginning with the first coefficient, are the coefficients of the quotient, except the last, which is the remainder. Thus in dividing 2x7—2x6+x3+x-1 by x- - 3, the coefficients are 2 4 12 36 109 327 982 2945 Hence the quotient is 2.6+ 4x2 + 12x1 +36x3 + 109x2 + 327x+982, and the remainder is 2945. To divide by x+a, or x-(-a), use —ɑ as a multiplier. Thus to find the quotient of x3- 21⁄23 + 4 divided by (x−1)3 (x + 1)2 (x-5), the process is as follows, it being seen beforehand that the answer is of the second degree. We omit all the work which is of no use in the final result. 8th 9th 389 2152 33 When we divide a3-23+4 by x-1 the three first terms of the quotient are +x+x5: two divisions more by x-1 give x2 + 3x1— 6x3. We then divide twice following by x+1, still preserving only three terms, and the result is 3+2+3x, and the final division by x-5 brings out x+6x+33 for the three first terms of the final quotient, that is, for the whole quotient. For verification the order of the divisions may be varied. When all the roots of the denominator are unequal, that is, when the quantity to be reduced is of the form x(x—a)—1 (x—b)—1 (x—c)—1 where a, b, c, &c., are all unequal, the process for finding the reduced form of the remainder is very easy. Strike out (x-a)-1 and Next make xa in the form thus mutilated; let this result be a. strike out (x-b)-1, and make x=b; let the result be B; and so on. Then the fraction x(x-a)-1 (x—b)−1. is the quotient, if any, + A (x − a)−1 + B (x —b)−1+ Thus to reduce the expression The succession of fractions continually approximating to the given 26(x-1), (x-2) (x+1), we set down all the work, as follows; the 1 1 2 15 47 62 109 280 389 1447 1836 5119 When the roots of the denominator are not all equal, the way of proceeding which is most easy in the simple cases which generally occur in practice, will be best caught from an example. Suppose, for instance, that (x+1)÷ (x-1)2 (x-2) (x+1) is the fraction to be reduced. Begin with the denominator (x-1) (x-2) (x+1), and by the preceding rule, obtain the result, x4+1 1 =x+2 + 17 + 1 1 3x+1 |