The

This is one among many instances of the existence of ancient usages in England, which are the same or nearly the same as the usages of nations that we call barbarous. Tacitus (Germania,' c. 26) says of the ancient German mode of agriculture: "The lands, in proportion to the number of cultivators, are occupied by all in turns, which presently they divide among themselves according to their rank (merit). The extensive plains offer facilities for division. They change the cultivated fields yearly; and there is still a superfluity of land." meaning of Tacitus is not clear. The following passage in Cæsar's account of the Gauls (vi. 22) is more distinct: "They pay no attention to agriculture, nor has any man a fixed quantity of land and boundaries of property: but the magistrates annually assign to the clans and tribes who have come together, as much land as they please and where they please, and in the next year they compel them to move to another spot." Herodotus (ii. 168 says that each member of the military caste in Egypt had a certain portion of land assigned to him; but they enjoyed the lands in a rotation, and the same persons did not continue in the enjoyment of the same lands. Strabo (p. 315) mentions a custom amongst the Dalmatians of making a division of their lands every eight years. "The third class is that of grazing lands, where the rights of parties are settled and defined, the ordinary stinted pasture. The commonable lands are subject to very great variety and peculiarity; for instance, in some of these lands the right of grazing sheep at all belongs to a man called a flock-master, and he has the power, during certain months of the year, of turning his own sheep exclusively on all the lands of the parish; or, according to particular circumstances, his right is limited and restricted to turning sheep upon a certain portion of it, with a view to giving parties an opportunity of putting in a wheat crop. In those parishes where there is a flock-master who has a right of lepasturing his sheep during a certain portion of the year over all the land of the parish, it is clear that no one can sow any wheat without having made a bargain with him for shutting up his own particular fields, or some portion of them."

"There is a very large extent of woodland in this kingdom that is commonable, strange to say, where certain individuals have a right during the whole year, to turn on stock, the owner of the wood having no means of preserving his property except by shutting out other commoners' stock by custom for some two or three years after felling. There is that right, as also the old right of estover, which is a very great inconvenience, namely, where parties have the right of cutting house-bote, and plough-bote, and fire-bote, and so on in woods belonging, quá wood, to another party. There is a great deal of land subject to that ruinous custom. There are many varieties of these commonable lands, but these are the most prominent and remarkable of them."

Under such a system as this, it is obvious that these common fields must be ill cultivated. The intermixed lands cannot be treated according to the improved rules of good husbandry. It is stated that the simple re-distribution of intermixed lands, now held in parcels so inconvenient in form and size as to be incapable of good husbandry, would in many instances raise the fee-simple value of the lands from 158. or 178. an acre to 308.

It was the opinion of witnesses examined before the parliamentary committee of 1844, on Commons' Inclosure, that judicious inclosure would make a large portion of common lands much more productive. Open arable lands are so intermixed that effectual drainage is nearly impossible. One witness said: "I have had occasion to go over two small properties, about 150 acres each; one I found in 301 different pieces, and another in a little more than a hundred. I mention this to show how the lands are frequently intermixed; they are therefore farmed at much greater expense; and it is impossible to drain them on the present improved mode of drainage, inasmuch as other parties are occupying the furrow by which the water should pass off." In the Midland counties, where there are these open arable fields, the course is two crops and a fallow, and every third year the flocks run over the whole field. The same witness considered that a fourth of all the arable land was totally unproductive. In cases where common arable fields have been subdivided and allotted, "the great improvement is, that in the first place every man has his allotment, and he deals with it as he pleases; he drains it, and crops it upon a proper course of cropping; he puts it in seed and keeps sheep upon it; he grows turnips and clover, or whatever he thinks proper." The same witness was of opinion that the average improvement in the value of common fields which had been inclosed was not less than 25 per cent. Indeed, the evidence that was produced before the committee established to a degree beyond what otherwise would be credible, the immense inconvenience and loss which arise from the system of intermixed lands, and their being also subject to commonage.

As to Common Rights, that is, rights of pasture and so forth on commons or waste lands, they are described generally under COMMON,

ARTS AND SCI. DIV. VOL. IV.

In 1836 an act (6 & 7 Wm. IV. c. 115) was passed for facilitating the inclosure of open and arable fields in England and Wales. The preamble to the act is as follows:-"Whereas there are in many parishes, townships, and places in England and Wales, divers open and common arable, meadow, and pasture lands and fields, and the lands of the several proprietors of the same are frequently very much intermixed and dispersed, and it would tend to the improved cultivation and occupation of all the aforesaid lands, &c., and be otherwise advantageous to the proprietors thereof, and persons interested therein, if they were enabled by a general law to divide and inclose the same," &c. Inclosures have been made under the provisions of this act, but the powers which it gives are limited, for the "act applies solely to lands held in severalty during some proportion of the year, with this exception, that slips and balks intervening between the cultivated lands may be inclosed." The lands which cannot be inclosed under the provisions of this act are "the uncultivated lands, the lands in a state of nature, intervening between these cultivated lands, beyond those that are fairly to be considered as slips and balks." However, it was stated in evidence before the committee of the House of Commons in 1844, that a large extent of common and waste land had been illegally inclosed under the provisions of the act, and the persons who hold such lands have no legal title, and can only obtain one by lapse of time. The chief motive to this dealing with commons appears to have been, that they thus got the inclosure done cheaper than by applying to Parliament for a private act.

In 1844 a select committee of the House of Commons was appointed "to inquire into the expediency of facilitating the inclosure and improvement of commons and lands held in common, the exchange of lands and the division of intermixed lands, and into the best means of providing for the same, and to report their opinion to the House." The committee made their report in favour of a general inclosure act, after receiving a large amount of evidence from persons who were well acquainted with the subject. The extracts that have been given in this article are from the printed evidence that was taken before the select committee.

In pursuance of the recommendation of the committee, an Act of Parliament was passed in 1845 (8 & 9 Vict. c. 118), the object of which is thus stated in the preamble: "Whereas it is expedient to facilitate the inclosure and improvement of commons and other lands now subject to the rights of property which obstruct cultivation and the productive employment of labour, and to facilitate such exchanges of lands, and such divisions of lands intermixed or divided into inconvenient parcels, as may be beneficial to the respective owners; and it is also expedient to provide remedies for the defective or incomplete execution and for the non-execution of powers created by general and local acts of inclosure, and to authorise the renewal of such powers in certain cases," &c.

It is not within the scope of this article to attempt to give any account of the provisions contained in the 160 sections of this act; but a few provisions will be noticed that are important in an economical and political point of view.

The 11th section contains a comprehensive description of lands which may be inclosed under the act, in which the New Forest and the Forest of Dean were excepted, but even with these portions have been since inclosed, and the new plantations are fenced for a certain time. The 14th section provides that no lands situated within fifteen miles of the city of London, or within certain distances of other towns, which distances vary according to the population, shall be subject to be inclosed under the provisions of this act without the previous authority of parliament in each particular case. The 15th section provides against inclosing town greens or village greens, and contains other regulations as to them. The 30th section provides that an allotment for the purposes of exercise and recreation for the inhabitants of a neighbourhood may be required by the commissioners under the act, as one of the terms and conditions of an inclosure of such lands as are mentioned in § 30.

The 108th section makes regulations as to "the allotment which upon any inclosure under this act shall be made for the labouring 3 н

poor," and (sect. 109) "the allotment wardens (appointed by sect. 108) shall from time to time let the allotments under their management in gardens not exceeding a quarter of an acre each, to such poor inhabitants of the parish for one year, or from year to year, at such rents payable at such times and on such terms and conditions not inconsistent with the provisions of this act, as they shall think fit." Section 112 provides for the application of the rents of allotments; the residue of which, if any, after the payments mentioned in this section have been defrayed, is to be paid to the overseers of the poor in aid of the poor rates of the parish.

Sections (147, 148) provide for the exchanges of lands not subject to be included under this act, or subject to be inclosed, as to which no proceedings for an inclosure shall be pending, and for the division of intermixed lands under the same circumstances.

Under section 152 commissioners are empowered to remedy defects and omissions in awards under any local act of inclosure, or under the 6 & 7 Will. IV. c. 115; and under section 157, the commissioners may confirm awards or agreements made under the supposed authority of 6 & 7 Will. IV. c. 115, if the lands which have been illegally inclosed or apportioned or allotted, shall be within the definition of lands subject to be inclosed under this act. In accordance with this act two commissioners, with a secretary, were appointed, under whose management its provisions have been extensively carried into operation. The commissioners make yearly reports to parliament of the proceedings which have taken place, and one or more acts are passed every year confirming the inclosures submitted by the commissioners for approval. In their 13th annual report, presented in 1858, the commissioners state that the number of applications of all kinds for inclosures since the passing of the acts had been 2351, and that the whole acreage of inclosures confirmed prior to this report was 226,010 acres, while the acreage of inclosures in progress amounted to 262,418 acres. The provisions of this act seem to be well adapted to remedy the evils that were stated in the evidence before the select committee; and there can be no doubt that agriculture has been greatly improved, the productiveness of the land increased, and employment given to labour by this judicious and important act of legislation. INCOMMENSURABLE, INCOMMENSURABLES,

THEORY

OF. The application of arithmetic to any science of concrete magnitude supposes a certain magnitude to be taken as unity, and all other magnitudes to be expressed by the number of times or parts of times which they contain this unit. Such an application, therefore, made in the usual manner, requires the assumption of this proposition, that all magnitudes are either fractions or multiples, or compounded of fractions and multiples, of any magnitude that may be named. This proposition is not true; for instance, we shall presently prove that if the side of a square be called 1, no number or fraction whatsoever will exactly represent the diagonal. But we shall also prove that it may be made as nearly true as we please: for instance, that we may find a line as nearly equal to the diagonal as we please, which shall be a definite arithmetical fraction of the side. Quantities which are so related that when one is capable of being represented in terms of a certain unit the other is not, are called incommensurables. The reason is as follows: any two whole numbers or fractions of the same unit must have a common measure; thus, all whole numbers have the common measure

have the common measure

1

which is contained exactly aq times in bq the first, and bp times in the second. Conversely, any two magnitudes which have a common measure can be arithmetically represented by the same unit for if A and B have the common measure M, and if this measure be contained 7 times in a and 10 times in B, then it is evident that by taking м as the unit, a is represented by 7 and B by 10. If, then, there be two magnitudes which cannot be represented by means of the same unit, they cannot have any common measure whatsoever, and are therefore incommensurable. It also follows from the preceding, that any two commensurable magnitudes must be to one another in the proportion of some one whole number to some other whole number.

To prove that there are such things as incommensurable magnitudes, we shall take the 117th (and last) proposition of the tenth book of Euclid, which demonstrates that the diagonal and the side of a square are incommensurable. Let D be the diagonal and s the side, and if they be not incommensurable let a and x be the whole numbers to which they are proportional; that is, let м be a common measure, and let D and s severally contain M, a and x times. Then the square on D will contain the square on м aa times; and the square on s will contain the square on M xx times. But the square on D is double of the square on s; therefore aa is twice xx. Now, let a and x have no whole common measure except unity, which may be supposed, for if they have a common measure, we may divide both by it, which will give two whole numbers in the same proportion, and so on until no common measure is left. Then, because a times a is double of x times x, a times a is an even number; whence a is an even number, for if a were odd, a times a would be odd. Therefore, x is not an even number, for if it were, a and z would have the common measure 2; whence x is an odd number. Let & be the half of a, which is a whole number,

since a is even; whence a = 2k, and aa = 4kk, which is also 2xx, and thence it follows that xx=2kk. Therefore, xx is an even number, and x also; for if x were odd, xx would be odd: whence x is even. But it was just now proved to be odd; so that the same number is both odd and even, which is absurd. This contradiction follows whenever we suppose s and D to be in the proportion of any two whole numbers; consequently, s and D are not in the proportion of any two whole numbers, and therefore are incommensurable, for if they were commensurable they would be in the proportion of some two whole numbers.

We have next to prove that any two magnitudes whatsoever, being incommensurable, may be made commensurable by as small an alteration

as we please in either. Let A and B be two incommensurable magnitudes, and let K be a third magnitude of the same kind, which may be as small as you please, provided only that it be given and known. [INDEFINITE.] Now, some aliquot part of a must be less than K; if not the hundredth, try the thousandth; if not the thousandth, try the millionth, and so on. Whatever K may be, it is possible to divide a into equal parts, each of which shall be less than K. Let м be such an aliquot part of A, and having divided A into its parts, set off parts equal to м along в. Then A and B being incommensurable, B will not contain M, the measure of a, an exact number of times, but will lie between two multiples of M, say BG and B L. From this it is obvious that B does not differ from either BG or BL by so much as GL, and therefore not by so much as K. But B G and BL are both commensurable with A, since all three are multiples of M. Here, then, are BG and B L, the first a little less than B, and the second a little greater, neither differing from B by so much as K, but both commensurable with A. Thus it is also evident that two whole numbers may be found which shall be as nearly as we please in the same ratio as two given incommensurable quantities.

The difficulty thus inherent in the application of arithmetic to concrete magnitude is not met with in practice, because no case can arise in which it is necessary to retain a magnitude so closely that no alteration, however small, can be permitted. But in exact reasoning, where any error, however small, to be avoided, it is obvious that the arithmetic of commensurable magnitudes, and the arithmetic (if there be such a thing) of incommensurable magnitudes, must not be confounded. The difficulty was overcome by Euclid, in the manner pointed out in the arti le PROPORTION, SO completely and effectually that nothing has been a lded to his solution of it except unsuccessful attempts to evade it. Those who avoid the fifth book of Euclid generally substitute either the tacit assumption that all magnitudes are commensurable, which is not true, or some play upon words, which a person who feels the rigour of Euclid places on the same shelf with nature's horror of a vac ium and other explanations of the same kind. We could even point out a celebrated work on geometry which

expressly rests on being able to make its errors too small to be per

ceived by the senses, and asks for no other reception of propositions which involve incommensurables.

The doctrine of incommensurable quantities was carried by Euclid to an extent which would excite as much admiration as any portion of his writings, if the tenth book were generally known and read as the production of a person unassisted by algebra. [IRRATIONAL QUANTITIES.]

INCOMPATIBLES, in Materia Medica, applied to those articles which are considered improper to be united in the same prescription. Strictly speaking, the term applies only to introducing into the same formula articles which exert a chemical action on each other, and so produce a result or compound of a useless or hurtful kind. Two or more articles may be introduced into a prescription, which by their combination neutralise the properties of each other. The resulting compound may be perfectly insoluble in the juices of the stomach, so as to be inert or hurtful by its insolubility. But this result is often sought on purpose to neutralise acrid or corrosive substances; such as when chalk or lime from a wall is given to a person who has swallowed oxalic acid. [ANTIDOTES.] But entire loss of power does not invariably result from combining substances which chemists deem incompatible; thus chalk and opium form a more powerful astringent than either singly given; and opium and acetate of lead in warm water form a fomentation of much use in erysipelas. But that utterly inert compounds often resulted from bringing together numerous ingredients is certain, as seen in the Polypharmacy of the ancients, of which the famous Mithridate is an instance. Even this has been surpassed in modern times, some prescriptions of Huxham containing above four hundred ingredients. On the opposite hand, excessive simplicity is perhaps too much aimed at in the present day. Dr. Paris' Pharmacologia' may be advantageously referred to; and Translation of the 'London Pharmacopoeia,' by Richard Philips.

INCONCINNOUS INTERVALS, in Music, are sounds which agree

with no scale, therefore are disagreeable to the ear, and never used in any kind of composition.

INCREMENT and DECREMENT. When two quantities are considered together, one of which is greater or less than the second, the latter is said to be the former with an increment or decrement. In the older English writings the calculus of differences is called the method of increments. This phraseology refers to the supposition of magnitudes being generated by continued increase or decrease as in the method of fluxions, so that two different magnitudes are spoken of as the same thing in different states, and of course at different times. Some difficulty to the beginner may be occasionally avoided by his stopping to interpret "let a become a +h" as follows: let us, having considered the value of a function of, proceed to consider the alteration which will arise if x+h be written instead of x." INCUBATION, ARTIFICIAL. [POULTRY.] INCUMBENT. [BENEFICE.]

66

INDEFINITE means "not given or defined in magnitude." Thus, a definite straight line is that of which the extremities are known; an indefinite straight line (in length) is one of which the direction is given, and which may be supposed to have any length, or which can be lengthened if necessary, without contravening any of the conditions of the problem. Thus, Euclid, in the first book, constructs an equilateral triangle upon a definite straight line, and shows how to draw two lines making with one another the same angle as that made by two given indefinite straight lines.

There is, however, a reprehensible use of the word indefinite, which is found in many mathematical works; namely, the employment of it to avoid the odium which attaches to the word infinite. Thus we hear of making a magnitude indefinitely great, of an indefinitely small arc being equal to its chord, of the circle being a polygon of an indefinitely great number of sides. In all these cases it would be better, with a proper definition, to use the word infinite at once.

A want of proper distinction between definite and indefinite sometimes leads to confusion. For instance, it is said that if a straight line be halved, if its half be then halved, and if fresh portions be continually taken, each of which is the half of the preceding, the result will at last become less than "any line which can be named." This is not true if the line which is to be named be indefinite; that is, if we may at any part of the process make it as small as we please; for it is obvious that whatever a line may be, a smaller line can be named. But it is true of a definite line, made definite, or given in length, at the beginning of the process: name any line, however small, but such as you name let it remain; then, by continually halving any other line, however great, you must at last arrive at a line which is less than the length you named. The phraseology of a line "less than any line which can be named has often caused a difficulty by not specifying the time at which it is to be named. The language used by Euclid himself is as follows (book x., prop. 1), and is free from the ambiguity in question: "Two unequal magnitudes being given, if from the greater be taken away its half, and from what is left its half, and if this be done continually, a magnitude will at last be found which is less than the lesser of the two given magnitudes."

INDELIBROME. [INDIGO.] INDENTURE. [DEED.] INDEPENDENTS, or CONGREGATIONALISTS, the name of a sect, class, or denomination of English Protestant Dissenters, one of the three who united form the Three Denominations, the other two being the Presbyterians and the Baptists.

When the principle of resistance to the power which maintained at least an outward and specious uniformity of Christian practice and opinion had received encouragement and was successful, it was not to be expected that nations who recognised that principle would agree among themselves respecting what should be done in their new condition of religious freedom. In England the politicians of the time soon succeeded in establishing a national church with pastors and bishops, and the church has been maintained in that form and order from the time of the Reformation, with the slight exception of the period of the Commonwealth. But there were many people in England who objected to several things which made a part of the constitution of that church; and as their objections consisted very much in the desire of what they considered a greater degree of purity in its forms, they were called in derision Puritans and Precisians, in which allusion was also included to the greater strictness with which they observed their religious duties, and their supposed peculiar preciseness in respect at once to an exactness of conformity to scripture precedent and to the obligations of a severe morality.

These persons were not all of one mind within themselves. Many uniting with these distinguishing characteristics the principle that, there being no scriptural authority for the Episcopal order, the government of the church or the superintendence of its ministers ought to be vested not in an individual, but in synods and presbyteries; these formed the Presbyterians. There were others who would have no union or government of the church, who regarded each congregation of faithful men as being in itself a church, and when properly constituted with deacons and a pastor forming a body which was independent of every other, and competent to its own direction and government with out any interference from presbyteries, bishops, or from the state itself; this is the pure principle of English Independency.

Robert Brown, a clergyman of the reign of Elizabeth, is generally reputed to be the first person in England who publicly avowed this opinion, and acted upon it by the establishment of various such separate churches, which however had no enduring existence. There is some question whether he retained his opinions to the last: but it is certain that after he had given no small trouble to the authorities in the church, he was presented to the living of Achurch in Northamptonshire. He closed a long and very troubled life in the jail at Northamp ton, or very soon after he had left it, in 1630. [BROWN, ROBERT, in BIOG. DIV.]

Other persons, and some of them of celebrity in the history of the Puritans, adopted the opinion, but were restrained from acting upon it by the laws then in force for maintaining the Church of England as then established. But when Episcopacy was abolished and Monarchy had been overcome, there was a large party of these Independents which suddenly presented itself, who had a great share in the struggle then being made, and who were the means of preventing the establishment of a Presbyterian church in England, which it was the object of by far the larger portion of the Puritan body taking part in the contest to form. Cromwell belonged to the Independents. Dr. John Owen, dean of Christ Church, who was also for a time vicechancellor of the University of Oxford, is considered as the chief ornament of this denomination at the time (the Cominonwealth) when it first became considerable. [OWEN, DR. JOHN, in BIOG. DIV.]

What the issue might have been of the struggle between the principle of Independency and the principle of Presbyterianism cannot now be told, the king being soon restored, and with him the Episcopal church. In 1662 the Act of Uniformity was passed, the object of it being to exclude from the ministerial office in the Church of England divines of either of those opinions. The act required a direct acknowledgment of the principle of Episcopacy. The effect of it was, that about 1900 ministers retired from the places they held in the church. Some make them 2000. These are the ministers whom Dissenters mean when they speak of "the illustrious two thousand," "the ejected ministers," or "the Bartholomew worthies." During the reign of Charles II. every effort was made to prevent these persons continuing to exercise their ministry. But it was all in vain. They, or at least the greater part of them, persisted in preaching, notwithstanding the certain penalties of imprisonment and fine. However, the Revolu tion of 1688 freed them from these penalties; one of the first acts of the new government being to grant toleration to them, that is, to allow them to open meeting-houses, or chapels, and to conduct the services under the protection of the law.

The Independents were inconsiderable at that time as compared with the Presbyterians. Both however (and the Baptists also) built chapels for themselves and formed themselves into congregations, called the Presbyterian congregations and the Independent congregations; and each denomination had its own board or fund.

The Act of Toleration' was passed in 1689, and for the seventy years succeeding that date the Independent denomination dwindled (as indeed did the whole body of Dissenters), and it was in a very low condition when the state of things arose which we have now to describe.

About the middle of the 18th century there was an extraordinary revival of religious zeal under the influence created especially by the Wesleys and Whitefield. The Dissenters, like the Church, had adopted pretty generally the principle that to inculcate the moral duties, to present the paternal government of God as a source of consolation and of hope, to hold out the prospect of future accountableness and of eternal life, to show the evidence on which we receive Jesus Christ as the minister and messenger of his heavenly Father, were the principal subjects on which it was the duty of Christian ministers to insist. This it was easy to represent as an abandonment of the distinctive truths, as they are sometimes regarded, of Christianity; and many persons, under the preaching above alluded to, were disposed so to regard it, and to seek a ministry by whom these distinctive truths would be made more prominent. Most of these persons joined themselves to the Wesleyan Methodists, or to the Whitefieldian Methodists (since better known as the Countess of Huntingdon's Connexion); but there were many who declined to unite themselves with any of these bodies, and formed themselves into separate churches upon the Independent principle. These new societies incorporating with themselves the small remains of the old Independents of England,-who, in some instances had, throughout the period by some called the period of Religious Indifference, adhered to the original opinions of the Puritan body at large, which were Calvinistic, and had continued to make those opinions prominent in the public services,- -or joining themselves to such decayed and decaying churches, gradually increased in numbers and influence, and constitute at the present day the large body of Dissenters called Independents or Congregationalists.

From the accession of George I., in 1714, when the London dissenting ministers of the three denominations (Presbyterian, Independent, and Baptist) presented an address to the king [WILLIAMS, DANIEL, D.D., in BIOG. DIV.], the three bodies have been accustomed to act together, by their appointed deputies, in reference to great public questions. Most of the old Presbyterian denomination in England having in the course of time adopted Unitarian sentiments, their repre sentatives at length withdrew from the board. The distinctive appellation of "The Three Denominations" is however still kept up, the