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imagine so many parallelograms to be erected thereon, either circumscribing the curvilineal figure, or inscribed in it; then the finding the sum of all these parallelograms is the business of the method of increments. But if the parts of the abscissa be taken infinitely sinall, then these arallelograms degenerate into the curve; and then it is the business of the method of fluxions to find the sum of all, or the area of the curve. So that the method of increments finds the sum of any number of finite quantities; and the method of fluxions the sum of any infinite number of infinitely small ones: and this is the essential difference between these two methods.” Again: “There is such a near relation between the method of fluxions and that of increments, that many of the rules for the one, with little variation, serve also for the other. And here, as in the method of fluxions, some questions may be solved, and the integrals found, in finite terms; whilst in others we are forced to have recourse to infinite series for a solution. And the like difficulties will occur in the method of increments, as usually happen in fluxions. For whilst some fluxionary quantities have no fluents but what are expressed by series, so some increments have no integrals but what infinite series afford; which will often, as in fluxions, diverge and become useless.” By means of the method of increments, many curious and useful problems are easily resolved, which scarcely admit of a solution in any other way. As, suppose several series of quantities be given, whose terms are all formed according to some certain law which is given; the method of increments will find out a general series, which comprehends all particular cases, and from which all of that kind may be found. The method of increments is also of great use in finding any term of a series proposed: for the law being given by which the terms are formed, by means of this general law the method of increments will help us to this term, either expressed in finite quantities, or by an infinite series. Another use of the method of increments is to find the sum of series, which it will often do in finite terms. And when the sum of a series cannot be had in finite terms, we must have recourse to infinite series; for the integral being expressed by such a series, the sum of a competent number of its terms will give the sum of the series required. This is cquivalent to transforming one series into another, converging quicker; and some.
times a very few terms of this series will give the sum of the series sought. Set Emerson's Increments. INCUBUS, or nught mare, in medicine, the name of a disease, which consists in a spasmodic contraction of the muscles of the breast, usually happening in the night, and attended with a very painful difficulty of respiration and great anxiety. INCUMBENT, a clerk diligently resi. dent on his benefice with cure; and called incumbent of that church, because he does or ought to apply himself sedulously to discharge the duty of his cure. INCURVATION of the rays of light, their bending out of a rectilinear or straight course, occasioned by refraction INDEFINITE, or INDETERMINATE, that which has no certain bounds; or to which the human mind cannot affix any. Des Cartes makes use of this word in his phi. losophy instead of infinite, both in numbers and quantities, to signify an inconceivable number, or a number so great that an unit cannot be added to it; and a quantity so great as not to be capable of any addition. Thus, he says, the stars visible and invisible are in number indefinite; and not as the ancients held infinite ; and that quantity may be divided into an indefinite number of parts, not an infinite number. INDEfl NITE is also used, in the schools, to signify a thing that has but one extreme ; for instance, a line drawn from any point and extended infinitely. Indefinite, in grammar, is understood of nouns, pronouns, verbs, participles, articles, &c. which are left in an uncertain indeterminate sense, and not fixed to any particular time, thing, or other circumstance. INDENTED, in heraldry, is when the out-line of an ordinary is notched like the teeth of a saw. INDENTED line, in fortification, the same with what the French engineers call redent; being a trench and parapet running out and in, like the teeth of a saw and is much used in irregular fortification. - INDENTURE, is a writing, containing a conveyance between two or more, indented or cut unevenly, or in and out, on the top or side, answerable to another writing that likewise comprehends the same words. Formerly, when deeds were more concise than at present, it was usual to write both parts on the same piece of parchment, with some words or letters written between them, through which the parchment was cut, either in a straight of
indented line, in such a manner as to leave half the word on one part, and half on the other; and this custom is still preserved in making out the indentures of a fine. But at last, indenting only hath come into use, without cutting through any letters at all, and it seems at present to serve for little other purpose than to give name to the species of the deed. INDEPENDENTS, or CoNGREGATIONAlists, in church history, a sect of Protestant Dissenters, which first made its appearance in Holland in the year 1616. Mr. John Robinson appears to have been the founder of this sect. The appellation of Independents was applied to, and adopted by, this denomination of Christians, from their maintaining that all Christian congregations are so many independent religious societies, having a right to be governed by their own laws, without being subject to any further or foreign jurisdiction. This term was publicly acknowledged in the year 1644, by those English Dissenters, who held similar sentiments respecting church government to the Independents in Holland; but on account of the ill use that many made of the term, by a perversion of its original meaning and religious designation, the English Independents renounced it, and adopted that of Congregationalists, or Congregational brethren. The term Independent is still, however, applied to various sects of Protestant Dissenters, and seems justly applicable to almost every sect of nonconformists in this country. The doctrines of the Independents are the same as those of the Bhow Nists. It is said, that the only difference between these sects was, that the Brownists were illiberal in their views concerning other denominations, while the Independents entertainca enlarged conceptions of church communion, and allowed that other churches, though different from them in points of discipline, might properly be called Christian churches. It is, however, to be feared that the Independents, properly so called, being Calvinists as to points of faith, do not cherish very liberal sentiments concerning the salvation of those who differ from them in most of their articles of belief. A spirit which seems to be a natural effect of the creed of the Geneva Reformer. See BaowN 1sts and PREs BYTERIANs. INDETERMINATE, in general, an apo given to whatever is not certain, xed, and limited; in which sense it is the same with indefinite. Ixpetraxtix. AtE problem, is that which
admits of many different solutions and auswers, called also an unlimited problem.
In questions of this kind, the number of unknown quantities concerned is greater than the number of the conditions and equations by which they are to be found; from which it happens, that generally some other conditions or quantities are assumed, to supply the defect, which, being taken at pleasure, give the same number of answers as varieties in those assumptions. If, for instance, it were required to find the value of two square numbers, whose difference is equal to a, a given quantity. Here if x* and y denote the squares, then x*—y”—a, which is only one equation for finding two quantities. Now, by assuming some other unknown quantity, as z, so that z=r-Hy=
the sum of the roots; then is ar
are the two roots having the difference of their squares equal to a given quantity a, and are expressed by means of an assumed quantity z; so that there will be as many answers to the question, as there can be taken values of the indeterminate quantity z. Mr. Leslie, in the transactions of the Royal Society of Edinburgh, has given a aper on this subject, the object of which is to resolve the complicated expressions which we obtain in the solution of indeterminate problems into simple equations, and this is done by means of a principle, which, though extremely simple, admits of a very extensive application. Let Ax B be any compound quantity equal to another, CxD, and let m be any rational number assumed at pleasure; it is manifest that, taking equimultiples, Axm B=CXm D. If, therefore, we suppose that A = m D, it must follow that
of a lower dimension are obtained. If these be capable of further decomposition, we may assume the multiples n and p, and form four equations still more simple. By the repeated application of this principle, an higher equation, admitting of divisors, will be resolved into those of the first order, the number of which will be one greater than that of the multiples as:
arbitrary, and if rational, * and y must also be rational. Hence the resolution of these two equations gives the values of r and y, the numbers sought, in terms of m, moa-Hb d moa-b 2m * andy--5. INDEX, in anatomy, the same with the fore finger. See FING ens. INDEx, in arithmetic and algebra, shows to what power any quantity is involved, and is otherwise called exponent. See Expo NENT. Innex of a logarithm, that which shows of how many places the absolute number, belonging to a logarithm, doth consist; and of what nature it is, whether an inteer or fraction. Thus, in this logarithm, .523421, the number 2 standing on the
left hand of the point is called the index;
because it shows that the absolute number, answering to the above logarithm, consists of three places: for the number is always one more than the index. If the absolute number be a fraction, then the index of the logarithm hath a negative sign, marked thus, 2.523421. See LoGARITHM.
INDEx of a globe, the little style or gnomon, which being fixed on the pole of the globe, and turning round with it, points out the hours upon the hour circle. See GLoBE.
INDIA rubber. See CAoutchouc.
INDICATIVE, in grammar, the first mood, or manner, of conjugating a verb, by which we simply affirm, deny, or ask something.
INDICTION, in chronology, a cycle of fifteen years. See CHRoxology.
INDICTMENT, is a written accusation of one or more persons, of a crime or misdemeanor, preferred to, and presented on oath by, a grand jury. An indictment may be found on the oath of one witness only, unless it be for high treason, which requires two witnesses; and unless in any instance it is otherwise specially directed by acts of parliament. The sheriff of every county is bound to return to every session of the peace, and every
commission of oyer and terminer, and of general gaol delivery, twenty-four good and lawful men of the county, some out of every hundred, to enquire, present, do, and execute all those things, which, on the part of our lord the King, shall then and there be commanded therein. As many as appear upon this pannel are sworn of the grand jury, to the amount of twelve, at the least, and not more than twenty-three, that twelve may be a majority. This grand jury is previously instructed in the articles of their enquiry, by a charge from the judge on the bench. They then withdraw from court, to sit and receive indictments, which are preferred to them in the name of the King, but at the suit of any private prosecutor; and they are only to hear evidence on behalf of the prosecution; for the finding an indictment is only in the nature of an enquiry or accusation, which is afterwards to be tried and determined; and the grand jury are only to enquire, upon their oaths, whether there be sufficient cause to call upon the party to answer it. The grand jury may not find part of an indictment true, and part false; but must either find a true bill, or ignoramus, for the whole; and if they take upon them to find it specially, or conditionally, or to be true for part only, and not for the rest, the whole is void, and the party cannot be tried upon it, but ought to be indicted anew. - * All capital crimes whatsoever, and all kinds of inferior crimes, which are of a public nature, as misprisions, contempts, disturbances of the peace, oppressions, and all other misdemeanours whatsoever, of a public evil example, against the common law, may be indicted, but no injuries of a private nature, unless they in some degree concern the King. And generally, where a statute prohibits a matter of public grievance to the liberties and security of a subject, or commands a matter of public convenience, as the repairing of the common streets of the town, &c., every disobedience of such statute is punishable, not only at the suit of the party #. but also by way of indictment, or contempt of the statute, unless such method of proceeding shall manifestly appear to be excluded by it. Yet, if the party offending have been fined in an action brought by the party (as it is said he may in every action, for doing a thing prohibited by statute,) such fine is a good bar to the indictment, because by the fine the end of the statute is satisfied; otherwise he would be liable to a second fine for the same offence.
If several offenders commit the same offence, though, in law, they are several offences in relation to the several offendders, yet they may be joined in one indictment; as if several commit a robbery or murder. No indictment for high treason, or misprision thereof, (except indictments for counterfeiting the King's coin, seal, sign, or signet.) nor any process or return thereupon, shall be quashed for mis-reciting, mis-spelling, false or improper Latin, unless exception concerning the same be taken, and made in the respective court where the trial shall be, by the prisoner or his council assigned, before any evidence given in open court on such indictment; nor shall any such mis-reciting, mis-spelling, false or improper Latin, after conviction on such indictment, be any cause of stay, or arrest of judgment; but nevertheless any judgment on such indictment shall be liable to be reversed on writ of error, as formerly. An indictment accusing a man in general terms, without ascertaining the particular fact laid to his charge, is insufficient; for no one can know what defence to make to a charge which is uncertain, nor can plead it in bar on abatement of a subsequent prosecution: neither can it appear, that the facts given in evidence against a defendant, on such a general accusation, are the same of which the indictors have accused him; nor can it judicially appear to the court what punishment is proper for an offence so loosely expressed. No indictment can be good, without expressly showing some place wherein the offence was committed, which must appear to have been within the jurisdiction of the court. * There are several emphatical words of art, which the law has appropriated for the description of an offence which no circumlocution will supply ; as feloniously, in the indictment of any felony; burglariously, in an indictment of burglary, and the like. And an indictment on the black act, for shooting at any persou, must charge that the offence was done wilfully and maliciously By 10 and 11 William, c. 23, it is enacted that no clerk of assize, clerk of the peace, or other person, shall take any money of any person, bound over to give evidence against a traitor or felon, for the discharge of his recognizance, nor take more than two shillings for drawing any bill of indictment against any such felon, or pain of five pounds to the party, grieved, with full costs. a defective bill, he shall draw a new one VOL. VI.
And if he shall draw.
gratis, on the like penalty. With respect to drawing indictments for other misdemeanors, not being treason or felony, no fee is limited by the statute, the same therefore depends on the custom and ancient usage. Every person charged with any felony or other crime, who shall on his trial be acquitted, or against whom no indictment shall be found by the grand jury, or who shall be discharged by proclamation for want of prosecution, shall be immediately set at large in open court, without payment of any fee to the sheriff or gaoler; but in lieu thereof, the treasurer, on a certificate signed by one of the judges or justices, before whom such prisoner shall have been discharged, shall pay, out of the general rate of the county or district, such sum as has been usually É. not exceeding thirteen shillings and our-pence. By these words, immediately set at large, the reader must not understand that this actually takes place immediately after the throwing out of the bill. It is usually done after the assizes or sessions are over, and when the judge or justices proceed to the gaol-delivery, as it is called. This affords an opportunity for the preferring a new indictment against the party, if there should be occasion; and it is upon this ground, that the detention of a prisoner, after rejecting the indictment by the grand jury, is countenanced. It is, however, in many cases, a hardship. The sheriffs of London, in A D. 1808, Sir Richard Phillips, Knt. and Mr. Alderman Smith, very much to their credit, endeavoured to procure the judges at the Old Bailey to discharge prisoners immediately, but the practice having long continued as above stated, the judges have been averse to altering it. Upon a certificate of an indictment being found, for an assult or other misdemeanor, and much more for a felony, at the sessions, a warrant is issued, on the application of the prosecutor, to take the party into custody, and he may be held to bail by a justice of the peace, or a judge; and it is usual, in expectation of such a warrant, to put in bail, and obtain a supersedeas to the warrant previously. This was not formerly the practice, upon indictments or informations in the court of King's Bench An act has passed to enable the court to issue warrants, and hold to bail, upon indictments or informations filed. This act is principally objectionable, as it may be used as the means of harassing persons, prosecuted harshly and vindictively by the Attorney General K k
for libels, &c. It is either a useless act, since the justice of the country has been safely conducted for centuries without it, or it is an act of great importance to the liberty of the subject. But an action cannot be brought by the person acquitted against the prosecutor of the indictment, without obtaining a copy of the record of his indictment and acquittal; which, in prosecutions for felony, it is not usual to grant, if there be the least probable cause to found such prosecution upon; for it would be a very great discouragement to the public justice of the kingdom, if prosecutors, who had a tolerable ground of suspicion, were liable to be sued at law whenever their indictments miscarried. But an action on the case, for a malicious prosecution, may be founded on such an indictment whereon no acquittal can be, as, if it be rejected by the grand jury, or be coram non judice, or be insufficiently drawn; for it is not the danger of the plaintiff, but the scandal, vexation, and expense, upon which this action is founded. However, any probable cause for preferring it is sufficient to justify the defendant, provided it do not appear that the prosecution was malicious. And it is necessary to show something more than the mere not prosecutis, in order to raise the inference of malice. INDIGESTION. See MEDICINE. INDIGO, a dye prepared from the leaves and small branches of the indigofera tinctoria. See the next article. Indigo is distinguished into two kinds, the true and the bastard. Though the first is sold at a higher price on account of its superiority, it is usually advantageous to cultivate the other, because it is heavier. The first will grow in many different soils; the second suceeds best in those which are most exposed to the rain. Both are liable to great accidents. Sometimes the plant becomes dry, and is destroyed by an insect frequently found on it; at other times, the leaves, which are the valuable part of the plant, are devoured in the space of twenty-four hours by caterpillars. This last misfortune, which is but too common, has given occasion to the saying, “that the planters of indigo go to bed rich, and rise in the morning totally ruined.” This production ought to be gathered in with great precaution, for fear of making the farina that lies on the leaves, and is very valuable, fall off by shaking it. When gathered, it is thrown into the steeping-vat, which is a large tub filled with water. Here it un
dergoes a fermentation, which in twenty. four hours at furthest is completed. A cock is then turned to let the water run into the second tub, called the mortar or pounding tub. The steeping-vat is then cleaned out, that fresh plants may be thrown in ; and thus the work is continued without interruption. The water which has run into the pounding tub is found impregnated with a very subtle earth, which alone constitutes the dregs, or blue substance, that is the object of this process, and which must be separated from the useless salt of the plant, because this makes the dregs swim on the surface. To effect this, the water is for. cibly agitated with wooden buckets that are full of holes, and fixed to a long handle. This part of the process requires the greatest precautions. If the agitation be discontinued too soon, the part that is used in dyeing, not being suffi. ciently separated from the salt, would be lost. If, on the other hand, the dye were to be agitated too long after the complete separation, the parts would be brought together again, and form a new combination; and the salt reacting on the dregs would excite a second fermen. tation, that would alter the dye, spoil its colour, and make what is called burnt indigo. These accidents are prevented by a close attention to the least alterations that the dye undergoes, and by the precaution which the workmen take to draw out a little of it from time to time in a clean vessel. When they perceive that the coloured particles collect by se. .#. the rest of the liquor, they eave off shaking the buckets, in order to allow time to the blue dregs to precipi. tate to the bottom of the tub, where they are left to settle till the water is quite clear. Holes made in the tub, at different heights, are then opened one after another, and this useless water is let out. The blue dregs remaining at the bottom having acquired the consistence of a thick muddy liquid, cocks are then opened, which draw it off into the settler. After it is still more cleared of much superfluous water in this third and last tub, it is drained into sacks; from whence, when water no longer filters through the cloth, this matter now becomes of a thicker consistence, and is put into chests, where it entirely loses its moisture. At the end of three months the indigo is fit for sale. It is used, inwashing, to give a bluish colour to linen: painters also employ it in their water colours; and dyers cannot make fine blue without indigo. The an