versary's ball, otherwise the player loses a point. This is a great advantage, and is reckoned between even players to be equal to reeeiving about eight or nine points. Carambole is a game newly introduced from France. It is played with three balls, one being red, which is neutral, and is placed upon a spot on a line with the stringing nail (i. e. that part of the table whence the player strikes his ball at first setting off, and which is generally marked with two brass nails). Each antagonist at the first stroke of a hazard, plays from a mark which is upon a line with it at the other end of the table. The chief object at this game is, for the player to hit with his own ball the two other balls: which is called a carambole, and by which the player wins two. If he put in the red ball he gets three, and when he holes his adversary's ball he gets two; so that seven may be made at one stroke, by caramboling and putting in both balls. This game resembles the losing, depending chiefly upon particular strengths, and is generally played with the cue. The game is sixteen up; nevertheless it is reckoned to be sooner over than the common game. The next object of this game, after making what we have distinguished by the carambole, is the baulk; that is, making the white ball, and bringing the player's own ball and the red one below the stringing nail, from whence the adversaries begin. By this mean the oppoment is obliged to play bricole from the opposite cushion; and it often happens that the game is determined by this situation. that when this game is played against the common game, the advantage for the latter, between equal players, is reckoned to be about six. The player at the one-hole, though it seems to those who are not judges of the game to be a great disadvantage, has in fact the best of it; for as all balls that go into the one hole reckon, the player endeavours to lay his ball constantly before that hole, and his antagonist frequently finds it very difficult to keep one or other ball out, particularly on the leads, when the oneplayer lays his ball (which he does as often as he can) on the brink of the hole; leading for that purpose from the opposite end, which in reality he has no right to do; for the lead should be given from the end of the table at which the hazard is made; but when a person happens to be a novice, this advantage is often taken. The four-game, consists of two partners on each side, as the common winning game; who play by succession after each hazard, or two points lost. The game is fifteen up; so that the point or hazard is an odd number, which makes a miss at this game of more consequence than it is at another; being as much at four, six, or eight, as it is at five, seven, or nine, at the single game. Hazards are so called because they depend entirely upon the making of hazards, there being no account kept of any game. Any number of persons may play by having balls that are numbered: but the number seldom exceeds six, to avoid confusion. The person whose ball is put in, pays so much to the player according to what is agreed to be played for each hazard; and the person who misses, pays half the price of a hazard to him whose ball he played at. The only general rule is, any ball a hazard for the next player, which may be in a great measure avoided, by always playing upon the next player, and either bringing him close to the cushion, or putting him at a distance from the rest of the balls. The table, when hazards are played, is always paid for by the hour. The Russian Carambole is a game that has still more lately been introduced from abroad, and is played in the following manner the red ball is placed as usual on the spot made for but the player, when he begins, not to lay purpose; or after having been holed, never places his ball on any particular place or spot; he being at liberty to put it where he pleases. When he begins to play, instead of striking at the red ball, he leads his own gently behind it, and his antagonist is to play at which he thinks proper; if he plays at the red ball and holes it, he scores three as usual towards the game, which is twenty-four instead of sixteen points; and the red ball is put upon the spot again: at which he may strike again, or take his choice which of the two balls to push at, always following his stroke till both balls are off the table. He is entitled to two points each time that he caramboles, the same as at the other game; but if he carambole and put his own ball into any hole, he loses as many as he might have got had he not holed himself, for example, if he strike at the red ball, which he holes, at the same time caramboles and holes himself, he loses five points; and if he hole both balls when he caramboles, and likewise his own, he loses seven, which he would have got if he had not holed his own ball. In other respects it is played like the common carambole game. The bar-hole is so called from the hole being barred which the ball should be played for, and the player striking for another hole; Fortification billiards consists of: first, forts made of wood, in the form of castles, which are to have lead put in them for the purpose of making them heavy, so that in playing the balls may not be moved from their places. In the front of each fort, at the bottom, is an arch, full wide and high enough to admit the ball, which is to be put through it to attack the fort. Within the arch of each fort a small bell is hung, which must be made to ring by the adversary's attacking ball, otherwise the fort cannot be taken. Secondly, the pass through which each of the adversary's attacking balls must enter, before a fort can be taken. Lastly, the grand batteries, and ten flags or colours. Two of the forts, called the grand forts, are made larger than the rest, and have an arch cut through them of the size the others have. Five of the forts, including one of the grand forts, one of the batteries, and five of the flags or colours are usually painted red, and the forts and battery are pointed like brick-work, which colour denotes them to be English; on each fort one red colour is to be hoisted on the centre of the front thereof. The other five forts, grand fort included, battery and colours, are a white; the forts and battery pointed with black-like stone, are called French, one white colour to be hoisted on each as before mentioned. The pass, which serves for the purpose of both parties' attacking balls to go through, is made in the form of the grand forts, but rather longer for distinction, has an arch of the size of the grand forts, and is painted of different colours; viz. one of the ends where the arch is, red, continuing half way on each side, and the same ame on the top; the other end of the arch white, and continuing in the same colour over the other half as before. Two colours are likewise hoisted on the pass, viz. one red and the other white; the red boisted at the English end, and the white at the French end. The pass is placed in the centre of the table, the red end to face the English forts, and the white end the French forts. The limits of each party's quarter, is from the end cushion, where his forts are placed, to his pass on each side of the table. The red or English forts possess one end of the table, and this is called the English quar ter. The white or French forts possess the other end of the table, and this is called the French quarter. The two forts in each quarter in the first acle from the pass are to be taken first, which pre therefore called the advanced forts. The two forts in the second angle are to be taken next, which are called the reserved forts. Lastly, the grand fort, with the battery placed before the same, is the last to be taken. The height of the advanced and the reserved forts is five inches and a half, the breadth and kngth of the advanced forts five inches to the quare, and the length of the reserved forts five inches and a half; and the back of them to be rounded off. The height of the grand forts is five inches and a half, the breadth and length six inches and a quarter. The batteries are made in a triangular form, the height of them three inches, the breadth at the extremity two inches and a half, and the length three inches and a half. The height of the pass is five inches and a half, the breadth six inches and a quarter, and the length seven inches. The height of the concave in the forts where the attacking ball must enter is three inches, the breadth two inches and a half, the depth two inches and three quarters. The bell within the arch in each fort must be hung one inch and a half within it. The balls to be played with at this game are one inch and three eighths diameter. BILLINGSGATE, or BILINSGATE, between London-bridge and the Custom-house, in Tower-street, is a great fish-market, the largest in England, and harbour for small ves, sels laden with fish. The sale is regulated by act of parliament, and it was made a free port, in 1699, for their sale every day in the week but Sunday. The gross and profane language frequently used in this market, is now com monly distinguished by the name of the place itself: so that when a person descends to low ribaldry and profanity, it is said such a one is versed in the dialect of Billingsgate. See Bi LINGSGATE. BILLON, in the history of coins, a composition of precious and base metals, where the latter predominate. Wherefore gold under twelve carats fine, is called billon of gold; and silver under six pennyweights, billon of silver. BI'LLOW. 3. (bilge, German.) A wayẹ swoln, and hollow (Denham), To BI'LLOW. v. n. (from the noun.) To swell, or roll, as a wave (Frior). BILLOWY. a. Swelling; turgid (Thom son). "BILOBATE LEAF. In botany, divided into two lobes. See LOBE and LOBATE. BILOCULAR PERICARP, in botany, more properly two-celled; divided into two cells internally; as in hyoscyamus, sinapis, nicotiana, &c. Some seeds are also two-celled, as in cornus, xanthium, valeriana locusta, cordia. BILSDON, a small town in Leicestershire, with a market on Friday. Lat. 52. 36 N, Lon. 0.51 W. BILSON (Thomas), a learned English pres late. He was born at Winchester, and educated at the school there, from whence he removed to New college, Oxford, where he proceeded to the degree of D. D. His first preferment was the mastership of Winchester school, and afterwards he was made warden of Winchester college. In 1585 he published a timely treatise of the true difference between christian subjection and unchristian rebellion, dedicated to queen Elizabeth; and in 1593 he printed another on the perpetual Government of Christ's Church, which procured him the bishopric of Worcester, in 1596; from whence he was shortly after removed to the see of Winchester, with the rank of privy counsellor. In 1604 he published a famous book, on Christ's Descent into Hell; and the same year he was appointed one of the disputants at the Hampton-court conference. He had also a chief hand in the present translation of the Bible; and died in 1616. His remains were deposited in Westminster-abbey, BILSTON, a village in Staffordshire, two miles SE. of Wolverhampton. It has a navigable canal, communicating with the Staffordshire and Worcestershire canals, and several great rivers. Near it are large mines of coal, iron-stone, &c. Here are furnaces for smelting iron-ore, forges, and slitting-mills; also, manufactures of japanned and enamelled goods, and buckle chapes, This place has much in, creased of late years, and now contains nearly 7000 inhabitants. BIMEDIAL LINE, in mathematics, is the sum of two medials. Euclid reckons two of these bimedials, in pr. 38 and 39, lib. x.; the first is when the rectangle is rational, which is contained by the two medial lines whose sum makes the bimedial; and the second when that rectangle is a medial, or contained under two lines that are commensurable only in power. BIMINI, one of the Bahama islands, near the channel of Bahama; it is about thirty-six miles in circuit. On account of the shoals, it is difficult of access. Lat. 25. N. Lon. 79. 30 W. BIN. s. (binne, Saxon.) A place where bread, corn, or wine is reposited (Swift). BINACLE. See BITTACLE. BINAROS, a town of Spain, in Valencia, remarkable for good wine. Lat. 40. 33 N. Lon. 0. 35 E. BINARY NUMBER, that which is composed of two units. BINARY ARITHMETIC, that in which two figures or characters, viz. 1 and 0, only are used; the cipher multiplying every thing by 2, as in the common arithmetic by ten: thus, 1 is one, 10 is 2, 11 is 3, 100 is 4, 101 is 5, 110 is 6, 111 is 7, 1000 is 8, 1001 is 9, 1010 is 10; being founded on the same principles as common arithmetic.-This sort of arithmetic was invented by Leibnitz, who pretended that it is better adapted than the common arithmetic for discovering certain properties of numbers, and for constructing tables; but he does not venture to recommend it, for ordinary use, on account of the great number of places of figures requisite to express all numbers, even very small ones. BINARY MEASURE, in music, is that which is beaten equally. See BEATING TIME and TIME. BINATE LEAF. (binatum folium.) In botany, digitatum foliolis duobus terminatum. Having a simple petiole connecting two leaflets at the top of it: a species of DIGITATE LEAF, which see. Biuated peduncles, peduncles growing in pairs; as in capraria, and oldenlandia zeylanica. BINBROKE, a a market on Wednesday. Lon. 0. 0. a town in Lincolnshire, with Lat. 53. 30 N. BINCHESTER, a village on the river Were, near Durham. It has a place in this Dictionary, because it appears to have been the Roman Vinovium. 1. To BIND. v. a. preterit bound; participle pass. bound or bounden. (binean, Saxon.) To confine with bonds; to enchain (Job). 2. To gird; to inwrap (Proverbs). 3. To fasten to any thing (Joshua). 4. To fasten together (Matthew). 5. To cover a wound with dressings and bandages (Wiseman). 6. To oblige by stipulation, or oath (Pope). 7. To compel; to constrain (Watts). 8. To confine; io hinder (Shakspeare). 9. To make costive (Bacon). 10. To restrain (Felton). 11. To bind to. To oblige to serve some one (Drydea). 12. To bind over. To oblige to make appearance (Addison). To BIND. r. n. 1. To contract; to grow stiff (Mortimer). 2. To be obligatory (Locke). BIND. S. A species of hop (Mortimer). BINDER. s. (from to bind.) 1. A man whose trade it is to bind books. 2. A man that binds sheaves (Chapman). 3. A fillet; a shred cut to bind with (Wiseman). BINDING. s. (from bind.) A bandage (Tatler). BINDING, in the art of defence, a method of securing or crossing the adversary's sword with a pressure, accompanied with a spring from the wrist. The great objection made by some people, particularly those time-catchers, against the frequent use of binding, is, that when a man, in performing it, cleaves too much to his adversary's sword, he is liable to his adversary's slipping of him, and consequently of receiving either a plain thrust, or one from a feint. BINDING is a term in falconry, which implies tiring, or when a hawk seizes. BINDING JOISTS, in architecture, those into which the trimmers of staircases, &e. are framed. BINDING-NOTES, in music, imply two or more notes on the same line or the same space, separated by a single bar, but linked by a semicircle, and which though written or printed twice, are not to be separated, but sustained like a single sound. See SYNCOPATION. BINDWEED. In botany. See CONVOL VULUS. BINDWEED (Black). See TOMUS. BINDWEED (Rough). See SMILAX. BINGALLE. In medicine. Sce CASSU MUNIAR. BINGEN, an ancient town of Germany, in the archbishopric of Mentz. Lat. 49. 49 N. Lon. 8. 0 E. BINGHAM, a town in Nottinghamshire, with a market on Thursday. Lat. 52. 58 N. Lon. 0. 51 W. In BINGHAM (Joseph), an English divine of great erudition. He was born at Wakefield, in Yorkshire, in 1663, and admitted of University college, Oxford, in 1684, where he took his degrees in arts and was elected fellow. 1695, he preached a sermon before the univer sity on the doctrine of the Trinity, which some of the auditors conceived not to be strictly or thodox, and in consequence he found it expedient to resign his fellowship. About this time he was presented to the rectory of Headbourne-worthy, near Winchester, and soon after vindicated himself fully in a visitation sermon. In 1708, he published the first volume of his Origines Ecclesiastica, which he completed in 1722, the whole making 10 vols. 8vo. In 1712 bishop Trelawney gave him the rectory of Havant, near Portsmouth. He died in 1723, and was buried in the churchyard of Headbourne-worthy. He left a widow and six children, two sons and four daughters, but scantily provided for. BINN. See BIN. BINOCLE, or BINOCULAR TELESCOPE, is one by which an object is viewed with both eyes at the same time. It consists of two tibes, each furnished with glasses of the same power, by which means it has been said to shew objects larger and more clearly than a Binomial Line, or Surd, is that in which at least one of the parts is a surd. Euclid enumerates six hands of binomial lines or surds, in the 10th book of his Elements, which are exactly similar to the six residuals or apotomes there treated of also, and of which mention is made under the article APOTOME. Those apotomes, however, become binomials by only changing the sign of the latter term trom minus to plus, which therefore are as below. Euclid's 6 Binomial Lines. First binomial 3 + √5, 2d binomial 18+4, 3d binomial √/24+ √18, 4th binomial 4+ √3, 5th binomial√/6 +2, 6th binomial √6 + √2, To extract the Square Root of a Binomial, as of a+b, or No+ Nb. Various rules have been given for this purpose. The first is that of Lucas de Burgo, in Summa de Arith. &c.; which is this: When one part, as a, is rational, divide it into two parts such, that their product may be equal to 4th of the number under the radical b; then shall the sum of the roots of those parts be the root of the binomial sought: or their difference is the root when the quantity is residual. That is, if &+e=a, and ce=46; then i √c+ √e=√a+√ the root sought. As if the bromial be 23+448; then the parts of 23 are 16 and 7, and their product is 112, which is 4th of 448; therefore the sum of their roots 4+7 is the root sought of 23+/448. De Burgo gives also another rule for the same extractions, which is this: The given binomial being, for example, √+ √b, its root will be ANO+ANG - 6+ √ & N-b.-So in the foregoing example, 23+/448, › here √√c=23, and √√√448; hence 11, and c-b √3. √3 = 3+1 the cube root, the same as before. Dr. Wallis's rule for the cube root of a binomial a± mbor am-b. In these forms the greatest rational part m is extracted out of the radical part, leaving only b the least radical part possible under the radical sign. He then observes that if the given quantity have a binomial root, it must be of this form nb, with the same radical b. Then to find the value of c and n, he raises this root to the 3d power, which gives 3 + 3cn2b±3c2n + n3b. /b, which must be amb the given quantity; hence putting the rational part of the one quantity equal to that of the other, and also the radical part of the one equal to that of the other, gives c3+3cn2b=a, and 3c2n+n3b=m. Then assuming several values of 1, from the last equation he finds substituted in the first equation, make it just, they the value of c; hence if these values of c and n, are right; but if not, another value of n is assumed, and so on, till the first equation hold true. teger or the half of an integer, For example, if the And it is to be noted that n is always either an incube root of 135/1825 be required, or 135 ± 73; here a=135,m=78, and 6=3; hence 32+ suming n=1, this last equation becomes c2+1= n3b=m is 3c2n + 3n3 = 78, or c2+n3 26; then as26, from whence c is found = 5; which values of e and a being substituted in the first equation c3 + 3cnba, makes 53+3.5.3 170, but ought to be 135, shewing that is too great, and consequently taken too little. Let n therefore be assumed = 2, so shall 2c+8=26, and c=3; and the first equation becomes 33+3.3.22.3=27.5 135a as it ought, which shews that the true value of n is 2, and that of e is 3; hence then the с = cube root of 13578/3 or 6 ±n√/b is 3 ± 2√/3 or 312. And in like manner is the process instituted when the number in the radical is negative, as the cube root of 8130/-3, which is 13. Sir I. Newton's rule for any root of a binomial ab. In his Universal Arith. is given a rule for the square root of a binomial, which is the same as the 2d of Lucas de Burgo, before given; and also a general rule for any root of a binomial, which is this: Of the given quantity ab, let a be the greater term, and the index of the root to be extracted. Seek the least number n whose power nc can be divided by aa-bb without a remainder, and let the quotient be y; compute a+b.g in the nearest integer number, which call r; divide aq by its greatest rational divisor, calling the quotient; and let the nearest integer number above be 25 2479 root can be extracted. And this rule is demonstrated by s'Gravesande in his commentary on Newton's Arithmetic: he has also given many numeral examples illustrative of the rule. amples may likewise be found in Newton's Universal Arith. and in Maclaurin's Algebra. (Hut ton's Dict.). Ex Impossible or Imaginary Binomial, is a binomial which has one of its terins an impossible or imaginary quantity; as a +-b. In the foregoing article are given several rules for the roots of binomials. Dr. Maskelyne, the astronomer royal, has also given a method of finding any power of an impossible binomial, by another like binomial. This rule is given in his Introduction prefixed to Taylor's Tables of Logarithms, pa, 56; and is as follows. The logarithms of a and b being given, it is required to find the power of the impossible binomial a±√ whose index is that is, to find m (≈+360o); 1. cos. m -(≈ + 720°); &c. there will arise several distinct answers to the question, agreeably to the remark above. Binomial curve, is a curve whose ordinate is expressed by a binomial quantity; as the curve whose ordinate is x2 x 6 x da. Binomial theorem, a general method of raising a binomial to any given power, or of extracting any given root thereof by an approximating intimite series, which was first given by sir I. Newton. Mr. Briggs, the celebrated improver of logaraising any power of a binomial independently of rithms, was the first who pointed out the way of any of the others. But whatever was known of this matter, by Briggs or any others, related only to pure or integral powers, no one before Newton having thought of extracting roots by infinite series. He happily discovered that, by considering powers and roots in a continued series, roots being as powers having fractional exponents, the same binomial series would equally serve for them (a√62)" by another impossible binomial; all, whether the index should be fractional or inand thence the value of m n n (a + √−b2) + (a−√—b2) which is always possible, whether a or b be the greater of the two. Then Solution. Put =tang. z. tegral, or whether the series be finite or infinite. The truth of this method however was long known only by trial in particular cases, and by induction from analogy; nor does it appear that even Newton himself ever attempted any direct proof of it: however, various demonstrations of the theorem have since been given by the more modern mathematicians, some of which are by means of the doctrine of fluxions, and others, more legally, from the pure principles of algebra only. Euler One of the first demonstrations of this kind that appears to have been given is that of James Bernoulli, which is to be found in his curious treatise, called Ars conjectandi. This, however, is only applied to the case of integral and affirmative powers; and is nearly the same with that which was afterwards given by Mr. John Stewart, in his Commentary on Newton's Quadratures. Dr. Hutton has published, in the first volume of his Mathematical Tracts, an ingenious demonstration, in which he supposes the truth of the binomial and multinomial theorems for integral powere as previously and perfectly established. gave a simple demonstration in the Memoirs of the Petersburgh Academy, vol. xix. for 1774: the substance of this demonstration is inserted in Bossut's Algebra, and in the Complement to Lacroix's Algebra. A demonstration upon analogous principles was given by Dr. A. Robertson, in the Philosophical Transactions for 1806. Mr. Sewell published a demonstration depending upon the principle of vanishing fractions in the Phil. Trans, for 1806; and the substance of this is inserted by Lacroix in the Complement to his Algebra. But roughly satisfactory demonstration which we have the only complete, and, in our estimation, thoyet seen, was pointed out by M. Lagrange, in his Theorie des Fonctions Analytiques, published in 1797; and is correctly exhibited in Mr. Manning's Algebra, vols. i. and ii. Though we cannot help thinking that a satisfactory demonstration may yet be struck out, which, while it rests upon the most unexceptionable principles, shall be inore simple and obvious than any which has hitherto appeared. This theorem was first discovered by sir I. Newton in 1669, and sent in a letter of June 13, 1676, to Mr. Oldenburgh, secretary to the Royal So |