effect; and to increase this a reflector is added. M is a kind of conical chimney, which stands over the lamp and includes the light: the conical part 1 of this is turned within, and well polished to reflect the light downwards. By this means the card is well illuminated, and if the shutters a aa are put down, no light is shown which would be perceived by an enemy at sea; for the only aperture 6 directs the light upwards, and that in a direction where it will not fall on any part of the ship. The lamp is trimmed or lighted by taking out the box from the ring at the top of the lantern; the lamp of course comes with it, and there is a hole on one side to give access to it: if it is to be filled with oil, the box is turned upside down, and the gimbal ef keeping the lamp horizontal, it may be filled or taken out, to clean the conical reflector, and the lens also if they require it. Gilbert's double binnacle lamp, is exhibited in fig. 5. DEFG in the lantern and L the lamp; AA are two condensing lenses, and BB two others, so adjusted to these as to throw the light, after being brought to a focus, upon the two mirrors M M, from which it is reflected strongly upon the two cards N S, NS; by a proper adjustment of the two lenses, the light is made to converge and diverge in any degree, and thereby to illuminate the card only. The contrast of this intense light with every thing else dark, renders the card so exceedingly bright, that in the darkest night the steersman sees it more distinctly than in the brightest day; at the same time not a single ray of light escapes that can be seen by an enemy. Messrs. Gilbert have also a single binnacle lamp upon the same principles. But the sum or the difference of the two terms. difference is also sometimes named a residual. and by Euclid an apotome. The term binomial was first introduced by Robert Recorde; see his Algebra, p. 462. BINOMIAL CURVE, is a curve whose ordinate is expressed by a binomial quantity; as the curve whose ordinate is r2 xb+drele. Stirling, Method. Diff. p. 58. BINOMIAL, IMAGINARY, or BINOMIAL, IMPOSSIBLE, is a binomial which has one of its terms an impossible or imaginary quantity; as a + √ b. Dr. Maskelyne has given a method of finding any power of an impossible binomial, by another like binomial, in his introduction prefixed to Taylor's Tables of Logarithms; which is as follows: The quantities a and b being given, it is required to find the power of the impossible binomial a± whose index is m n m n that is, to find (a± √ — b2) value of (a + √ — b2) + (a ——62) which is always possible, whether a or b be the greater of the two. Solution. Put tang. *, then a m 2n sin' m z). BINNOCH, a Scots patriot, in the days of Robert Bruce, who, during the usurpation of Edward I., assisted Bruce in recovering the castle of Linlithgow from the English. Having been Hence a + employed to supply the castle with hay, he introduced armed men in his cart without suspicion, who soon made themselves masters of it, and opened the gates to their countrymen. He was rewarded with some lands, which descended to his posterity, the Binnings of Wallyford, who bore for their arms a hay wain, with this motto Virtute doloque. BIN’OCLE, n. s., from binus and oculus, a kind of dioptric telescope, fitted so with two tubes joining together in one, as that a distant object may be seen with both eyes together. Harris. BINOCULAR, adj. Lat., from binus and oculus, having two eyes. Most animals are binocular, spiders for the most part octonocular, and some senocular. Derham. BINOCULAR TELESCOPE, a kind of dioptric telescope fitted with two tubes, joined in such a manner that one may see a distant object with both eyes at the same time. See OPTICS. BINODIS, in entomology, a species of formica, a native of Egypt: color black; head rufous; with two tubercles on the petiole. BINOMIAL, a quantity consisting of two terms or members, connected by the sign + or -, viz. plus or minus; as a + b, or 3a 2c, or a2 + b, or x2 — 2 √ c, &c.; denoting the VOL. IV. m n n m X 2 cosin =(bxcosec z) x2 cos. z, where the first or second of these two last expressions is to be used according as z is an extreme or mean arc; b a or rather because is not only the tangent of z but also the tangent of z + 360°, z + 720°, &c.; therefore the factor in the answer will have several values, viz. in the tenth book of his elements, which are ex- 5, 4th 43, 2d 18 + 4, 5th 62, 3d24 +18, 6th 6 + √ 2. BINOMIAL ROOT, in algebra, a root composed of only two parts connected with the signs plus or minus.-Harris. BINOMIALS, RULES FOR EXTRACTING THE SQUARE ROOTS OF; such as of a + b or c +b. Various rules have been given for this purpose. The first is that of Lucas De Burgo, in his Summa de Arith. &c. which is this: When one part, as a, is rational, divide it into two such parts, that their product may be equal toth 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 c + e = a, and ce=4b; then is c + √e = √ a + b the root sought. As if the binomial 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, c + √b, its root will be √ √ C + } √ c − b + √ & √ c -√c-b -So in the foregoing example, 23 +448, herec 23, and √ b = √ 448; hence } √c 11, and c − b = § √ 232 — 448=} √81 = 4; therefore+ § √ c − b = √✓ 11 + 4 = √16 = 4, and√ √ c - 4 √7; consequently 47 is the root sought, as before. Again, if the binomial be 18+ .0; here c 18, and b = 10; therefore = √18 = √2; hence, and √ √ √ c− b = √ √ 2 √2 = √ √2=/; √5+1 consequently is the 1/2 root of 18+ 10 sought. And this latter rule has been used by all other authors, down to the present time. 2 of Sir Isaac Newton's applies to all roots of hi nomials whatsoever. BINOMIALS, SIR ISAAC NEWTON'S RULE FOR ANY ROOT OF. Of the given quantity a± b, let a be the greater term, and c the index of the root to be extracted. Seek the least number n, whose power n°. can be divided by a abb without a remainder, and let the quotient beq; Compute a + b. √g in the nearest integer number, which call r; divide a q by its greatest rational divisor, calling the quotient s; BINOMIALS, RULES TO EXTRACT THE CUBIC AND OTHER HIGHER ROOTS OF. This is useful in resolving cubic and higher equations, and was introduced with the resolution of those equations by Tartalea and Cardan. The rules for such extractions are in a great measure tentative. Tartalea, Bombelli, Gerard, Demoivre, and Dr. Wallis, have given rules for extracting the cube roots, &c. of binomials: for which we shall refer to the works of these authors, as the following rule and let the nearest integer number above 25 n be the root ts ± √ t2 s2 n be t: so shall 2c√9 sought, if the root can be extracted. And this rule is demonstrated by s'Gravesande in his And Commentary on Newton's Arithmetic. many numeral examples, illustrating this rule, are given in s'Gravesande's Algebra, above-mentioned, p. 100; as also in Newton's Univers. Arith. p. 53, 2d edit. and in Maclaurin's Algebra, p. 118. Other rules may be found in Schooten's Commentary on the Geometry of Des Cartes, and elsewhere. BINOMIAL SURD. See BINOMIAL LINE. denote the celebrated theorem invented by Sir BINOMIAL THEOREM, in algebra, is used to Isaac Newton for raising a binomial to any power, or for extracting any root of it by an approximating infinite series. Stifelius and others, about the beginning of the sixteenth century, knew how to raise the integral powers, not barely by a continued multiplication of the given binomial, but by means of a table of numbers formed by continual addition, which showed by inspection the co-efficients of the terms of any power of the binomial contained within its limits; but still they could not, independent of a table, and of any of the lower powers, raise any power of a binomial at once by determining its powers from one another only, viz. the second term from the first, the third from the second, and so on as far as we please, by a general rule, and much less could they extract general algebraic roots in infinite series by any rule whatever. Dr. Hutton has shown, in the Historical Introduction to his Mathematical Tables, p. 75, that Mr. Henry Briggs, about the year 1600, was the first who taught the rule for generating the terms successively one from another of any power of a binomial, independent of those of any other power, for he gives the binomial theorem in words, wanting only the algebraic notation in symbols, as it stands at this day, and as it was extended by Newton to roots or fractional exponents long after Briggs had given it in the case of integral powers. The theorem being thus plainly taught by Briggs, about the year 1600, (says Dr. Hutton,) I am surprised, that a man of such general reading, as Dr. Wallis was, could possibly be ignorant of it, as he plainly appears to have been, from the eighty-fifth chapter of his Algebra, where he fully ascribes the invention to Newton; and adds, that he himself had formerly sought for such a rule, but without success; or how Mr. John Bernouilli, not half a century since, could the mth power of 1 + would be 1, m m m &c. till we come which will be the last figurate numbers, by Mr. James Bernouilli, in the third chapter of his Treatise de Arte Conjectandi, published in 1713, about eight years after his death, but probably written in the latter years of the preceding century. That part of his book relating to the properties of figurate numbers, from which the binomial theorem may be deduced, was published at his native place Basil or Basle in Switzerland, in 1692. Various other demonstrations have been given of this very important theorem by raore modern mathematicians, some of which are by means of the doctrine of FLUXIONS (which see), and others more legally from the pure principles of algebra only; among the best of that kind is that given by Dr. Hutton in his Mathematical Tracts, vol. i. where a full account is given of what others have done upor this subject. A demonstration of it is also giver. from algebraical principles, by Mr. Abram Robertson of Oxford, in the London Philosophical Transactions for 1795, part ii. The binomial theorem is engraved on Sir Isaac Newton's monument in Westminster Abbey as one of his greatest discoveries. The Newtonian theorem, in one of its most sim n.n- -1.n 2 2.3. n.n- 1 2 p2 + 1 2.3 term. But how he discovered this proposition, ple forms is 1+p" = 1 + np + covery of Newton's appears first to have been given to the world, but without a demonstration, by Dr. Wallis in his Algebra, chap. 85, in the year 1685, though it was inserted in Sir Isaac's first letter to Mr. Oldenburgh, the secretary to the Royal Society, dated July 13th, 1676; and the said letter was shown to Mr. Leibnitz, and probably to other mathematicians of that time, yet it remained for some years without a demonstration, either in the case of integral powers or At last it was demonstrated, in the case of integral powers, by means of the properties of roots. 1+p = 1 + sn + } s2 n2 + new, sn1, &c. 1 has done so much for the improvement of the зcience of mathematics. BINOMI'NOUS, adj Lat., from binus and nomen, having two names. BINOMIUS, from bis, and nomen, name; in middle age writers, denotes a person with two names. Most Christians anciently were binomii, as having had other names in their heathen state, which they changed at their conversion. Be sides, it was an ancient custom for parents to give names to their children immediately after they were born, and sometimes other ones afterwards at their baptism; one of which frequently became a cognomen, or surname. It was a constant practice to assume a new name at baptism, as the religious still do in the Romish church, on their reception into the monastic state; and Jewish proselytes at their circumcision. BINTANG, an island in the Eastern seas, on the coast of Malacca, at the entrance of the straits of Sincapore. It is about thirty-five miles in length, and eighteen in breadth, surrounded by rocky islets. The island produces gold dust; its principal town is Reheo or Rio, which was taken and destroyed by the Portuguese in 1527, and is governed by a sultan. Long. 104° 30′ E., lat. 1° 2′ N. BIOBIO, or BIOPHIO, a large river of Chili; rises in the Andes, and after running through veins of gold and fields of sarsaparilla, enters the South Sea, near the city of Conception, opposite the Isle of Avequirinà, in lat. 37° 0′ S. It is the northern boundary of Chili, and its entrance known by two remarkable hills, called the Teats of Biobio, which are situated at the north, betwixt it and the bay of Conception, and serve to both as land-marks for navigators. It is about one mile across at the mouth, has good depth of water in the middle, and the tide rises about seven feet and a half at the full and change of the moon. BIOCOLYTE, from Bia, violence, and raλvw, to hinder; in the Byzantine empire, an order of officers appointed to prevent the violence frequently committed by the soldiers. They were suppressed by the emperor Justinian. BIOGRAD, a town of Dalmatia, anciently Alba Maritima, and the former residence of the kings of Croatia. It has a good harbour at a distance from the town, secured by several small islands, and is considered as belonging to the county of Zara; being nearly twenty miles south-east of the town of that name. BIOGRAPHER, From Boc, life, and BIOGRAPHY, ypapw, I write. BiograBIOGRAPHICAL. phy is that portion of literature which is devoted to the lives of individuals. Our Grub-street biographers watch for the death of a great man, like so many undertakers, on pur pose to make a penny of him. Addison. In writing the lives of men, which is called biography, some authors place every thing in the precise order of time when it occurred. Watts. The necessity of complying with times, and of sparing persons, is the great impediment of biography. Johnson. BIOGRAPHY is undoubtedly the most entertaining and instructive kind of history. It admits of all the painting and passion of romance; but with this capital difference, that our passions are more keenly interested, because the characters and incidents are not only agreeable to nature, but strictly true. Few books more attract or are more proper to be put into the hands of young people, than those of a biographical kind. The history of this study is that of the whole republic of letters. We would only suggest that groupes of biographical memoirs, chronologically arranged, are much wanted as a kind of appendix to all our popular histories. BIOLYCHNIUM, from βιος, life, and λύχιος, light; a substantial fire, flame, or heat, which some ancient physicians supposed inherent or actually lodged in the heart, and remaining there as long as life lasts. Some would have it to have been the human soul, others the animal spirits, and others the Deity, that did the office of a biolychnium, and was the spring of all the actions and motions of the body. Casp. Hoffman and Couringius have written treatises on the ancient doctrine of the lamp of light or innate heat. BION, a bucolic poet, a native of Smyrna, lived at the same time with Ptolemy Philadelphus, whose reign reached from the fourth year of the 123d Olympiad to the second year of the 133d. He was an incomparable poet, if we may believe the lamentations of his disciple Moschus. few pieces which are left do not contradict this testimony. His BION, surnamed Borysthenites, because be was a native of Borysthenes, was a philosopher of a great deal of wit, but very little religion: he flourished about the 120th Olympiad; but falling sick, like many other profane persons, he became superstitious. BIORKO, an island of Sweden, in the Malar, three miles from Stockholm, where anciently stood Birca. BIOTA, in zoology, a name introduced by Dr. Hill for the POLYPE, which see. BIOTHANATI, from Bia, violence, and Oavaros, death, in some medical writers denotes those who die a violent death. The word is also written, and with more propriety, biathanati sometimes biæothanati. In a more particular sense, it denotes those who kill themselves. In this sense the word is used both by Greek and Latin writers. See SUICIDE. Also a name of reproach given by the Heathens to the primitive Christians, for their constancy and forwardness to lay down their lives in martyrdom. Bİ'OVAC, n. s. Br'HOVAC, BIVOUAC. Fr. from wey wach, a double guard, Germ. See BIVOUAC. Bringing forth two at a birth. BIPARTITE, adj. Lat. from binus and parlior. Having two correspondent parts; divided into two. BIPARTITION, n. s. from bipartite. The act of dividing into two; or of making two correspondent parts. No serpent, or fishes oviparous, have any stones at all; neither biped nor quadruped oviparous have any exteriourly. Brown's Vulgar Errours. BIPEDAL, adj. Lat. bipedalis. Two feet in length; or having two feet. BIPEN NATED, adj. Lat. from binus and penna, having two wings. All bipennated insects have poises joined to the body. Derham. BIPENNIS, a two-edged axe, used anciently by the Amazons in right; as well as by the seainen, to cut asunder the cordage of the enemy's vessels. It was a weapon chiefly of the oriental nations, made like a double axe, or two axes joined back to back, with a short handle. Modern writers usually compare it to our halberd or partizan; from which it differs in that it had no point, or that its shaft or handle was much shorter. BIPET'ALOUS, adj. Lat. of bus, and Gr. TεTaλOV. A flower consisting of two leaves. BIPLICATA, in conchology, a species of Voluta. It is of a tapering shape, smooth, white spotted with yellow, and dotted with black; lip acute; pillar with two plaits. BIPUNCTARIA, in entomology, a species of phalana, geometra; anterior wings cinereous, undulated with brown; a dusky band in the mid dle, and two black dots. BIPUNCTATA, a species of apis, inhabiting Siberia. This insect is black, hairy, and with two yellow belts; the first with two lateral black dots. Also a species of aranea, with a black, globose abdomen, marked with two excavated dots. BIPUNCTATUM, a species of opatrum, color brown, thorax cylindrical and immarginate, with two hollows. BIPUNCTATUS, a species of bruchus, inhabiting Helvetia, It is cinereous; wing-cases brown, with an ocellar, black dot at the base of each. Also, a species of carabus, of a brassy color, with two dots on the wing-cases. BIPUNCTELLA, in eutomology, a species of phalana, tinea, wings cinereous brown, with a marginal white spot. Also a species of phalana, tinea, with fuscous wings, with a common dentated white stripe; thorax snowy-white with two black spots. This is tinea echiella of Schmetterl. BIPUSTULATA, a species of cantharis, malachius, of a green color; apex of the wing-cases red. This is thelephorus viridianeus nitidus of Degeer. BIPUSTULATUM, a species of opatrum, inhabiting Pomerania. Its form is narrow and elongated; color ferruginous: wing-cases slightly grooved. BIPUSTULATUS, a species of attelabus; found in North America: color black, with a rufous spot at the base of each of the wing-cases. BIQUAD'RATE, Į n. s., in algebra. The BIQUADRATICK. fourth power, arising from the multiplication of a square number or quantity by itself. Harris. BIQUADRATIC EQUATION, is that which rises to four dimensions, or in which the unknown quantity rises to the fourth power; as x + ar3 + bx2 + cx + d=0. The first resolution of a biquadratic equation was given in Cardan's Algebra, chap. 39, being the invention of his pupil and friend, Lewis Ferrari, about the year 1540. This is effected by means of a cubic equation, and is indeed a method of depressing the biquadratic equation to a cubic, which Cardan demonstrates and applies in a great variety of examples. We subjoin Euler's Rule for BIQUADRATICK EQUATIONS, The celebrated Leonard Euler gave, in the sixth volume of the Petersburgh Ancient Commentaries, for 1738, an ingenious and general method of resolving equations of all degrees, by means of the equation of the next lower degree, and among them of the biquadratic equation by means of the cubic; and this last was also given more at large in his Treatise on Algebra translated from the German into French in 1774, in 2 vols. 8vo. The method is this: Let ra — a r2 bx c 0, be the given biquadratic equation, wanting the second term. Take ƒ}, g=aa + &c, and h = bb; with which values of f, g, h, form the cubic equation, s2 — f2+gz-h=0. Find the three roots of this cubic equation, and let them be called p, q, r. Then shall the four roots of the proposed biquadratic be these following, viz. p+ √9 + √", When bis positive; √p+q - √ r p- 1 + √ ? 4th. pr. 1st. 2d. 3d. - When bis negative: √p + √9 √ P −√ p + √ q + √? The following is Mr. SIMPSON'S RULE. Mr. Simpson gave also a general rule for the solution of biquadratic equations, in the second edition of his Algebra, p. 150, in which the given equation is also resolved by means of a cubic equation, as well as the two former ways; and it is investigated on the principle, that the given equation, is equal to the difference between two squares; being indeed a kind of a generalisation of Ferrari's method. Thus, he supposes the given equation, viz. + pr3 + 9x2 + rx + s = x2 + §px + Al - B+C2; then from a comparison of the like terms, the values of the assumed letters are found, and the final equation becomes A-9A2 + k A — } l = 0 where k pr—s, and 1= r2 + s· } p2—q· The value of A being found in this cubic equation, from it will be had the values of B and C, which have these general values, viz. = pA-r B √ 2 A + §p2 —g and C=-- Hence, 2 B finally, the root will be obtained from the assumed equation r2 + 1px + A2 B x + C =0, or x2+1 px + A = ± Br± C, in four several values. Mr. Simpson subjoins an observation, viz. that the value of A, in this equation, will be commensurate and rational, and therefore the easier to be discovered, not only when all the roots of the given equation are commensurate, but when they are irrational and even impossible.' But this has since been found erroneous, as the instances in which it holds true, |