B I O 



m.m— i.m — 2 



on. 



or" I, or q =. 



lid fo 



2-3 



From which it follo ws, th at a-\'xY = oT + m.a"'~ ' x + 



ni,m—\ m.m — \,m — 2 „ , ,. • r ii , 

 • • .rt^-'.v' &c. univeifally, 



c' + 



2-3. 



whatever may be the value of m, whether integral or frac- 

 tional, pofitive or negative, as was to be (hewn. 



The demonllration here given (wliich is fimilar to that in 

 vol. ii. of Manning's Algebra) is founikd upon the principles 

 firll laid down by la Grange, in his " 'I'heorie des Fonftions 

 Analytiqucs;" to which admirable work the reader is referred 

 for farther information on this fubjed, as w^l! as for what- 

 ever regards the do£lrine of expanded funilions in general, 

 which is there treated of in a way worthy the genius of the 

 author. 



BINOMIUS, from ih and ncrmn, name, in MiMk ylge 

 I'/riliis, denotes a perfon with two names. 



Moll Clirillians anciently were called binomii, as having 

 had other names in their heathen Hate, which they changed 

 at their converfion. Befides, it was an ancient tullom for 

 parents to give names to their children immediately after they 

 were born,' and fometimes other diflVrent ones aitervvarJs at 

 their baptifm ; one of which frc((nently became a cogromcn, 

 or fnrname. In realitv, it was a conllant prartice to affume 

 a new name at baptifm, as the religious tlili do in the 

 Romilh church, on their reception into the monalfic llate ; 

 or tlie Jewilh profelytes at their circumcifion. 



BINOTA TA, in Er.tomohgy, a fpccicsof Chry so\?Et,A, 

 found in Denmark. This is teflaceous, with the wing-cafes 

 ferruginous at the bafe. Gmelin. 



BisOTATA, a fpecics of Cicada [IL-nlr^ich Folincen). 

 This infecl inhabits New Holland ; the thorax is fligiitly 

 armed, and produced bfhind ; abdomen fliort aiid tcllaceous; 

 with a black fpot at the bafe of the wing-cafes. Fabricius, 

 &c. 



BINOTATUS, an European fpecies of Carabus, of 

 a black colour, with two red fpots in front of the head, and 

 the antennx yellow at the bafe. Fabricius. 



BiNOTATiis, a fpecies of Cimex {Reduv'nis), found in 

 Surinam. It is black above, with a rufous dot at the apex 

 of each of the wing-cafcs. Fabricius. 



BLNTAM. in Geography, one of the chief towns cf the 

 kin:;dom of Yohor or Jor. See ^TAI.ACCA. 



BINTAN, one of the fmall iflands at the fouth end of the 

 ftraits of Malacca, and nearly north from Lingan ifland. 



BINTH A, in yIncUnt Gergrup!}y, a town of Libya inte- 

 rior, rear the Niger, according to Ptolemy. — Alfo, a place 

 iu Afia, in OfrhoLne, according to the Notitia Imperii. 



BINTSCHAY, in Gcj^raphy, a town of Bohemia, in 

 the circle of Boltflaw, 7 miles north-cad of Turnau. 



BINWY Head, a cnpc of Ireland, on the north-weft 

 Coal! of the county of Mayo. N. lat. 5.}.'- :o'. W. long. 



BIOBIO, or Viovio, a river of Chili, in South Ame- 

 rica. Its lor.rce is among the filver mines, in the moun- 

 tains called Sierra hel'uJii ; it receives the ftreanis of the Hue- 

 qutn and Tolpar, before it reaches Santa Pee, where it 

 firll becomes navigable, and from whence, to its mouth, for 

 the dillance of 33 leagues, its courfe is nearly from eaft 

 to will. 



The new city of Mocha, peopled by inhabitants who re- 

 moved from Coaci-ption twelve years ago, is fit'.;ated on its 

 north bank, about twelve miles from the fea, and is a depot 

 for tiie filver from the mines of Nim'no, and for the gold 

 with which itis lands abound. 



B I O 



The mouth of this river Is in 8.^1.36*45', W.Iong. 

 73° 28' ; and its entrance known by two remarkable hills, 

 called the Teats of Biobio, which are Ctuated at the north, 

 betwixt it and the bay of Conception, and fervc to both 33 

 land-marks for navigators. 



The river is ?.bout one mile acrofs at the mouth, has good 

 depth of water in the middle, and the tide rifes about fcven 

 feet and a h.ilf at the fall and change of the moon. 



BIOCOIA'TjE, in the By-zantme Empire, an order of 

 officers appointed to prevent the violences frequently com- 

 mitted by the foldiers. The word is compounded of ^.-., 

 vis, -yislcnce, and xiX-jv, / hiinler ; and (hould rather be writ- 

 ten liscolytx. 



The biocolvtx appear to have been much the fame with 

 the French archers of the Marlhalfea. They were fuppreffed 

 bv the emperor Jullir.ian. 



' BIOCULATA, in Entomology, a fpecies of EpheN7B!«a, 

 defcribed by GeofFroy, Linn-cus, fee. The wings are 

 white, reticiilated ; on the head two yellow tubercles. Fabr. 

 Inhabits wet places in Europe. The tail cf this kind is fur- 

 nilhcd with two hrilllcs as long as the body- 



BiocULATA, in Natural ITiflory, a fptcics of Hikudo, of 

 an elongated form and cinereous colour, with two eyes. 

 Gmel. Mi;'.!. This is h'lriido fiagnalis of Linnaus. Fn. 

 Suec. &c. Found in wet hollows and rivukts very coin- 

 mou. Length of this creature is nine lines. The female 

 bears about forty eggs at a time, which are furroundcd by 

 a pellucid circle ; at firll thcfe are cinereous, afterwards 

 brown ; and the young, after exclufion, adhere by their taili 

 to the belly of the female. Gmel. 



Biocui-ATVS, in Entsir.o'o^y, a fpecies of Cryptoce- 

 PHALUS [CrioceritJ, found at the cape of Good Hope. It 

 is teftacL'ous ; thorax immaculate ; on the wing-cafes two 

 ocellar white fpots. Fabricius. 



BIOCZ, in Gc-o^raphy, 3 town of Poland, in the palati- 

 nate of Cracow, north of the Carp.-.thian mountains. N. lat. 

 49° 48'. E. long. 21^40'. . , r , 



BIOEA, in Ancient Geography, a fea-port in the fouth- 

 ern part of the illand of Sardinia, according to Ptolemy. 



BIOOLIO, in Geography, a tov>-n of Italy, in the lordfhip 

 of Vtrcelli, 23 miles N.W. of Vercelli. 



BIOGRAPHER, formed from the Greek /5.o?, life, and 

 -/fx^ij, / defcribe, an author who writes a hiilor)-, or life, of 

 one or more perfons. Such were Plutarch, Corn. Nepos, 

 J;c. 



BIOGRAPHY, the art cf dcfc.-ibing or writing lives, 

 is a branch or fpecies of hillory more entertaining, as well as 

 more ufeful in many refpe£ts, than general hillory, as it re- 

 prefents great men more diftinftly, unincumbered with a 

 crowd of other ac\ors, and defcendirg into the detail of their 

 aftions and charafttr, their virtues and tailings, gives more 

 light into human nature, and leads to a more iiitim.ate ac- 

 quaintance with particular perfons than general hiftory allows. 

 A writer of lives may defcend, with propriety, to minute cir- 

 cumllancts and famihar incidents. From him it is expeAed 

 to give the private as well as the public lite of thofe whofe 

 ailions he records ; and it is from private life, from familiar, 

 dumeftic, and feemingly trivial occurrences, that we often 

 derive the moll accurate knowledge cf the real charr.fter. 

 The fuhjecis of biography are not only the lives cf public or 

 private perfons, who have been eminent and beneficial to the 

 world in their refpedlive ftat'ons, but thofe alfo of perfons no. 

 torious for their vice and profligacy ; which may ferve, when 

 jullly characlerizid, as wan.ings to others, by exhibiting 

 the fatal cor.fequenccs which, fooucr or later, g-.nerally fol- 

 low licentious practices. As for thofe, who expofed their 

 hves, or o'.herv.-ife employed their lime and labours for the 



fervice 



