TRIAMYLAMIXK. 



TRIBE 



o 



application nude, and grounds shown for supposing that justice has 

 not been done between the parties ; and that the case is of sufficient 

 importance to warrant such a further expense. These grounds are 

 rarioiu, such as a misdirection by the judge, a verdict against evidence, 

 excessive damages, ftc. 



TKIAMYLAMINE. [OBCVUOD BiUB.1 



TRIANGLE, a figure having three angles, and consequently three 

 rides : this consequence is usually made the definition ; and the same 

 thing occurs in Euclid, whose word is rptyvnr in the Elements, though 

 it is rfl*\tvfw in the definitions prefixed. 



A triangle may be drawn upon any surface, and having any sort of 

 lines for its sides : but it is uot usual to consider any except plane 

 triangles drawn on a plane with right-lined sides, and ipheriral triangles 

 drawn on a sphere with arcs of great circles for the aides. The SPHE- 

 RICAL triangle has been already considered ; and the formula) connected 

 with the plane triangle have been given in MENSURATION. There is 

 much connected with this article in ANGLE, PARALLEL, SIHII.AU, 

 TRANSLATION, TRANSVERSAL, TRIGONOMETRY, Ac., so that we have no 

 need to make this article of a length proportioned to the importance of 

 its subject in geometry. In fact, triangles are as much the elements 

 of all figures as the letters are of words, being the figures of the 

 amallest possible number of sides, and into which all figures can be 

 divided. 



The two most important properties of the triangle are, that the sum 

 of its angles is always two right angles, and that the area is half that 

 of a rectangle of the same base and altitude. Both of these proposi- 

 tions admit of such practical verification as would make them perfectly 

 intelligible to those who do not understand geometry. Take the 

 greatest angle B A c of a triangle cut out in paper, and fold the 

 paper so that A may rest on B c at F, the part ABE folding over n F >:. 

 Then it will be found that by further folding E c L can be brought over 

 E p L, and SDK over F D K, so that the three angles of the triangle 



B K F L C 



K F D, D F E, and E F L are so placed that the first side of the first and 

 the last side of the List, K F and F L, are in the same straight line, and 

 the three make up the two right angles KFA, A PL. Again, the 

 triangle B A c is either the sum or difference of the two right-angled 

 triangles r A c, FAB, which are the halves of the rectangles F A c u, 



A O H 



V 11 



F A B o, the sum or difference of which is the rectangle B o n c : whence 

 the triangle U the half of the rectangle B a H c, of the same base and 

 altitude as the triangle. 



The three lines which bisect the angles of a triangle meet in one 

 point, which is the centre of the inscribed circle ; and the three per- 

 pendiculars which bisect the three sides also meet in one point, which 

 is the centre of the circumscribed circle. Moreover, the three lines 

 drawn from the vertices bisecting the sides meet in one point, which 

 is the centre of gravity of the triangle : as also do the three perpen- 

 diculars drawn from the vertices to the sides. All these propositions, 

 except the second, can be proved by the same process, namely, by 

 showing that the segment* of the sides satisfy the theorem given in 

 TRANSVERSAL. 



The number of isolated theorems which might be given on this 

 subject U very large, but there is little unconnected with the trigono- 

 metrical formula: which is of use in application. 



TRIANOULA and TRIANGULUM AUSTRALE (constellations). 

 The first (the Triangles) is a northern constellation, surrounded by 

 Perseus, Andromeda, Aries, and Musca. It is one of the old constel- 

 lations, but there is only one triangle in Arotns. Hevelius added the 

 second. The second, or Southern Triangle, is a constellation of Bayer, 

 lying between Ara, Centaurus, sod the South Pole. 



The principal star* are as follow : 



TRIASOULA. 



No. In CsUlotns 

 Wo. In Catalofm of r 



Chsneter. oTFUmstsed. Association. Mafnitndo. 



a 2 MO 4 



4 CM 4 



TllIAXOfLl'M Al'STRALE. 



No. In Ctulogns 



No. In CaUlogne of British 



Character. ofLaeaille. Association. Macnitndt. 



y 11M7 5005 2 



1311 52SS 3 

 a 1381 5578 2 



TRIANGULAR COMPASSES. [COMPASSES.] 



TRIANGULAR NUMBERS. [NUMBERS, APPELLATIONS OF.] 



TRIANGULATION, a name given to the net-work of triangles 

 with which the face of a country is covered in a TRIOONOMEI 

 SmvKY. 



TRIARSENIO-METHYLAMINE. Synonymous with IHnutliyl- 

 artinf. [OiUiANOMKTAU.ic BODIES. Anenic Seriei.] 



TR1ATOM1C KTHKUS. [ KTHKIIS.] 



TRIBE (Irilna, <pi>A4). All the states of antiquity of which we have 

 any records were divided into a certain number of tribes, consisting of 

 the great bodies of citizens of which the state was composed. These 

 tribes, however, were of two different kinds, either genealogical (ytniial) 

 or local (TOITIKOJ). (Dionys. Hal., iv. 14.) The former, which must be 

 considered as the more ancient of the two, were the different nut i. >n.il 

 elements of which a state was made up, that is, each was a distinct 

 people, though akin to the others, and each traced its origin to some 

 mythical ancestor; whence Dionysius calls such tribes genealogical. 

 The .other, or local tribes, to which a later origin must be assigned, 

 and which in most cases superseded the old genealogical tribes, were 

 artificial local divisions made for political and other purposes, and any one 

 of them might contain people who, according 1 to the genealogical divi- 

 sion, would belong to different tribes. Thus we find in the hist" rv "f 

 Attica that the four original tribes were done away with after tin- 

 institution of the ten local tribes by Cleisthenes ; and at Rome the 

 three ancient Romulian tribes which retained their political importance 

 almost to the time of the decemviral laws, died away and ceased to 

 be noticed in Roman history save as an archaic institution, the thirty 

 local tribes established by toe Servian constitution having gradually 

 superseded them. At Sparta alone the three original Doric tribr 

 Hylleans, Painphylians, and Dymanians, were retained without any 

 change. Genealogical tribes may, in many instances, as was originally 

 the cose at Rome, have inhabited different districts, so that they were 

 at the same time local tribes ; but this is merely an accidental circutn- 

 ttaDM, 



The number of the genealogical tribes was different in the different 

 states of antiquity, and depended upon various circumstances, such as 

 the number of national elements brought together to form a state, or 

 the partiality of a race of men for particular numbers which were used 

 as typical. Thus we find that the Doric states were originally divided 

 into three, the Ionic into four, and the Romans into three tribes, and, 

 later on, into a multiple of these numbers. Although these tribes only 

 contained freemen, they were uot always on a footing of equality ; but 

 the most ancient one, to which the others had only been added at some 

 time by treaty or contract, always retained, at least for a time, a supe- 

 riority over the others, and reserved for itself rights and privileges 

 which were denied to the others. Such was the case with the Hylleans 

 at Sparta, the Eupatrids at Athens, and the Ramues at Rome. K-u-li 

 tribe was usually subdivided into smaller bodies, as at Athena into 



aTpicu and ytvri, at Sparta into wjBai and TptaKaSc*, and at U<>nu' int<> 

 curia: and gentes. The number of senators and of the great officers of 

 a state likewise bore a certain relation to the number of tribes or their 

 subdivisions. The bond of union between auch tribes was more or less 

 loose according to circumstances ; and the history of Attica gives UH an 

 instance of their being at war with one another. Each tribe hod usually 

 its separate religious observances and festivals, and the same was the 

 case with its subdivisions. 



All the tribes of which a state consisted formed the sovereign people 

 (as at Rome the populus), which in many oases ruled over a su 

 population superior in numbers (trtpioutoi, plebeians). When a Greek 

 state sent out a colony to a foreign country, it appears to have U . n 

 customary to divide the new state into the name number of tribes as 

 that which existed in the mother-city, and the names also were 

 retained: in some cases, as at Cydonia and Halicarnassus, both of win. M 

 were Doric colonies, we only find mention of one tribe, which may 

 arisen from the fact that only members of one tribe of the niothei 

 took part in the establishment of the colony. (Wachsmuth, ' llel- 

 lenische Altherthumskunde,' ii. 1, p. 15, 4c.) 



In regard to the later or local tribes, it is clear from the name itself 

 that each inhabited a distinct district, containing either one or more 

 townships, which were called in Attiea Siinoi, and at Rome rin or //./<. 

 Every citizen belonging to a tribe was obliged to have his name regis- 

 tered in a township of his tribe, though ho was not bound to reside in 

 the same in which he was registered and to which he belonged. 



Each tribe, whether genealogical or local, managed its own affairs, 

 and was headed by a tribune (<t>v\afxs). The same was the case with 

 the subdivisions of a tribe. 



We have here only given a brief outline of the subject in general, 

 as a more detailed account of the tribes in the dill'erent states of ant i- 

 <|iiity i given in the articles ATHINS: STAHTA ; HOME, in GKOO. 

 Div. ; CLEISTUENES; BERTH'S TUU.IUB, in Bioo. Uiv. ; DORIANH 





