TR1SECTIOX OK THK ANGLE. 



THirMl'H. 



light increase of descriptive power granted to pun geometry would 

 overcome the difficulty entirely. In modern analysis there it no more 

 trouble in trisecting an angle than in finding a cube root : the trigono- 

 metrical tables m.lve the question immediately to a certain number of 

 plafin* of decimal*, and the calculation of a series and the solution of a 

 cubic equation may be made to serve for any number of place* of 

 decimals. In order to show this, let a be the cine of a given angle, 

 and * the sine of its third part : common trigonometry readily give* 

 the equation 



Now can be found from the angle by meant of the aeries fur tin- 

 nine ; and the solution of the cubic equation is then easy enough. 

 [IxvoLCTiox.l The three roots of the cubic equation are respectively 

 the sines of the third part of the given angle, and of 120* ami _lo 

 more than that same third part. The cause of the geometrical 

 difficulty is seeu in the cubic equation, which, as appears above, is 

 assentisl to the problem : no root of a cubic equation was ever exhibited 

 by Euclid's geometry alone, unless that cubic equation were algebrai- 

 cally reducible to one of a lower degree, which could be solved without 

 the extraction of cube root*. 



The old geometers soon reduced the question to depend upon the 

 :' ;. .:..-. - - I : \. , . . . < < ti. . -. .;. iol : :. . ngl . and 

 K the diagonal r"""c through the junctions of u, c, and A, D : also let 

 the angle K A be the one which is to be trisected. Through the jxiint 

 common to B, c, draw a line F passing through D and A produced, in 

 such a manner that the part between D and A produced is twice K in 

 length. Then it is easily shown that the angle FA is the third part of 

 E A. Through the point c D draw an equilateral hyperbola, of which 

 the asymptotes are A and B. A chord of this hyperbola, set off from 

 C D towards A produced, and equal in length to twice E, will be a parallel 

 to the line r required. Admit then the hyperbola among the curves 

 of geometry, and the difficulty ceases. Again, if with two-thirds of 

 any given line A as a major axis, an hyperbola be described whose 

 asymptotes make an angle of 120 ; and if with A as a base, and a point 

 on the branch of the hyperbola adjacent to the single third of A as a 

 vertex, a triangle be described, the larger of the angles adjacent to A 

 will always be double of the smaller. Consequently, one of the 

 external angle* will be triple of one of it* internal and opposite angles : 

 so that by describing on the straight line A a segment of a circle con- 

 taining the supplement of any given angle less than 180, that circle 

 will cut the branch of the hyperbola in a point which, being joined 

 with the further extremity of A, will give an angle equal to the given 

 angle. 



Again, if from any point of a circle a straight line be drawn cutting 

 the circle again, and then a diameter produced, in such manner that 

 the portion externally intercepted between the diameter produced and 

 the circle is equal to the radius, the angle formed by that line and the 

 diameter produced is the third part of the angle made by tho two 

 radii, of which one passes through the first point of tho circle 

 mentioned, and the other is on tho diameter which was produced. 

 The construction can be effected by the CONCHOID of Nicomedes, 

 which curve, if granted, gives the means of drawing a straight line of 

 given length between any straight line and a curve, so that when pro- 

 duced it shall pass through a given point 



Either of the curves known by the name of QUADHATHIX may be 

 made to trisect an angle, as obviously may any curve which assigns a 

 straight line equal to a given arc : for a straight line may be easily 

 trisected. The SPIRAL of Archimedes obviously gives another solution. 

 But there is one particular curve known by the name of the trinrctrir, 

 which, among curve* not geometrical, is peculiarly possessed of this 

 property. It is one of the TBOcnoiDAL curves having the deferent and 

 epicycle equal, the motion in the latter being direct and equal to one- 

 half of that in the epicycle. Or, add and subtract the radius of a 

 circle from every one of the chords which passes through a point in 

 it* circumference, and the result will be a looped curve, which is the 

 one in question. Let A be the point where the branches unite, and 

 AH the axis of the loop: describe a circle with A as a centre. and , :, 

 as a radius ; take a point r in the loop, and let A p and B p produced 

 meet the circle in o. and it. Then the arc B R is three times B Q. 



Many other mod** of trisection have been proposed, some of great 

 geometrical beauty ; but the preceding are those to which it is most 

 likely the student will meet with reference* in his reading. Many 

 false bisections have also been prnjioscd by jwrsons who thought they 

 could conquer the geometrical difficulty. There is not so much to 

 ex POM in this class of trisections a* in the one of .piadr.itur. 

 circle which corresponds to it. Thar* ha* never been so much of 

 romance applied to this problem, no explanations of theological point* 

 have been made to arise out of it, no mode of converting the heathen 

 aasertod to be a necessary consequence, no Number of the Beast taken 

 into the calculation. We shall only notice one false trisection, because 

 it will afford a useful remark. In May, 1830, an Austrian 

 announced his having obtained the geometrical solution in the ' 

 Service Journal,' and various comments appeared in that periodical, 

 running through various months up to March, 1832. In January, 

 1832, an actual attempt at solution appeared, the work of a British 

 officer then abroad. This at first sight ap|>earod to be a geometrical 

 solution; and what is more, it teat a geometrical solution, and it might 



have cost a practised mathematician a moment's doubt whether the 

 problem was not actually solved. But, owing to a mistake, a con- 

 n was made, which amounted to n-. piiiin^ that two side* of a 

 certain triangle should bo together equal to the third, the consequence 

 of which was that the vertex of this triangle was brought down upon 

 the base. Now the angle to be trisected was one of the angles at the 

 base of this triangle, or ry/ lo n;tkuiii ; an angle which no geometer 

 would refuse to declare capable of Euclidean division into three 

 part*, each of course equal to nothing. Algebra generally furnishes 

 some proof of the absurdity of tin- > n.litiniu of a problem when they 

 contradict one another : but this i not the cos* with geometry. A 

 latent assumption which restrict* the generality of a solution always 

 produces iU. effect in the former science ; whereas in the latter such 

 an assumption might be made port of a demonstration, and produce it* 

 consequencci!, without pointing out that those consequences are not 

 true of the general figure which was drawn. The accurate use 

 ruler and compasses will sometimes correct an error of this sort (and 

 would have done so in the instance before us), l>ut nut alteay* 

 tions have been proposed before now which give so nearly the third 

 part of an angle, that ordinary drawing will not serve to detect their 

 I. Any one who imagines he has discovered a geometrical 

 trisection should take care to submit his -i to an algc 



verification ; that is, if any person possessing algebra enough to do so 

 should ever In- in mieh a case. 



THISUCCINAMIDE (N,[C,H 4 O t ] s ). An unimportant organic 

 substance bearing the same relation to succinaniide as triuthy loiuiuo 

 bears to ethylamine. [ORGANIC BASKS.] 



TRITHIONIC ACID. [Sfu-iicn.] 



TKITV1,. [PBOPTL.1 



TKITYLAMIV .I.AMIXK.] 



T1UTVI.KNK. [I'liopvi: 



TKITYL-SULPHUUIC ACID. Synonymous with Propyl-mlphtu !c 

 acid. [Piicr 



TRIUMPH (Triumphus) is in general a solemn procession for the 

 purpose of celebrating a victory. Such processions and solemnities 

 have been customary in all warlike nations, but they have never i 

 m> prominent a feature in the history of a people a* among tho Roman.". 

 In u Roman triumph, the general who had gained a victory of 

 cient importance to entitle him to this honour, entered the 

 Rome in a chariot drawn by four horses ; he was preceded by tin; 

 captives and spoils, and followed by his army. The whole train passed 

 along the Via Sacra up to the Capitol, where the general sacrii 

 bull to Jupiter. Such a triumph was the highest honour that a 

 military commander could look for ; it was granted by the senate after 

 any victory either by sea or by land, provided it was thought suffi- 

 ciently important to deserve it. 



When a general had gained a victory or had accomplished tin 

 of his mission, he sent in a report to the senate, which thru usually 

 decreed a public tl i<>.) The general returned to 



Rome, cither with bis army, or appointed a time when it was to meet 

 him there ; but he did not enter the city, and a meeting of tin- 

 was held outside the walls, usually in the temple of Bellon.i, f.ir the 

 purpose of examining the general's claims to a triumph. The principal 

 conditions upon which a triumph was granted, and vhi.-h wen- 

 established partly by custom, and partly l>y law, are as follows: 1. 

 That the general should have held one of the great offices of tl. 

 republic, that is, the dictatorship, com nlr-hip, or prn. 

 he should have been invested with one of these offices at th> 

 when he gained the victory, and that it should not 

 the day of the triumph. Tin however wa* set aside 



early period, and in cases where the term of office had e\pii,d tin-. 

 senate used to gran ." that is, a pi" 



his imperium or authority OH I In- day of the triumph. 3. 



That the victory should have been gained under the auspices and with 

 the troops of the general who claimed a triumph. -1. That tho 

 -CM gained by the victory and the ntimbrr of the enemies slain 

 Hhoiild come up to the amount prescribed by law. 6. That the victory 

 h. uM h.ivc been gained over a foreign enemy, and not in a civil wai -. 

 (i. Th.it the dominion of the Roman people should have been extended 

 by the victory, and that it should not be a mere n-i i losses 



previously sustained. 7. That the war should be actual!;. 

 by it, so as to enable the army to quit the enemy's country. 



These rules however were not always strictly observed, and various 

 deviations from them are recorded. Kven the sanction of the 

 ceased to be thought necessary as early as the fifth eentuiy 

 Christ, and the people in the Coiniti.-i Tribute assumed the right to 

 grant triumphs (Liv., iii. 63 ; Dionys., xi. 5) ; and there are in-- 1 

 of generals triumphing in defiance of the senate and the peopl 



mes a general to whom a triumph in the city was refused, used 

 to celebrate it on the Alban Mount. (Uv.,.\lii. LM.) If howe\. 

 senate granted it, a sum of money was voted as a contribution t< 

 defraying the expenses of the triumph, and the ; neral was for the day 

 of his triumph invested with mi in the city. Dining thu 



triumphal procession, the general, standing in his chai 

 purple toga embroidered with gold ; his brow was adorned with a 

 wreath of bay (laurus), and in his hand he carried a sceptre with the 

 Roman eagle. On reaching the temple of Jupiter ho <1< 

 wreath in the lap of the god. Banquets and other cutcrUin 



