gives judgment, stating his conclusions on the law and facts involved.
There remain certain modes of trial not obsolete but rarely used. Such are impeachment of the House of Commons before the House of Lords; and in the case of a charge of treason or felony by a person having privilege of peerage, trial on indictment before the House of Lords, or in vacation before the court of the lord high steward. Trials by certificate, by inspection and by record, are obsolete.
The decisions on a trial at first instance are reviewed by appeal (q.v.), or in trial cases heard before a jury by application for a new trial, where the judge has not directed the jury correctly as to the law or has permitted them to consider inadmissible evidence, or the jurors have in their verdict acted without evidence or against the weight, i.e. the quality not the quantity of the evidence. Under the Criminal Appeal Act 1907 the decisions in criminal trials on indictment, whether on matters of law or of fact or on mixed questions of law or fact, are reviewable by the court of criminal appeal; but that court has no power to order a retrial of the case before a jury.
Scotland.—Jury trial was introduced into Scotland for certain classes of civil cases in the 19th century but is not much used. In criminal cases it is used where summary jurisdiction has not been conferred.
Ireland.—The law of Ireland as to trials is in substance the same as in England, except as to appeals in criminal cases.
United States.—In the United States the system of trial is that of the English common law as varied by Federal and state legislation. (W. F. C.)
TRIANGLE, in geometry, a figure enclosed by three lines; if the lines be straight the figure is called a plane triangle; but
if the figure be enclosed by lines on the surface of a sphere it is a
spherical triangle. The latter are treated in Trigonometry;
here we summarize the more important properties of plane
triangles. In a plane triangle any one of the angular points can
be regarded as the vertex; and the opposite side is called the base.
The three sides and angles constitute the six elements of a
triangle; it is customary to denote the angular points by capital
letters and refer to the angles by these symbols; the sides are
usually denoted by the lower case letter corresponding to that of
the opposite angular point. Triangles can be classified according
to the relative sizes of the sides or angles. An equilateral triangle
has its three sides equal; an isosceles triangle has only two
sides equal; whilst a scalene triangle has all its sides unequal.
Also a right-angled triangle has one angle a right angle, the side
opposite this angle being called the hypotenuse; an obtuse angled
triangle has one angle obtuse, or greater than a right
angle; an acute-angled triangle has three acute angles, i.e.
angles less than right angles. The triangle takes a prominent
place in book i. of Euclid; whilst the relation of the triangle
to certain circles is treated in book iv. (See Geometry:
§ Euclidean.)
The following is a summary of the Euclidean results. The angles at the base of an isosceles triangle are equal and conversely; hence it follows that an equilateral triangle is also equiangular and conversely (i. 5, 6). If one side of a triangle be produced then the exterior angle is greater than either of the two interior opposite angles (i. 16), and equal to their sum (i. 32); hence the sum of the three interior angles equals two right angles. (In i. 17 it is shown that any two angles are less than two right angles.) The greatest angle in a triangle is opposite the greatest side (i. 18, 19). On the identical equality of triangles Euclid proves that two triangles are equal in all respects when the following parts are equal each to each (a) two sides, and the included angle (i. 4), three sides (i. 8, cor.), two angles and the adjacent side, and two angles and the side opposite one of them (i. 26). The mensuration is next treated. Triangles on the same base and between the same parallels, i.e. having the same altitude, are equal in area (i. 37); similarly triangles on equal bases and between the same parallels are equal in area (i. 38). If a parallelogram and triangle be on the same base and between the same parallels then the area of the parallelogram is double that of the triangle (i. 41). These propositions lea to the result that the area of a triangle is one half the product of the base into the altitude. The penultimate proposition (i. 47) establishes the beautiful theorem, named after Pythagoras, that in a right angled triangle the square on the hypotenuse equals the sum of the squares on the other two sides. Two important propositions occur in book ii. viz. 12 and 13; these may be stated in the following forms: If ABC is an obtuse-angled triangle with the obtuse angle at C and a perpendicular be drawn from the angular point A cutting the base BC produced in D, then AB2 (i.e. square on the side subtending the obtuse angle)＝BC2 + CA2 + 2BC·CD (ii. 12); in any triangle (with the sarne construction but with the side AC subtending an acute angle B, we have AC2＝AB2 + BC2 − 2CB·BD (see Trigonometry).
Book iv. deals with the circles of a triangle. To inscribe a circle in a given triangle is treated in iv. 4; to circumscribe a circle to a given triangle in iv. 5. The centre of the first circle is the intersection of the bisectors of the interior angles; if the meet of the bisectors of two exterior angles be taken, a circle can be drawn with this point as centre to touch two sides produced and the third side; three such circles are possible and are called the escribed circles. The centre of the circum circle is the intersection of the perpendiculars from the middle-points of the sides. Concerning the circum circle we observe that the feet of the perpendiculars drawn from any point on its circumference to the sides are collinear, the line being called Simson’s line. We may here notice that the perpendiculars from the vertices of a triangle to the opposite sides are concurrent; their meet is called the orthocentre, and the triangle obtained by joining the feet of the perpendiculars is called the pedal triangle. Also the lines joining the middle point of the sides to the opposite vertices, or medians, are concurrent in the centroid or centre of gravity of the triangle. There are several other circles, points and lines of interest in connexion with the triangle. The most important is the “nine point circle,” so called because it passes through (a) the middle points of the sides; (b) the feet of the perpendiculars from the vertices to the opposite sides; and (c) the middle points of the lines joining the orthocentre to the angular points. This circle touches the inscribed and escribed circles. For the Brocard points and circle, Tucker’s circles—with the particular forms cosine circle, triplicate ratio (T.R.) circle, Taylor’s circle, McCay’s circles, &c., see W. J. M’Clelland, Geometry of the Circle; or Casey, Sequel to Euclid.
TRIANGLE, in music (Fr. triangle, Ger. Triangel, Ital. triangolo),
an instrument of percussion of indefinite musical pitch, consisting
of a triangular rod of steel, open and slightly curved at one corner.
The triangle, suspended by a loop, is played by means of a steel
stick with a wooden handle. Varied rhythmical effects and
different grades of forte and piano can be obtained. A sort of
tremolo or roll can be produced by striking each end of the triangle
alternately in rapid succession. When the triangle is
scored for on a separate staff, the treble clef is used, but it is more
often included with the bass drum on the bass stave. The tone
of the triangle is clear and ringing, but it should have no definite
pitch. The small triangles are the best. Beethoven, Mozart,
Weber and other great masters employed the instrument.
TRIASSIC SYSTEM, in geology, the lowest or youngest system of the Mesozoic era; it occupies a position above the Permian and below the Jurassic system of rocks. The principal formations of the type region, Germany, are the Bunter, Muschelkalk
and Keuper; these were for the first time grouped together under the systematic name “Trias” by F. von Alberti (1834).
A description of the rocks in these formations will be found under their respective headings. For a long time this German development of the strata was regarded as typical of the period; later, however, the discovery of another more fossiliferous phase in the Alps and Mediterranean region, and subsequently in Asia and elsewhere, led geologists to take a different view of the system as a whole. It was clearly seen that there existed two distinct phases of Triassic rock-building, the one continental (terrestrial and lagoonal), the other marine (pelagic).
