Cornish copper mines, especially in the neighbourhood of Redruth: it is also found as small, brilliant crystals very rich in faces in the white crystalline dolomite of the Binnenthal in the Valais, Switzerland, and under the name binnite was long considered as a distinct species. (L. J. S.)
TETRAHEDRON (Gr. τέτρα-, four, ἕδρα, face or base), in geometry, a solid bounded- by four triangular faces. It consequently
has four vertices and six edges. If the faces be all
equal equilateral triangles the solid is termed the “regular”
tetrahedron. This is one of the Platonic solids, and is treated
in the article Polyhedron, as is also the derived Archimedean
solid named the “truncated tetrahedron”; in addition, the
regular tetrahedron has important crystallographic relations,
being the hemihedral form of the regular octahedron and consequently
a form of the cubic system. The bisphenoids (the
hemihedral forms of the tetragonal and rhombic bi pyramids),
and the trigonal pyramid of the hexagonal system, are examples
of non-regular tetrahedral (see Crystallography). “Tetrahedral
co-ordinates” are a system of quadriplanar co-ordinates,
fundamental planes being the faces of a tetrahedron, and
co-ordinates the perpendicular distances of the point from
faces, a positive sign being given if the point be between
face and the opposite vertex, and a negative sign if not.
If (u, v, w, t) be the co-ordinates of any point, then the relation
u+v+w+t=R, where R is a constant, invariably holds. This system is of much service in following out mathematical,
physical and chemical problems in which it is necessary to
represent four variables.
Related to the tetrahedron are two spheres which have received much attention. The “twelve-point sphere,” discovered by P. M. E. Prouhet (1817–1867) in 1863, is somewhat analogous to the nine-point circle of a triangle. If the perpendiculars from the vertices to the opposite faces of a tetrahedron be concurrent, then a sphere passes through the four feet of the perpendiculars, and consequently through the centre of gravity of each of the four faces, and through the mid-points of the segments of the perpendiculars between the vertices and their common point of intersection. This theorem has been generalized for any tetrahedron; a sphere can be drawn through the four feet of the perpendiculars, and consequently through the mid-points of the lines from the vertices to the centre of the hyperboloid having these perpendiculars as generators, and through the orthogonal projections of these points on the opposite faces.
TETRARCH (τετράρχης), the ruler of a tetrarchy, that is, in the original sense of the word, of one quarter of a region.
Such were the tetrarchies of Thessaly as reconstructed by
Philip of Macedon and of Galatia before its conquest by the
Romans (169 B.C.). In later times the title of tetrarch is familiar
from the New Testament as borne by certain princes of the
petty dynasties which the Romans allowed to exercise a dependent
sovereignty within the province of Syria. In this
application it has lost its original precise sense, and means only
the ruler of part of a divided kingdom, or of a district too unimportant
to justify a higher title. After the death of Herod
the Great (4 B.C.) his realm was shared among his three sons:
the chief part, including Judaea, Samaria and Idumaea, fell
to Archelaus (Matt. ii. 22), with the title of ethnarch (Josephus,
Antiq., xvii. 11, 4); Philip received the north-east of the realm
and was called tetrarch; and Galilee was given to Herod
Antipas, who bore the same title (Luke iii. 1). These three
sovereign ties were reunited under Herod Agrippa from A.D. 41
to 44. In the same passage of Luke mention is made of Lysanias,
tetrarch of Abilene near Damascus, in the valley of the
Barada.
TETRASTOÖN (Gr. τέτρα-, four, and στοά, a portico), the term in architecture given to a rectangular court round which
on all four sides is carried a covered portico or colonnade; the
same as peristyle (q.v.).
TETRASTYLE (Gr. τέτρα-, four, and στῦλος, a column), the term in architecture given to a portico of four columns which
forms the main front of a temple (q.v.).
TETRAZINES, in organic chemistry, a group of compounds
containing the ring system
C·N·N C·N·N C·N·C
Ċ·N·Ṅ, Ṅ·N·Ċ, Ṅ·N·Ṅ;
only derivatives of the first two types are known. The members of the first series may be prepared by oxidizing osazones (i.e. dihydrazones of α-diketones), dihydrotetrazines resulting. Dihydro-derivatives of the second type are formed from hydrazine and imino-ethers (A. Pinner, Ber., 1893, 26, p. 2126; 1894, 27, p. 984); these easily oxidize to the corresponding tetra zines, which are stable towards acids; their dihydro-derivatives, however, are decomposed, the group—NH-NH-being eliminated as hydrazine and replaced by oxygen, with consequent formation of the five membered oxybiazole ring. Concentrated acids convert the dihydro-tetra zines into isodihydrotetrazines, thus:—
R·C·NH·NH > R·C·NH·N
N̈·N :C̈·R N̈-NH-C-R,
the N-alkyl derivatives of which type may be prepared by the action of alcoholic potash and chloroform on aromatic hydrazines. I .
Much discussion has circulated about the decomposition of diazo-acetic ester, from which A. Hantzsch and O. Silberrad (Bern, 1900, 33, p. 58) obtained what they considered to be a dihydrotetrazine, thus:—
CH·CO2R > HO2C-CH-Nzlf g Hc-NH-151-IQ
N=N -CH-CO2H N-NH-CH.
C. Bülow (Ber., 1906, 39, pp. 2618, 4106), however, showed this substance to be an N-aminotriazole, which necessitates the first decomposition product being an acid (I.), the conversion into the (I.) HO2C-(; ~NH~NH 9 (H) HOQC-ge-NZNH,
N-N: C-COQH ' N~N:C-CO2H
triazole derivative (II.) being due to the ring opening on the addition of the elements of water and then closing again to the five-membered ring with elimination of water again. The decompositions of diazoacetic ester were then again examined by T. Curtius and his students (Ber., 1907, 40, pp. 262, 350, 450, &c.), who showed that both triazole and tetra zine derivatives could be obtained from the Vbisdiazo-acetic acid which is formed by the action of alkali on diazo-acetic ester.
TETRAZOLES, in organic chemistry, a group of hetero cyclic
compounds, ” capable of existing in two isomeric series (formulae
1 and 2), although the methods of preparation do not always
permit discrimination between the possible isomers. They are
prepared by the action of nitrous acid on cyanamidrazone
(dicyanophenylhydrazine) and hydrolysis of the resulting
nitrile, from which I. A. Bladin by elimination of 'the phenyl
group (by nitration, reduction, &c.) and of carbon dioxide obtained
free tetrazole, CH4N2; from amidines by the action of
nitrous acid, followed' by the reduction of the intermediately
formed dioxytetrazotic acids with sodium amalgam; from
amidoguanidine by diazotization, the diazonium nitrate-» on
treatment with acetates or carbonates yielding aminotetrazole
(J. Thiele, Ann., 1892, 270, p. 1); from the action of nitrous
acid on phenylthiosemicarbazide; and by the action of arylazoimides
on aldehyde hydra zones (O. Dimroth, Ber., 1907, 40,
p. 2402). The tetrazoles behave as strong mono basic acids,
and are exceedingly stable. A series of tetrazolium bases
(formula 3) have been obtained by H. V. Pechmann (Ber.,
1894, 27, p. 2920) starting from formazyl compounds (formula 4),
which are oxidized by means of amyl nitrite andihydrochloric
acid. They are strong bases, which in aqueous solution, absorb
carbon dioxide readily. The free bases have not been isolated,
but their salts are well-crystallized solids.
N=N /N-NH N:CH RHN~N:CH
HN N:CH°rNN:CH RN NN RN:N
HO R
(1), (2). . (3), (4)
TETSCHEN, a town of Bohemia, Austria, 83 m. N.N.E. of Prague by rail. Pop. (1900) 9692, exclusively German. It is situated at the confluence of the Polzen with the Elbe, on the
right bank of the latter river opposite Bodenbach (q.v.), with
