Under the general heading "Fundamental Notions" occur the subheadings "Foundations of Arithmetic," with the topics rational, irrational and transcendental numbers, and aggregates; "Universal Algebra," with the topics complex numbers, quaternions, ausdehnungslehre, vector analysis, matrices, and algebra of logic; and "Theory of Groups," with the topics finite and continuous groups.
A theory of matrices has been constructed by Cayley in connexion particularly with the theory of linear transformation.
This was successfully accomplished by the use of flexible paper matrices, from which metal plates could be cast in shaped moulds to any desired curve.
In the 11th century a fairly large number of matrices were cut in ivory.
The common material for re ceiving the impressions from the matrices was beeswax, generally strengthened and hardened by admixture with other substances, such as resin, pitch and even hemp and hair.
Such antique gems as were adopted for matrices in the middle ages were usually set in metal mounts, on which the legends were engraved.
Among other official seals a very interesting type is that of the Lord High Admiral in the 15th century, several matrices of the seals of holders of the dignity having survived and being exhibited in the British Museum.
Observe the notation, which is that introduced by Cayley into the theory of matrices which he himself created.
Taber, on Matrices, Amer.
The oldest of all the matrices is lime, and many splendid examples of its use by the Romans still exist.
Other matrices are slag cement, a comparatively recent invention, and some other natural and artificial cements which find occasional advocates.
The method is essentially the same as that developed, under the name of " matrices," by Cayley in 1858; but it has the peculiar advantage of the simplicity which is the natural consequence of entire freedom from conventional reference lines.
Among his most remarkable works may be mentioned his ten memoirs on quantics, commenced in 1854 and completed in 1878; his creation of the theory of matrices; his researches on the theory of groups; his memoir on abstract geometry, a subject which he created; his introduction into geometry of the "absolute"; his researches on the higher singularities of curves and surfaces; the classification of cubic curves; additions to the theories of rational transformation and correspondence; the theory of the twenty-seven lines that lie on a cubic surface; the theory of elliptic functions; the attraction of ellipsoids; the British Association Reports, 1857 and 1862, on recent progress in general and special theoretical dynamics, and on the secular acceleration of the moon's mean motion.
Examples of seals, both matrices and impressions, are found among the antiquities of Egypt, Babylonia and Assyria.
On the clay stoppers of wine jars of the remote age which goes by the name of the pre-dynastic period, and which preceded the historic period of the first Pharaohs, there are seal impressions which must have been produced from matrices, like those of Babylonia and Assyria, of the cylinder type, the impress of the design having been repeated as the cylinder was rolled along the surface of the moist clay.
As already stated, the matrices of ancient Babylonian and Assyrian seals, usually cut on precious stones, are in cylinder form.
The Phoenicians, as was only to be expected of those traders and artisans of the ancient world, appear to have adopted both the cylinder of Assyria and the scarab of Egypt as have survived the numerous engraved stones or g pebbles, technically called gems, which served as matrices and in most instances were undoubtedly mounted as finger-rings or were furnished with swivels.
One of the oldest matrices is an intaglio in rock crystal, now preserved at Aix-la-Chapelle, bearing a portrait head of Lothair II., king of Lorraine (A.D.
Phoenician names are found cut both on cylinder matrices and on scarabs by the Phoenician engravers employed in Assyria and Egypt; and, when the cone-shaped matrix superseded the cylinder in Western Asia, the Phoenicians conformed to the change.
In the British Museum are the bronze matrices of seals of ZEthilwald, bishop of Dunwich, about Boo; of lElfric, alderman of Hampshire, about 985; and the finely carved ivory double matrix of Godwin the thane (on the obverse) and of the nun Godcythe (on the reverse), of the beginning of the 11th century.
In the later centuries also, particularly in the 14th century, they were set in seal matrices and finger rings.
But there are examples of elaborate matrices composed of several pieces, from the impressions of which the seal was built up in an ingenious fashion, both obverse and reverse being carved in hollow work, through which figures and subjects impressed on an inner layer of wax are to be seen.
It has usually been the custom to break up or deface the matrices of official seals when they have ceased to be valid, as, for example, at the commencement of a new reign.
But the legal maxim that corporations never die is well illustrated by the survival of the fine series, not complete, indeed, but very full, of the matrices of English corporations, beginning with the close of the 12th century.