quantity independent of coordinates; the quadratic expression has the form considered by Riemann. In Riemann's theory, following Gauss, account is taken of curves, called geodesies, which satisfy the condi- tion that the integral Jds, taken along with such a curve, shall be stationary according to the ordinary rules of Lagrange's Calculus of Variations, where ds is the square root of the quadratic expression referred to. Einstein's suggestion is that the path of a particle under the influence of what we call gravitating masses may be represented as such a geodesic, provided the coefficients in the quadratic form are chosen to depend suitably upon these masses; and this has proved capable of verification in the case of the planet Mercury, and in the case of a ray of light passing near to the Sun. An analogous sugges- tion has led Weyl not only to the equations belonging to the theory of gravitation, but also to those which express the phenomena of electromagnetism (and light). And it is very interesting from our present point of view to see the character of the modifications which Weyl has been led to make in Einstein's mathematical formulation in order to attain this end. For our present purpose we may state this in a twofold manner without entering into the logical connexions. In the first place, in Weyl's theory, instead of the quadratic form ds 2 being regarded as definite for two specified neighbouring events, a product e 2 is regarded as definite, where is a function variable from point to point, whose derivatives in regard to the coordinates are utilized to represent electromagnetic phenomena. As Weyl writes (Math. Zeitschrift, II. p. 397, 1918), " Riemann machte die . . . Annahme, dass sich Linienelemente nicht nur an derselben Stelle, sondern auch an zwei endlich entfernten Stellen ihrer Liinge nach miteinander vergleichen lassen. Die Moglichkeit einer solchen ferngeometrischen Vergleichung kann aber . . . nicht zugestanden werden." This is precisely in the spirit which has moved geometers increasingly since the publication of G. K. C. von Staudt's Geo- metric der Laee (1847). It introduces however evidently a wide ar- bitrariness, which Weyl limits by adopting as a datum the possibility of the translation of a vector given at one point to ancther neigh- bouring point without change of direction. This conception, adopted from T. Levi-Civita (see Levi-Civita, Palermo Rendiconti, XLII., I 9 I 7. PP- I 732O5. ar >d F. Severi, ibid., p. 254), is as follows: The two elements of direction defined by (a) the vector at the first point P and (b) the displacement from P to the neighbouring point P', define a family of geodesic directions through P, forming a surface; the parallel vector at P' is that whose direction on this surface makes with the direction PP' the same angle as that made by the vector at P. Evidently the assumption of the possibility of this determination of unchanged direction is fraught with large consequences or condi- tions. A suggestion subsequent to Weyl's (A. S. Eddingtpn, Proc. Roy. Soc. XCIX., 1921, pp. 104-122), begins with Levi-Civita's differential equations for parallel displacement of a vector, but work- ing backwards towards the quadratic differential form leads to a generalization of Weyl's formulation.
So much of detail in regard to these remarkable contemporary speculations seems necessary in order to compare the gconictrical aspects with those of older conceptions. In the so-called space of Einstein, still less in Weyl's space, there exist neither bodies, nor movement; and what are the fundamental geometrical conditions assumed prior to the establishment of the system of coordinates is as yet undetermined. The latter fact, which is equally true of any Cartesian space, may provisionally be evaded by regarding the space as being in point to point correspondence with a quadric manifold in a projective space of five dimensions; the former fact, which relates to the consideration of a quadratic differential expression, is most probably, if it proves finally to be possible to put the phenomena of physics into exact correspondence with geometrical considerations, suggestive of a physical theory which, given some fundamental relations of experience, shall be developed not by computation, but by descriptive methods. For the aim of geometry, towards which since von Staudt's time, much progress has been made, is such a descriptive conception of the relations of figures in space as may render computation unnecessary.
(b) General Theory of Surfaces. The older theory of circles and conies, or of rational curves in general, as also the theory of quadric surfaces, of cubic surfaces or of rational surfaces in general, can be placed in (l, correspondence with the geometry of lines, or of the planes, respectively; it deals ultimately with linear equations when viewed analytically. A consideration of cubic curves on a plane, or of the curve of intersection of two quadric surfaces, soon shows that these do not depend upon linear equations ultimately or more precisely that the points of a plane cubic curve cannot be put into ( I, I) correspondence with the points of a line. And it further appears that a quartic curve in a plane is again of a higher category, and cannot be put into (i, i) correspondence with a cubic curve. This fact first emerges clearly in Abel's great paper on the integrals of algebraic functions. The general theory of the so-called Higher Curves was then historically subsequent to the theory of algebraic functions and the integrals of these; though, when this theory had received a sufficient development, it proved possible to elaborate a descriptive theory of these curves embodying the results obtained by the earlier analytical methods. In geometry, entities which can be put into exact (1,1) correspondence are equivalent for geometrical purposes, and conversely, for purposes of a general theory, it is vital to know whether two entities have this equivalence or not. It is one of the most important recent developments of geometry to have made it clear that criteria can be given by which to determine whether two surfaces have this (1,1) correspondence. And it is interesting to remark that historically the development in this case has been on similar lines to that by which the corresponding result was obtained for curves; in the first place, over many years, Picard developed the theory of algebraic integrals associated with surfaces on lines as far as possible analogous to those which had been followed in the case of curves, therein in part carrying out a suggestion due to Clebsch and Noether, though the integrals which have proved most effective hitherto were not those suggested by Clebsch ; after this the geomet- rical aspect of the matter was investigated by Italian geometers, more especially Enriques, Castelnuovo and Severi, who have suc- ceeded in surpassing, in beauty and generality, even the distinguished contributions of their own countrymen to the theory of curves. It is impossible indeed to convey to a nongeometrical reader any idea of the interval which separates the development of geometry in Italy to-day from the development reached in England.
The new theory is under the disadvantage that an appreciation of it is impossible without sympathy and acquaintance with the theory of algebraic functions and their integrals, and it may be some time before detailed applications of it become the common property of mathematicians. But it offers a limitless scope for new work, its importance cannot be doubted, and its permanence is assured.
BIBLIOGRAPHY. For the questions suggested by the Einstein- Minkowski work, ample material arises in attempting to sift into logical coherence many of the current writings on Relativity. An ample bibliography of these concludes the work of Hermann Weyl, Raum-Zeit-Materie, Vierte erweiterte Auflage, Berlin, igzi. The English reader will find much stimulus to geometrical consideration in Eddington's volume, Space, Time and Gravitation (Cambridge, 1920); and should consult E. Cunningham's two fundamental volumes on Relativity, and A. A. Robb, A Theory of Time and Space (Cambridge, 1914).
Emile Picard's work is summarized in his book Theorie des fonctions algebriques de deux variables independentes (Paris, 1897- 1906), which concludes with a summary by MM. Castelnuovo et Enriques (t. II., pp. 485-522) of the results obtained by the Italian geometers up to 1906. Subsequent progress is recorded (and scat- tered) in the various mathematical journals, mainly of Italy.
(H. F. BA.)
MATHEWS, SIR CHARLES WILLIE (1850-1920), English lawyer, was born in New York Oct. 16 1850, the son of the actress Mrs. Davenport, who became irr 1857 the second wife of the comedian Charles James Mathews (see 17.887). The boy took his stepfather's name, and was sent to England to be educated at Eton. In 1868 he entered the chambers of Montagu Williams, the well-known criminal lawyer, as a pupil, and in 1872 was called to the bar. His rise was rapid, and he soon built up a wide connexion and became known as an extremely skilful cross-examiner. In 1886 he was made counsel to the Treasury, from 1893 to 1908 was recorder of Salisbury, and in 1908, on the retirement of Lord Desart, became director of public prosecutions. Mathews, who was knighted in 1907 and received the K.C.B. in 1911, was concerned in most of the important criminal cases and causes celebres of his time, among them being the Colin Campbell divorce suit ( 1 886) , the trial of the Jameson raiders ( 1 896) , and the prosecution of Lynch for high treason (1903). He was also well known in the theatrical world, being a constant attendant at first nights, and was besides an excellent after-dinner speaker and all-round sportsman. He died in London June 6 1920.
MATTEI, TITO (1841-1914), Italian musician and composer, was born at Campobasso, near Naples, May 24 1841. He became at an early age a professor at the Santa Cecilia academy of music at Rome, and subsequently had several successful European tours as a pianist. In 1863 he finally settled in London, where he remained for the rest of his life. He composed several hundred songs and pianoforte pieces, many of which became very popular. He died in London March 30 1914.
MATTER, CONSTITUTION OF (see 17.891). In the decade 1910-20 many important advances were made which gave much more definiteness and precision to our knowledge of the constitution of matter. The atomic theory of matter, which for long appeared to be of necessity unverifiable by direct experiment on account of the minuteness of the atom, received almost direct proof in a number of ways. Methods have been developed, for example, to detect the electrical effect of a single a particle from radium, and a single swift electron (see GASES, ELECTRICAL PROPERTIES OF).
The a particle has been shown to be a charged atom of helium
