
The Mathematical colloquium organised by the Edinburgh Mathematical Society this year for the first time, began its meetings yesterday in the rooms of the Mathematical Department of Edinburgh University. The success of the movement is strikingly shown by the fact that the number of members is 76, of whom no fewer than 20 are professors or lecturers in Universities or University Colleges, nearly 50 are engaged in Mathematical teaching in secondary schools throughout England and Scotland, and the rest are occupied with the practical application of mathematics, as in the Census, Ordnance Survey, and Meteorology Departments of the Government. Special interest attaches to this course, as it is the first of its kind to be held in Great Britain, and also because of the fact that the newlyequipped Mathematical laboratory of the University is to be used by Professor Whittaker in his demonstrations on some of the more important applications of mathematics. The Committee of the Edinburgh Mathematical Society have been successful in getting wellknown authorities to deliver the courses of lectures. Professor A W Conway, of the National University of Ireland, is taking for his subject "The Theory of Relativity and the New Physical Ideas of Space and Time;" Dr Sommerville, of St Andrews University, lectures on "NonEuclidean Geometry and the Foundations of Geometry;" and Professor Whittaker, Edinburgh University, gives a course of five lectures and demonstrations on "Practical Harmonic Analysis and Periodogram Analysis." By the courtesy of the University Court, several rooms have been set aside as reception and writing rooms, and these have been furnished for the comfort and convenience of members of the colloquium. The social side has also been attended to, and yesterday afternoon the members were invited to tea at the Edinburgh University Union, while for the remaining evenings of the week golf matches have been arranged at various courses in the neighbourhood.
The first lecture was given at ten o'clock yesterday morning by Professor Conway. Starting with the conception of relative motion, the lecture described the various attempts which were experimentally made to detect the relative motion of the earth and the surrounding aether. The failure of all such attempts led to the investigations of Larmor and Lorentz. The theory of relativity after Einstein started by asserting the impossibility of observing the absolute velocity of the observer. The lecturer proceeded to deduce the relations between distances and times in the frame of reference and those in another frame which was moving with respect to the former.
The second lecture was given at 11.30 by Professor E T Whittaker in the new Mathematical Laboratory, which is the first of its kind in the United Kingdom. Professor Whittaker, whose topic was "Periodogram Analysis and Practical Harmonic Analysis," said that these methods of analysis were designed to deal with the results furnished by observation in Meteorology, Astronomy, and other sciences. The observations showed that the observed quantity (such as the temperature, the barometric pressure, the height of the seawater in tides, the brightness of a variable star, etc.) fluctuated continually in a way that might be represented by a wavy curve. The mathematician received this wavy curve from the observer, and set to work on it in order to find out what laws governed its undulations. It was found that many of these curves were compounded from a number of distinct oscillations, each of which ran through its course in a definite period, and the mathematician was able to find out these oscillations by his analysis, and so to predict in many cases the future behaviour of the observed quantity.
The third lecture, on the subject of "NonEuclidean Geometry," was delivered at 2 p.m. by Dr Sommerville. After explaining how nonEuclidean Geometry arose from attempts to prove the axiom about parallel lines, the lecturer proceeded to give an exposition of the system of geometry which was discovered by Lobachevsky, in which Playfair's axiom was directly contradicted and the sum of the angles of a triangle was always less than two right angles.
The meetings of the Colloquium, organised by the Edinburgh Mathematical Society, were resumed yesterday in the Mathematical Department of Edinburgh University.
Professor A W Conway delivered his second lecture on "Relativity." In the course of the lecture the relations between the coordinates and time in the two systems of reference were deduced. It was shown that moving objects will appear to be distorted to a fixed observer, and a moving clock runs in advance of the local time in the other system. The question of velocity was next considered, and it was shown that the parallelogram of velocities does not hold good unless the velocities are small, and that the relative velocity of two points moving in the same line and in the same direction is always greater than the difference of the velocities.
Dr Sommerville's second lecture on NonEuclidean Geometry was devoted to the geometry of Riemann, in which parallel lines do not exist, and the sum of the angles of a triangle is always greater than two right angles. After showing how this hypothesis leads to an absolute polar system of points and planes in space, the lecturer gave some examples of the principle of duality. While there are no parallel lines in this geometry, lines in space may be equidistant, and a remarkable surface is obtained by revolving one line about another to which it is equidistant. This surface, discovered by W K Clifford, has the property that the geometry of shortest line upon it is the same as the geometry of Euclid.
Practical work began in the Mathematical laboratory of the University, when the seventy members of the Colloquium were introduced by Professor Whittaker to the process of periodogram analysis, and made practical application of the method to a mass of astronomical data.
Edinburgh Mathematical Colloquium continued its meetings yesterday in the mathematical department of Edinburgh University. Professor A W Conway delivered the third lecture on Relativity. The lecture dealt with the transformation of accelerations and the discussion of quantities which obeyed the same law of transformation as the space and time coordinates. It was then shown that the electrodynamical equations were transformed in equations of the same type, and examples were given to show how from the solution of a problem in electrostatics the solution of a problem dealing with moving distributions of electricity could be obtained. In his third lecture on NonEuclidean Geometry, Dr Sommerville elaborated the conception of the "absolute," the assemblage of points at infinity. It was shown how this figure, which in NonEuclidean Geometry was a conic, real or imaginary, degenerated in Euclidean geometry to a straight line and two imaginary points. The method of determining distance and angle with reference to the absolute was explained, and it was shown how this process reduced the whole of metrical geometry to projective geometry in relation to the absolute. In the second part of the lecture Dr Sommerville considered the question from the point of view of geometry on a curved surface, and showed how concrete representations of the NonEuclidean geometries were obtained by means of certain surfaces which possessed constant measure of curvature. In the new laboratory the members under Professor Whittaker's direction continued the consideration of the example on the magnitude (i.e., a measure of the brightness) of a variable star, and as a result of their periodogram analysis decided that there were two principal periods of 24 and 29 days. By means of this it was possible to predict the magnitude of the star on any future date. At midday the members were photographed in the quadrangle, and in the afternoon a number of the members visited Mortonhall Golf Course, while a number visited the Zoological Gardens on the invitation of Dr Knott, who entertained the company to tea.
The meetings of the Edinburgh Mathematical Colloquium were continued yesterday in the Mathematical Department of Edinburgh University.
The fourth lecture on "Relativity" was delivered by Professor Conway.
The question of nonNewtonian dynamics was introduced, and the fundamental formula deduced and shown to agree with the researches of Kaufmann on the velocity of Betarays from radium. The Newtonian measure of momentum  mass multiplied by velocity  has in this system to be divided by the square root of the excess of unity over the squared ratio of the velocity of the particle to the velocity of light.
In his fourth lecture, Dr Sommerville introduced the subject of the foundations of geometry. The problem was to establish a system of axioms, or assumptions, satisfying the tests of consistency, independence, and categoricalness, and such that the whole of geometry can be developed from these by pure logical deduction. The lecturer confined the discussion to projective geometry, and showed how the necessary assumptions were analysed into their primary constituents. The theorem of Desargues for triangles in perspective, and of Pascal for a hexagon inscribed in a conic appear to be fundamental. When the method of denial was applied to these as to the parallelpostulate, new forms of nonEuclidean geometry emerged. These were called nonDesarguesian and nonPascalian geometries.
Professor Whittaker, in his fourth lecture, described the theory of harmonic analysis whereby from the given data the whole effect of the various oscillations can be divided up into its constituent forms, and under his direction the members began the study of a special example on the brightness of the star R W Cassiopeia. In the afternoon some of the members paid a visit to Baberton golf course. On the invitation of Dr Dunlop a number paid a visit to the Census Office in the General Register House, and saw a number of the machines used in the analysis of the census returns.
The Colloquium, held last week under the auspices of the Edinburgh Mathematical Society came to on end on Friday. Professor Conway delivered his concluding lecture on "Relativity." The ideas of Minkowsky were explained. The transference to moving axes is formally equivalent to a rotation about an axis. The idea of time as a fourth dimension was introduced, and was shown to occupy in the theory a position symmetrical and incapable of being distinguished from the space coordinates.
In his concluding lecture, Professor Whittaker completed the harmonic analysis begun on the previous day, and explained the construction and the theory of Mader's Harmonic Analyser. The Brunsviga and other multiplying and dividing machines were on exhibition.
In his fifth and concluding lecture, Dr Sommerville continued the subject of the foundations of geometry. It was shown how the complete proof of the fundamental theorem of projective geometry requires an assumption of continuity, which in a curious way implies the theorem of Pascal and the commutative law of multiplication.
A vote of thanks to the lecturers and to the Mathematical Society, under whose auspices the Colloquium was held was proposed by Professor Steggall, and cordially responded to by the members.
At the close of the Colloquium in the afternoon the President, Mr Burgess, paid a tribute to the enthusiasm and energy with which the secretary, Mr Comrie, had thrown himself into the work of organising the meeting. The first Mathematical Colloquium in Britain had proved a complete success, a success that was chiefly due to the ability of the lecturers, each an authority on his own subject. No happier combination could have been made than that of the three lecturers, in whom it was remarked that the three nationalities were represented  English, Irish, and Scottish. Mr Burgess ventured the prediction that this, the first, Colloquium would not be the last, and suggested the possibility of a similar meeting at the time of the Napier celebrations next year. Visits were paid to the physical and chemical laboratories of the University and the HeriotWatt College.