1. 2. VOLUME 48. BELL, LOUIS.- On the Ultra Violet Component in Artificial Light. pp. 1-29. 2 pls. May, 1912. 40c. WALCOTT, HENRY P.- Alexander Agassiz. pp. 31-44. June, 1912. 30c. 3. PHILLIPS, H. B. and MOORE, C. L. E.-A Theory of Linear Distance and Angle. pp. 45-80. July, 1912. 50c. 4. 5. 6. 7. 8. 9. 10. 11. CHIVERS, A. H.- Preliminary Diagnoses of New Species of Chaetomium. pp. 81-88. July, 1912. 20c. KENT, NORTON A.-A Study with the Echelon Spectroscope of Certain Lines in the Spectra of the Zinc Arc and Spark at Atmospheric Pressure. pp. 91-109. 2 pls. August, 1912. 50c. KENNELLY, A. E., and PIERCE, G. W.-The Impedance of Telephone Receivers as affected by the Motion of their Diaphragms. pp. 111-151. September, 1912. 70c. THAXTER, ROLAND.- New or Critical Laboulbeniales from the Argentine. pp. Culture Studies of Fungi producing Bulbils and pp. 225-306. October. 1912, $1.50. BRIDGMAN, P. W.-Thermodynamic Properties of Liquid Water to 80° and 12000 Kgm. September, 1912, pp. 307-362. 70c. THAXTER, ROLAND. - Preliminary Descriptions of New Species of Rickia and Trenomyces. September, 1912. pp. 363-386. 40c. WILSON. EDWIN B., and LEWIS, GILBERT N. The Space-Time Manifold of Relativity. The non-Euclidean Geometry of Mechanics and Electromagnetics. November, 1912. pp. 387-507. $1.75. Proceedings of the American Academy of Arts and Sciences. VOL. XLVIII. No. 11.- NOVEMBER, 1912. THE SPACE-TIME MANIFOLD OF RELATIVITY. THE NON-EUCLIDEAN GEOMETRY OF MECHANICS AND ELECTROMAGNETICS. BY EDWIN B. WILSON AND GILBERT N. LEWIS. THE SPACE-TIME MANIFOLD OF RELATIVITY. THE NON-EUCLIDEAN GEOMETRY OF MECHANICS BY EDWIN B. WILSON AND GILBERT N. LEWIS. Introduction. 1. The concept of space has different meanings to different persons according to their experience in abstract reasoning. On the one hand is the common space, which for the educated person has been formulated in the three dimensional geometry of Euclid. On the other hand the mathematician has become accustomed to extend the concept of space to any manifold of which the properties are completely determined, as in Euclidean geometry, by a system of self-consistent postulates. Most of these highly ingenious geometries cannot be expected to be of service in the discussion of physical phenomena. Until recently the physicist has found the three dimensional space of Euclid entirely adequate to his needs, and has therefore been inclined to attribute to it a certain reality. It is, however, inconsistent with the philosophic spirit of our time to draw a sharp distinction between that which is real and that which is convenient,1 and it would be dogmatic to assert that no discoveries of physics might render so convenient as to be almost imperative the modification or extension of our present system of geometry. Indeed it seemed to Minkowski that such a change was already necessitated by the facts which led to the formulation of the Principle of Relativity. 2. The possibility of associating three dimensional space and one dimensional time to form a four dimensional manifold has doubtless occurred to many; but as long as space and time were assumed to be wholly independent, such a union seemed purely artificial. The idea of abandoning once for all this assumption of independence, although fore-shadowed in Lorentz's use of local time, was first clearly stated by 1 See, for example, H. Poincaré, La Science et l'Hypothèse. |