No native born citizen ever carried to Europe a more pronounced spirit of personal independence than he did. His stories of experiences with officials on the other side of the Atlantic were a source of much entertainment for his friends. In the later years of his life his thoughts turned more willingly to the other shores of the Atlantic, he had made warm friends there, and he looked forward with much satisfaction to the few weeks in Paris which generally were the end of his foreign excursions for the winter. Here in the company of kindred spiritsAssociates in the Institute of France and others he spent days of real enjoyment, speaking the language which belonged to his father if not to his mother and which never had become at all unfamiliar to him. The theatres of the better sort attracted him and his distance from the demands of his active life here left him free to indulge in his always temperate pleasures. Notwithstanding the very serious illness of his early life his originally slender but vigorous frame bore him safely through a life of more than the usual exposures in the varied hardships of a mining camp and journeys which were often perilous. He was spared the usual defects of advancing years and carried to the end a clear head, unimpaired senses and an active body. On Easter morning, March 27th, 1910, on board the Steamer Adriatic in mid ocean he passed from sleep to death without a struggle and the last great mystery was revealed to him who had dealt with the immensities of time and space in all the oceans of the globe. It was well known to some of Agassiz's friends that he had bestowed much thought upon a plan for giving to this Academy a more satisfactory house than any it had yet had. He had made provision in his will for a bequest to the Academy which would have given it a substantial aid in this direction. He, however, had promised himself a more immediate gratification of this wish and on 16 October, 1909, wrote to President Trowbridge offering to erect upon the land already owned by the Academy and the adjoining lot which he had recently purchased a building which should become, to use his own words, "a scientific and literary Club," while remaining the domicile of the Academy. He had caused plans to be prepared by Mr. S. F. Page, for a building to be erected on this spot not merely a house for the Academy but a home for its members, a place to which they would gladly at all times come, to which they might bring their friends and associates from other parts of the country or from foreign lands. It was quite clear to those who were most familiar with his plans, that the house was destined to have all the attractive features which he knew so well to give to his own dwellings. His sons in quick response to the father's wishes with a generous piety have carried out his plans. Mr. Page, the architect, had submitted his sketches to Mr. Agassiz and had had frequent conferences with him before he left the country in December, 1909. His death on March 27, 1910, of necessity caused some delay in the progress of the work, but the plans had been so fully developed that there seemed no doubt as to his intentions and the architect under the direction of the sons and of your committee has faithfully and successfully brought the building to completion. Kings and ambitious noblemen have in other lands and other times been patrons of learned societies and have provided sumptuous accommodations for them. Our house is believed to be the only abode of a scientific society built by a member of the body and devoted to the unrestricted uses of his fellows. If Agassiz had lived to see the completion of this house, it is safe to say that neither his name not his features would have appeared upon these walls. What his singular modesty would have forbidden to him living has been done in the one instance by the authorities of the Academy, and in the other by the loving hands of one of his own family. In the great Museum at Cambridge is the monument of two great men of science laboring in the service of science alone. Here in this pleasant house and home may their associates and successors for all time remember the gracious spirit of him who asked only of his fellows a kindly remembrance. May we not speak of him in the words which our own poet used in describing another of our greatest and best loved associates, The wisest man could ask no more of fate To feel mysterious Nature ever new; To touch, if not to grasp, her endless clue, O friend of this house and all who gather here, not of a day but for long years to come may your place still be here to welcome by this visible presence the generations of this Academy, till this solid structure which you have built and all that it contains shall sink in dust. Proceedings of the American Academy of Arts and Sciences. VOL. XLVIII. No. 3.-JULY, 1912. A THEORY OF LINEAR DISTANCE AND ANGLE. BY H. B. PHILLIPS AND C. L. E. MOOre. A THEORY OF LINEAR DISTANCE AND ANGLE. BY H. B. PHILLIPS AND C. L. E. MOORE. Presented May 8, 1912, by H. W. Tyler. Received May 6, 1912. INTRODUCTION. 1. IN a recent article1 we developed for the plane a theory of distance and angle such that points equally distant from a fixed point lie on a line and lines making a given angle with a fixed line pass through a point. On account of this property we have called this distance linear. In the present paper we extend this theory to higher dimensions. Because of the increased complexity, the synthetic method of the previous discussion cannot be used here and since we know none better we have adopted that of Grassmann. In the first part of the paper we have shown how the extensive quantities of Grassmann can be regarded as matrices and the progressive and regressive multiplication interpreted as simple operations performed upon these matrices. In this way we develop as much of the Grassmann analysis as is needed for our purpose. We then determine for any two spaces R, R' of the same dimension, a distance or angle RR' having the property that if this invariant is constant and either of the spaces fixed, the other satisfies a linear relation and such that for three spaces R, R', R" of a pencil RR'+R' R" + R" R = 0. Any distance between points that has these properties is expressible in terms of a hyperplane and a linear line complex. The plane is the locus of infinitely distant points and the complex the locus of minimal lines. If the complex does not degenerate, the hyperplane and line complex in n dimensions determine a point and n-2 other complexes forming altogether n elements which we use for a reference system. This system of elements forms a group under outer multiplication 1 An Algebra of Plane Projective Geometry, Proceedings of the American Academy of Arts and Sciences, Vol. 47, p. 737. |