Einstein's Theories of Relativity and Gravitation by Albert Einstein, is part of the HackerNoon Books Series. You can jump to any chapter in this book here. The Reference Frame for Space

The Reference Frame for Space

The mathematician, following the lead of the great French all-around genius, Descartes, shows us very clearly how to set up, for the measurement of space, the framework known as the Cartesian coordinate system. The person of most ordinary mathematical attainments will realize that to locate a point in a plane we must have two measurements; and we could probably show this person, without too serious difficulty, that we can locate a point in any surface by two measurements. An example of this is the location of points on the earth’s surface by means of their latitude and longitude. It is equally clear that if we add a third dimension and attempt to locate points in space, we must add a third measurement. In the case of points on the earth’s surface, this might be the elevation above sea level, which would define the point not as part of the spherical surface of the earth but as part of the solid sphere. Or we may fall back on Dr. Slosson’s suggestion that in order to define completely the position of his laboratory, we must make a statement about Broadway, and one about 116th Street, and one telling how many flights of stairs there are to climb. In any event, it should be clear enough that the complete definition of a point in space calls for three measurements.

The mathematician formulates all this with the utmost precision. He asks us to]* [pick out any point whatever in space and call it O. We then draw or conceive to be drawn through this point three mutually perpendicular lines called coordinate axes, [37]which we may designate OX, OY and OZ, respectively. Finally, we consider the three planes also mutually perpendicular like the two walls and the floor of a room that meet in one common corner, which are formed by the lines OX and OY, OY and OZ, and OZ and OX, respectively. These three planes are called coordinate planes. And then any other point P in space can be represented with respect to O by its perpendicular distances from each of the three coordinate planes—the distances x, y, z in the figure. These quantities are called the coordinates of the point.]272

[To the layman there seems something altogether naive in this notion of the scientist’s setting up the three sides of a box in space and using them as the basis of all his work. The layman somehow feels that while it is perfectly all right for him to tell us that he lives at 1065 (one coordinate) 156th Street (two coordinates) on the third floor (three coordinates), it is rather trivial business for the serious-minded [38]scientist to consider the up-and-down, the forward-and-back, the right-and-left of every point with which he has occasion to deal. There seems to the layman something particularly inane and foolish and altogether puerile about a set of coordinate axes, and you simply can’t make him believe that the serious-minded scientist has to monkey with any such funny business. He can’t be induced to take this coordinate-axis business seriously. Nevertheless, the fact is that the scientist takes it with the utmost seriousness. It is necessary for him to define the positions of points; and he does do it by means of a set of coordinate axes.

The scientist, however, is not interested in points of empty space. The point is to him merely part again of the conceptual machinery which he uses in his effort to run along with the external world. He knows there are no real points, but it suits his convenience to keep track of certain things that are real by representing them as points. But these things are in practically every instance material bodies; and in practically every instance, instead of staying put in one spot, they insist upon moving about through space. The scientist has to use his coordinate system, not merely to define a single position of such a “point,” but to keep track of the path over which it moves and to define its position in that path at given moments.

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This book is part of the public domain. Albert Einstein (2020). Einstein's Theories of Relativity and Gravitation. Urbana, Illinois: Project Gutenberg. Retrieved October 2022.

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The Reference Frame for Space

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