Russel Wallace : Alfred Russell Wallace (sic)
Now that a mathematician and astronomer like Dr. Pratt takes up the very same ground as Von Gumpach, it seems time that the matter should be clearly explained; and, with your permission, though neither an astronomer nor mathematician, I will endeavour to do so; and I have the more hope of succeeding because I once felt a difficulty as to the very same point myself.
The fact (universally stated in works on astronomy and geodesy) that degrees of the meridian increase in length towards the poles, on account of the earth's compression at the poles, is, indeed, one well calculated to mystify a mere mathematician, though it is clear enough to anyone who reflects on the various conditions involved in the problem. If we look at the diagram of a sphere, and the space from the equator to the pole be divided into equal parts subtending angles of one degree each at the centre, and we then flatten the poles by cutting off a portion with a curve of greater radius, it is evident that the distance from the pole to the centre of the sphere will be shorter than before, and therefore, that degrees of latitude, measured angularly from that centre, would really diminish in length from the equator towards the poles.
But in our actual rotating globe, the unequally curved surface is one of equilibrium, owing to the varying centrifugal force at different latitudes; and, as degrees of a meridian can only be measured upon the surface by tangents or perpendiculars to it (obtained by the spirit-level or the plumb-line), it follows that a degree at the pole, measured by an angular instrument from the earth's centre, would not represent a degree of latitude, because the curvature of the polar regions has its centre much further off than the earth's centre of gravity, and a degree measured on the surface would therefore be longer. The centre of curvature of the earth's surface rarely coincides with the centre of gravity, and a plumb-line will therefore not always point directly to that centre. It will do so only at the equator and the pole. Everywhere else adjacent plumb-lines will meet at points within or beyond the centre, according as the curvature of the surface is less or greater than the mean curvature of the globe. The flattened polar regions are, for the geometer, portions of a larger sphere; the protuberant equator (as far as latitude is concerned) is part of a smaller one; and degrees of the meridian measured on these parts must be respectively longer and shorter than what would be due to the mean curvature of the globe.
These considerations seem so very obvious, that I am almost afraid I have mistaken Dr. Pratt's meaning. I hope, however, that the explanation here given may be useful to some young astronomers, as I do not recollect seeing it in any popular work.
1. That a plumb-line everywhere points to the centre of the earth.
2. That the earth, at the sea level, is not a sphere.
If both are true, it follows that there are places where the plumb-line is not at right angles to the ocean surface, so that the water must there stand permanently out of level. In other words, the forces that determine the direction of the plumb-line, and those that determine the level of fluids, are not the same at the same points on the earth's surface. Will he explain this little difficulty in the way of his peculiar view?