In Dr. Pratt's letter in your journal of April 28, he seems to argue that modern astronomers and geometers are in error as to the true form of the earth. His words are: "In conformity with the assumed oblate figure of the earth, arcs of the meridian should progressively diminish from the equator to the poles. In fact, these arcs become longer with advance in this direction." And he goes on to advance a theory of some polar attraction in space which has drawn out the earth at the poles instead of flattening it, as is commonly, but he thinks erroneously, assumed. Von Gumpach has been long asserting the very same thing, and has importunately called the attention of our Government to the fact, that numbers of vessels are annually lost owing to the impossibility of calculating their true position, so long as navigators mistake the very figure of the globe they are travelling over. But his warnings have been all in vain. The Admiralty persist in refusing to alter the Nautical Almanack, and the philosopher thinks he has just cause of complaint because the Astronomer Royal will neither accept his conclusions nor point out the flaw in his argument.

    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.

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    Sir, --Dr. Pratt maintains two propositions, which are incompatible with each other:--

    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?