The Reminiscences of an Astronomer

by Simon Newcomb

The idea that the facts of nature are to be brought out by observation is one which is singularly foreign not only to people of this class, but even to many sensible men. When the great comet of 1882 was discovered in the neighborhood of the sun, the fact was telegraphed that it might be seen with the naked eye, even in the sun's neighborhood. A news reporter came to my office with this statement, and wanted to know if it was really true that a comet could be seen with the naked eye right alongside the sun.

"I don't know," I replied; "suppose you go out and look for yourself; that is the best way to settle the question."

The idea seemed to him to be equally amusing and strange, and on the basis of that and a few other insipid remarks, he got up an interview for the "National Republican" of about a column in length.

I think there still exists somewhere in the Northwest a communistic society presided over by a genius whose official name is Koresh, and of which the religious creed has quite a scientific turn. Its fundamental doctrine is that the surface of the earth on which we live is the inside of a hollow sphere, and therefore concave, instead of convex, as generally supposed. The oddest feature of the doctrine is that Koresh professes to have proved it by a method which, so far as the geometry of it goes, is more rigorous than any other that science has ever applied. The usual argument by which we prove to our children the earth's rotundity is not purely geometric. When, standing on the seashore, we see the sails of a ship on the sea horizon, her hull being hidden because it is below, the inference that this is due to the convexity of the surface is based on the idea that light moves in a straight line. If a ray of light is curved toward the surface, we should have the same appearance, although the earth might be perfectly flat. So the Koresh people professed to have determined the figure of the earth's surface by the purely geometric method of taking long, broad planks, perfectly squared at the two ends, and using them as a geodicist uses his base apparatus. They were mounted on wooden supports and placed end to end, so as to join perfectly. Then, geometrically, the two would be in a straight line. Then the first plank was picked up, carried forward, and its end so placed against that of the second as to fit perfectly; thus the continuation of a straight line was assured. So the operation was repeated by continually alternating the planks. Recognizing the fact that the ends might not be perfectly square, the planks were turned upside down in alternate settings, so that any defect of this sort would be neutralized. The result was that, after they had measured along a mile or two, the plank was found to be gradually approaching the sea sand until it touched the ground.

This quasi-geometric proof was to the mind of Koresh positive. A horizontal straight line continued does not leave the earth's surface, but gradually approaches it. It does not seem that the measurers were psychologists enough to guard against the effect of preconceived notions in the process of applying their method.

It is rather odd that pure geometry has its full share of paradoxers. Runkle's "Mathematical Monthly" received a very fine octavo volume, the printing of which must have been expensive, by Mr. James Smith, a respectable merchant of Liverpool. This gentleman maintained that the circumference of a circle was exactly 3 1/5 times its diameter. He had pestered the British Association with his theory, and come into collision with an eminent mathematician whose name he did not give, but who was very likely Professor DeMorgan. The latter undertook the desperate task of explaining to Mr. Smith his error, but the other evaded him at every point, much as a supple lad might avoid the blows of a prize-fighter. As in many cases of this kind, the reasoning was enveloped in a mass of verbiage which it was very difficult to strip off so as to see the real framework of the logic. When this was done, the syllogism would be found to take this very simple form:—

The ratio of the circumference to the diameter is the same in all circles. Now, take a diameter of 1 and draw round it a circumference of 3 1/5. In that circle the ratio is 3 1/5; therefore, by the major premise, that is the ratio for all circles.

The three famous problems of antiquity, the duplication of the cube, the quadrature of the circle, and the trisection of the angle, have all been proved by modern mathematics to be insoluble by the rule and compass, which are the instruments assumed in the postulates of Euclid. Yet the problem of the trisection is frequently attacked by men of some mathematical education. I think it was about 1870 that I received from Professor Henry a communication coming from some institution of learning in Louisiana or Texas. The writer was sure he had solved the problem, and asked that it might receive the prize supposed to be awarded by governments for the solution. The construction was very complicated, and I went over the whole demonstration without being able at first to detect any error. So it was necessary to examine it yet more completely and take it up point by point. At length I found the fallacy to be that three lines which, as drawn, intersected in what was to the eye the same point on the paper, were assumed to intersect mathematically in one and the same point. Except for the complexity of the work, the supposed construction would have been worthy of preservation.

Some years later I received, from a teacher, I think, a supposed construction, with the statement that he had gone over it very carefully and could find no error. He therefore requested me to examine it and see whether there was anything wrong. I told him in reply that his work showed that he was quite capable of appreciating a geometric demonstration; that there was surely something wrong in it, because the problem was known to be insoluble, and I would like him to try again to see if he could not find his error. As I never again heard from him, I suppose he succeeded.

One of the most curious of these cases was that of a student, I am not sure but a graduate, of the University of Virginia, who claimed that geometers were in error in assuming that a line had no thickness. He published a school geometry based on his views, which received the endorsement of a well-known New York school official and, on the basis of this, was actually endorsed, or came very near being endorsed, as a text-book in the public schools of New York.