Triumphs and Wonders of the 19th Century: The True Mirror of a Phenomenal Era / A volume of original, entertaining and instructive historic and descriptive writings, showing the many and marvellous achievements which distinguish an hundred years of material, intellectual, social and moral progress

by James P. Boyd

V. GEODESY.

It may be that many, who have read of the incredulity of all Europe when the voyages of navigators during the fifteenth and sixteenth centuries first demonstrated the sphericity of the earth, will be surprised to learn that this knowledge had been acquired almost two thousand years before, and had since then been forgotten. To Eratosthenes, a Grecian, belongs the honor of first making a measurement (about the year 230 B. C.) of the size of the earth, which, while very rude and inaccurate, used the same fundamental principle as is now employed by geodesists. But the appliances of those ancient Grecians and of the Arabians, who later carried on the work, were exceedingly crude. Even during the sixteenth and seventeenth centuries, when the French, English, and Dutch were working very hard on the problem, and were gradually obtaining results which came closer and closer to those now known to be correct, the appliances for measuring angles were so rough and inaccurate that it was only possible to assert that the earth is spherical, with a diameter of about 7900 miles. The seventeenth century was nearly past when Picard first used spider lines to determine the “line of collimation,” or the true line of sight, in a telescope. This marked a new era in methods of work, but the eighteenth century was about half gone when it was first authoritatively proven that the earth is not a sphere, but is more truly an “oblate spheroid,”—such a figure as would be obtained by flattening a sphere at the poles. Some idea of the accuracy of the work done, even at this stage, may be obtained by considering that the computed flattening is so slight that if we had a perfect reproduction of the earth, reduced to a diameter of 12 inches, the flattening would be less than 1/25 of an inch—almost imperceptible even to a trained eye. The very highest mountain would be considerably less than 1/100 of an inch in height on such a sphere.

The present marvelous state of the science is due to the great improvements which have been made in the construction and use of angle-measuring instruments and of “base bars;” also to the development of the mathematical theory and processes involved, notably that of the “method of least squares.” As an illustration of the accuracy attainable in the construction of theodolites, the writer recently made an elaborate test of the error of the centering of one of these angle-measuring instruments. Of course no direct measurement is possible. The result is based on a long series of observations, which, when combined according to certain mathematical principles, will give the desired result. The error was thus computed to be forty-two millionths of an inch. To realize what is meant when an angle is measured with a “probable error” of a few hundredths of a second of arc, it should be remembered that one second of arc on a circle 10 inches in diameter is less than 1/40000 of an inch. The accuracy which has been attained in the measurement of base lines is not easily realized by a layman. An engineer realizes the practical impossibility of measuring a line twice and obtaining precisely the same result to the finest unit of measurement. The initiated are therefore able to appreciate the achievement of measuring a base line having a length of over nine miles, with a “probable error” of less than one five-millionth of its length. The words “probable error,” as used above, have a scientifically exact meaning, but they may be taken by the uninitiated as representing a measure of the precision obtained.

At about the close of the last century the great mathematician, Laplace, had declared that the results of the surveys which had then been made were inconsistent with the theory that the form of the earth is exactly that of an oblate spheroid. That form would require that the equator and all parallels of latitude shall be true circles, and that all meridian sections shall be equal ellipses. Laplace showed that the discrepancies between the actual results obtained and the results which the theory would call for are too great to be considered as mere inaccuracies in the work done. With the extension, during this century, of the great geodetic surveys, carried on by the various governments of the world, more and more evidence has developed that the meridian sections of the earth are not equal, which is equivalent to saying that the equator is not a perfect circle. This has led to the next stage, which has been to prove that the form of the earth may be more closely represented by an “ellipsoid” than by a spheroid, that is, that every section of the earth is an ellipse. Several calculations have been made to determine the length and location of the principal axes of such a figure. But these calculations are considered unsatisfactory, because evidence has developed that the true form of the earth cannot be represented even by an ellipsoid. This figure is symmetrical above and below the equator. There are reasons for believing that the southern hemisphere of the earth is slightly larger than the northern, and that the form of the earth is more nearly that of an “ovaloid,”—a figure of which the ordinary hen’s egg is an exaggerated example.

All the above forms, the sphere, spheroid, ellipsoid, and ovaloid are geometrical forms which represent with more and more exactness the true form of the earth, but even this increasing exactness will not account for the discrepancies and irregularities which have been found at various places, and which cannot be explained on the ground of inaccurate work. Geodesists have been forced to the conclusion that the true form of the earth is not a regular geometrical form, but is a “geoid,” that is, like the earth and like nothing else, unless we admit the exaggerated comparison that it is “like a potato.” It should be understood that the words “form of the earth” do not refer to the actual surface of mountain, valley, or ocean bottom, but to the actual ocean surface, and to the surface which the free ocean would assume if it could penetrate into the heart of the continents. The astounding accuracy of the work done may be appreciated when we consider that the differences between the “geoid” and the more accurate mathematical forms are distances which should be measured in feet rather than in miles. For many purposes, it is sufficiently exact to consider the earth as a sphere. For some very precise work it is necessary to consider it as a spheroid. The more exact forms have little or no utilitarian value, and the vast amount of work that has been spent on these researches has been due to man’s thirst for knowledge as such,—due to the same enthusiasm which advances the sciences in fields which only broaden man’s knowledge of the world in which we live.