Popular Science Monthly/Volume 44/February 1894/The Shape of the Earth from a Pendulum
|THE SHAPE OF THE EARTH FROM A PENDULUM.|
IT was thought that a maximum paradox was reached when the quotation ex pede Herculem (from the foot, Hercules) forced its way into use. Hercules, in laying out the stadium, the length of the running course in the Olympian games, used his foot as the unit, and made the stadium six thousand feet long. From this distance, which was preserved, Pythagoras obtained the length of the foot of Hercules, and from an arbitrary ratio between the parts of the body deduced his height, thus restoring from the foot, Hercules.
But we can now propound a greater paradox, and say from a pendulum, the earth. Not the world that one can put in a sling, but the earth's shape. This striving after the shape of the earth has occupied men's attention for centuries; to know this shape they have braved the cold within the Arctic Circle, endured the heat of the equatorial regions, and penetrated India's malarial jungles. Peaks have been climbed, deserts traversed, and hostile tribes subjugated. To the theoretical side of this problem scores of the world's most profound mathematicians have devoted their time, while the practical side has been pushed ahead by the energies of countless troops of observers, artisans, and laborers, supported by the expenditure of millions upon millions of dollars.
While this great work is going on, looking toward a solution of this problem, with staffs of specialists in sixteen nations, employing instruments most complicated and refined, making, as it appears, an onslaught on the earth itself to compel it to yield to direct measurement, it now seems that from a modified form of the device which regulates our clocks—the pendulum—we may expect the most accurate knowledge regarding the earth's shape.
When Galileo deduced from observation that a pendulum is isochronal—that is, would make all its oscillations in the same interval of time whether the arc be long or short—he did not dream that the swinging lamp in the dome of Pisa's great cathedral in the year 1583 would be the prototype of the accurate geodetic instrument of three centuries later.
If the ball of a pendulum be drawn away from the vertical and released, its first impulse is to descend perpendicularly; but being held in restraint by the string, or connecting rod, it does the next best thing, and, keeping as near to this perpendicular direction as possible, it swings down a circular arc whose center is the point of support. When the lowest point of this arc is reached, an amount of energy has been stored up and the ball ascends the other side of the arc until this supply of energy is exhausted; then stopping for an instant the ball again descends, to ascend on the other side, thus adding oscillation to oscillation. Were it not for the resistance of the atmosphere and certain mechanical imperfections these arcs would be the same, but, what is more important, the times of oscillating are the same.
The rapidity with which the pendulum descends depends upon its length and the amount of this impulse to drop vertically. This impulse is known as gravity. Therefore, with a pendulum of constant length the time of oscillation will be dependent upon gravity, and thus time and gravity are determinable one in terms of the other.
Newton had shown that gravity on the earth's surface depended upon distance from the center of the earth, and also the diminishing effect of the revolution of the earth on gravity. To this theory other mathematicians made valuable contributions, notably Clairaut, who demonstrated that the relative lengths of the equatorial and polar radii could be ascertained directly from the force of gravity at the equator and at one of the poles. Then, since the gravity is obtained directly from the time in which a pendulum makes an oscillation and its length, it was necessary to simply swing a pendulum at the equator and at one of the poles to have at once the coveted ellipticity—that is, the ratio of the difference between the equatorial and polar radii to the equatorial radius.
Unfortunately, it has not been possible to swing a pendulum at one of the poles. This inability, however, is made of no moment by a law which gives the value of the polar gravity whenever the gravity of a given place is known, together with the latitude of the place.
From this it appears that the earth's figure becomes known through a determination of the length of a pendulum and the time required for it to make an oscillation at the equator (or near it) and at the pole (or as near to it as possible). If the same pendulum is used and the constancy of its length assured, it becomes necessary to make sure of the length of time required for an oscillation at these two places. Inasmuch as the pendulum appears to stop for an instant when it reaches the highest point in its arc, it is a difficult matter to determine with exactness the time of an oscillation; but if one counts the number of oscillations in an hour, in two hours, or in any number of hours, a simple division will give the time of one oscillation.
The figure of the earth desired is an ideal figure, such a figure as it would have if one could remove all the land now standing higher than the surface of the sea—were a sea to occupy the place of the land. Hence it is the sea-level earth whose figure we want. Newton's law of gravity would require that a pendulum, if raised above the level of the sea, would make its oscillation in a longer time than when swinging at sea level. Therefore it is necessary to know the elevation of the station in order to ascertain the force of gravity in that latitude on the ideal earth.
If the parallels were perfect circles and if observations were absolutely correct, it would be necessary to swing a pendulum at only two points on the earth's surface in order to determine its shape. However, the results obtained by combining observations two and two are not harmonious; not only because the observations may be affected by errors, but the attraction of dense matter immediately beneath a station might seriously impair the observations made there; and as we never know the exact constitution of the earth's crust at any point, it becomes necessary to eliminate, as far as possible, this uncontrollable error by making observations at many places.
The ideal pendulum would consist of a ball of symmetrical form suspended by a wire stiff and uniform. Like all ideal conditions, these are never attained, but a close approximation is sought. In seeking rigidity the pendulum rod must be so large in cross section as to make the instrument cumbrous. This was a serious feature when, in order to avoid slips in counting, it was not thought feasible to use a pendulum that made an oscillation in less than a second of time—that is, a pendulum about thirty-nine inches long. Again, as the pendulum was provided with sharp knife-edges on opposite sides near its upper end, shaped like a V, on which it swung, the greater the weight of the pendulum the more wear there would be on these knife-edges. This becomes a serious matter, as the length of the pendulum is estimated from the line of support furnished by these same knife-edges. Then, too, the swinging of a large and heavy pendulum was liable to induce a swinging motion in its support, unless the latter were exceedingly rigid, thereby vitiating the results.
Several years ago it was realized that the resistance of the atmosphere would vary with different conditions of moisture and density, and hence retard the pendulum more at some times than at others, more at some elevations than at others. Therefore it seemed necessary, in the absence of any well-accepted correction for these hurtful resistances, to swing the pendulum always under the same atmospheric pressure and surrounded by similar conditions as to moisture. This could be done only by inclosing the entire pendulum in a chamber in which the air could be maintained at the same density and dryness. One can readily see how difficult this would be with an apparatus more than four feet in length and weighing many pounds.
Although the shortcomings of the ordinary pendulum forced themselves into recognition one by one, still the readiness with which observations could be made, in comparison with direct measures of the earth, has caused it to be regarded as a most important geodetic instrument.
As early as 1735 observations were made at St. Domingo, Panama, and Quito, using a plummet suspended by a thread of the aloe; about the same time the party sent to measure an arc of the earth within the polar circle swung a pendulum within twenty-four degrees of the pole. Lacaille carried a pendulum to the Cape of Good Hope and the Isle of France, Legentil took one on his voyage to the Indian Ocean, Phipps on his voyage toward the north pole, and Malaspina while visiting the Spanish possession in the Western hemisphere-Biot, Arago, and Borda were perfecting the pendulum and measuring gravity at different places in France; the labors of Ross, Kater, Foster, and Sabine were giving to England the supremacy in matters pertaining to gravity determinations; while Bessel, in Germany, was busy investigating corrections for the weight of air by swinging a pendulum in a vacuum, then in gases of known elasticity.
The French, not willing to follow in the lead of others, sent out expeditions under Freycinet and Duperrey, who brought back pendulum data that still find their places in the discussion of the earth's figure. These were followed up by Sawitsch in Russia, Plantamour in Switzerland, Basevi in India, and Peirce in the United States.
During all this time attention was given chiefly toward perfecting the mechanism of the pendulum without changing materially its form. It became heavier rather than lighter; the supports were correspondingly more cumbersome; the knife-edges subjected, because of increased weight, to greater danger of dulling, while theory was continually devising corrections because of atmospheric pressure and viscosity. The defects in structure took on an exaggerated magnitude, and the chance to discover absolute corrections appeared hopeless when the rapid advance in physical science set a limit of error to direct observation, and it looked almost as if the pendulum would be a doomed instrument of investigation.
Just in this emergency Superintendent Mendenhall, of the United States Coast and Geodetic Survey, called to his aid his experienced assistants to so modify the form of the pendulum as to bring it into its proper sphere of usefulness. Skilled as a physicist, it was not possible for him to waste time stumbling through the mistakes detected by the experience of others. He started anew where they had stopped.
The first point reached was the important one. By an application of the principle of coincidences first employed by Foucault in 1850 in determining the velocity of light, it became possible to ascertain the number of oscillations made by a pendulum within an interval of several minutes without counting them. This at once suggested that the danger of making a slip in counting the oscillations, should they be as frequent as two in a second, might be avoided, and thus a short or half-second's pendulum be employed. This shortening resulted, of course, in a lightening, and each ounce of diminution added to the safety of the knife-edge, thereby contributing to the permanency of the pendulum. Nor was this all: the parts now became of such wieldy size that the whole could be incased in a chamber sufficiently air-tight to maintain a constant atmospheric pressure either by exhausting a portion of the air near sea level or forcing air in when stations at great altitudes are occupied.
With a pendulum so compact one can visit places heretofore inaccessible with the larger forms, and require distant islands and inhospitable climes to give a voice in determining the earth's shape. Large land areas are needed for the measurement of arcs, and hence less than one fourth of the earth only is available to determine geodetically its shape. But now each party sent out on a voyage of discovery or to observe astronomic phenomena can take one of these compact pendulums along and make stations within the bounds of the three fourths so that they may not be encompassed by a figure dictated by the minority.
Now that differential methods are used almost universally that is, comparing the times of oscillation of the same pendulum at different places—it is essential that the length may continue to be what it was when swinging at the base station, or station where absolute gravity had been determined. Supposing that due correction has been made for such changes in length as would be occasioned by differences of temperature, the only possibility for variation in length could come from disarrangement or wear of the knife-edge. Any chipping of this knife-edge—made of agate—could not be rectified, and dullness could not be removed without making in so doing a new pendulum, thereby destroying its differential value. In swinging, this agate V rests on a steel plane, and as this plane, forming no part of the pendulum proper, is less liable to injury or derangement, the idea occurred to the survey officers to let these parts change places. So now we have a pendulum with a slit in the upper end of its rod, having for its upper surface a plane of hard steel. This plane rests on the agate knife-edge which projects into the slit. If now the agate becomes dull or injured it can be repaired or a new one substituted, and the pendulum remain the same.
As already stated, the usual procedure has been, when observing with a pendulum, to note the number of oscillations made in a given interval of time; then, by dividing this interval by the number, the time of one oscillation is obtained. Of course, the clock or chronometer which furnished this time might be running too fast or too slow. However, it has been customary to determine the rate of the timepiece by making an astronomic observation before swinging the pendulum and then again after. This would give the amount of time gained or lost in the interim, but does not prove that this amount, or even more, was not the change within the interval of swinging, or that the keeping of time was not perfect while the pendulum was swung, and the error occurred either before or after.
From this uncertainty arose the need to eliminate time error, and it has been most ingeniously met in the survey pendulum. Here two pendulums are employed—one at one station and one at another, connected by a telegraph wire. Each is made to record its own coincidence with a beat of one and the same chronometer, so that if the chronometer has a constant rate for a minute or two it is sufficient. The chronometer at the other station is then used to eliminate such errors as might arise in the transmission of signals. In this way the relative time of the oscillations of two pendulums is known with absolute accuracy, and from these relative times relative gravity is obtained, and from relative gravity we have relative distances to the earth's center, or the shape of the earth.
In this enlightened age it is not necessary to enumerate reasons why we should know the shape of the earth. It enters as a potent factor in astronomic computations; it is indispensable in map-making, and no boundary line can be drawn without its aid. Besides carrying on the special cartographic functions prescribed for it by law, the United States Coast and Geodetic Survey has lost no opportunity to improve our knowledge of this important factor, and under no régime has the survey so completely filled this dual purpose as under the superintendency of Prof. Mendenhall.
We do not measure the earth with a span, but with a pendulum one span in length we find its shape.
The publication of an Index to the Names and Authorities of all Known Flowering Plants and their Countries, which was contemplated by Mr. Darwin, has been undertaken at the Clarendon Press, under the direction and supervision of Sir J. D, Hooker. Part I of the work is now ready, and Part II is well advanced.