# Popular Science Monthly/Volume 19/October 1881/How the Earth Is Weighed

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* HOW THE EARTH IS WEIGHED.
By Dr. OTTO WALTERHÖFER.
EFFORTS have been made, at all times in which the spirit of investigation can be said to have existed, to ascertain the condition of the interior of the earth. There has been no lack of unfounded assumptions on the subject, and fanciful hypotheses were held even down to a period in which correct conclusions had been reached upon ​it—to the beginning of the nineteenth century. Alexander von Humboldt relates that he and Sir Humphry Davy were several times invited by Captain Symmes to join an expedition into the interior of the earth, which was represented as a hollow sphere having a large opening at the eighty-second parallel of north latitude. The idea of the existence of a hollow space within the earth was set at rest by the measurement of the average density of the planet, and the contrary view was advanced that the globe is a mass of great specific gravity. The constituency of this mass, whether it is fluid or solid, with only local bubble-like spaces, filled with fluid matter, has not been determined; but the calculations that have been made contradict the theory of a wholly fluid interior.
Several methods have been adopted for ascertaining the mean density of the earth, to the older of which a more accurate method has been added within a few years. An account of the methods hitherto adopted, and the results obtained by them, is here given.
Determination from the Deflection of the Plum-Line.—Newton first suggested that the specific gravity of the earth could be ascertained by means of the plumb-line, but he made no effort to apply his suggestion. The thought was a sequence of his law of gravitation, on which all the methods that have been employed have been based. That law declares that all bodies exert an attractive force upon each other in direct proportion to their masses and in inverse proportion to the square of the distance of their centers of gravity from each other. Accordingly, a body hanging by a line, which over a level surface would be drawn by the earth's attraction into a direction with reference to its point of suspension, the prolongation of the line of which would pass through the center of the earth—that is, would be perpendicular, or plumb—would be attracted and turned away from the perpendicular by a mass like a mountain in the neighborhood. If, now, the amount of this diversion and the size of the mass exercising the deflecting influence were known, then the mass of the earth, and from this in connection with the shape and size of the earth, its mean density, could be computed. The diversion of the plummet from its perpendicular direction is, however, too minute to make a direct measurement possible, and the following method has, therefore, been adopted: In Fig. 1, let K L be a part of the surface of the earth, and G an isolated mountain. A plumb-line at the point A, at the foot of the mountain, and one at B, several miles from it, would take such directions in case the earth were a perfect sphere that the prolongation of the lines would intersect each other at the center of the earth, and form the angle x, with the sides C Z and C Z″, Z and Z″ representing the zeniths at A and B. The zenith-distance v, of any suitable fixed star S, in the neighborhood of Z, may be easily obtained by direct measurement. Let also the zenith-distance of the same star at the point B, which is equivalent to the angle u, be determined. The lines ​A S and B S', representing the direction of the star S, as seen from the points A and B, may, in consequence of the immense distance of the star from the earth, be regarded as parallel. On account of the proximity of the mountain G, the plumb-line does not take the direction
Fig. 1.
A Z, but is deflected toward the mountain, so that it gives the direction A Z' as the apparent vertical, and Z' as the apparent zenith. On this account, the zenith-distance of the star is increased by the angle a, to a degree that is represented by the angle m. The prolonged plumb-lines B Z" and A Z' consequently do not form the angle x at the center of the earth, but another angle, y, which differs from x by the magnitude a, wherefore, a $=$ x $-$ y. If, now, we imagine the line of direction A S prolonged backward, an equivalent of the angle u is formed at T, and by the lines A C' and A T the angle m, equal to the observed zenith-distance at A. But u being the external angle of a triangle, $=$ m $-$ y, or y $=$ m—u; and since a is equal to x $-$ y, if we substitute for y the difference m $-$ u, a $=$ x $+$ u $-$ m. The angles u and m have been obtained by observation as zenith-distances of the fixed star S S', and we have only to obtain the value of the angle x, which is deduced from a trigonometrical measurement of the arc A B. The mass of the ​mountain which diverts the lead is found by a calculation of its form, magnitude, and density, and the mean density of the earth is afterward obtained by a calculation based upon the following data: Let A C (Fig. 2) represent the amount and direction of the attraction which the mountain exercises on the plummet, A B that of the earth upon the same; then A G represents the resultant attraction to which the lead is subjected. If, further, we make R represent the distance of the earth's center, and r that of the center of gravity of the mountain, from the lead, and M and m respectively, the masses of the earth and of the mountain, then we have, according to the law of attraction, $\scriptstyle {\frac {M}{R^{2}}}:{\frac {m}{r^{2}}}::AB:AC$, or since $\scriptstyle AC=BG,{\frac {M}{R^{2}}}:{\frac {m}{r^{2}}}::AB:BG.$ From this proportion the mass and density of the earth are deduced by a series of mathematical formulas which it is not necessary to give in detail here. Fig. 2. Proceeding by this method, Maskelyne and Hutton undertook, between 1774 and 1776, the first efforts to estimate the specific gravity of the earth. They conducted their experiments near Mount Shehallien in Perthshire, Scotland, and found that the lead was deflected by the mountain to the amount of fifty-three seconds, whence they calculated the mean density of the earth to be 4·7. Making use of the observations of these two philosophers, Playfair and Seymour, after corrected calculations of the density of Shehallien, obtained a mean density of 4·7113. Although no theoretical objections can be offered to the manner in which these observations were applied, great exactness can not be claimed for the results, because the calculations of the mass of the mountain, of its mean density, and of the distance of its center of gravity from the lead, were based on estimates, and liable to errors.
Determination by Means of the Pendulum.—A pendulum which is forced out of the vertical direction tends to resume it as soon as the deflecting force is removed. Its momentum carries it beyond the vertical position, and it therefore swings back and forth in times proportionate to its length. The durations of single oscillations of the same pendulum may be considered to be equal to each other if the departure from the vertical does not exceed five degrees. The cause of the oscillations is gravity, or the attractive power of the earth. Since this force diminishes as the square of the distance from the earth's center increases, its amount at different elevations above ​the surface may be exactly calculated. The time required for the vibration of the pendulum is, in consequence of the same law, longer at heights above, shorter at points below the surface of the earth, than at the surface itself; hence it is easy to calculate the time of an oscillation at any given elevation. It is necessary, however, in order that the time calculated in this manner may agree with the result actually observed, that the surface of the earth at the given point shall be plane, and form part of an exact sphere. Mountains near the place of observation cause the attraction on the ball to be stronger than is contemplated in the calculation, and make the oscillations more rapid. The difference between the calculated and observed rate of oscillation will give the amount of influence which the mountain exerts. From this, the relative masses of the mountain and the earth being known, the mean density of the earth may be calculated by a series of formulas similar to those by which it is computed in the method just described. This method is liable to the same defects as the former one; that is, that the elements of the mountain on which the calculations are based are estimated, not accurately measured.
Carlini, Biot, and Matthieu employed it in 1824, Carlini selecting Mont Cenis as his point of observation, the other philosophers performing their experiments at Bordeaux. Their calculations gave a mean density of 4·83. Two other philosophers, Julius and E. Schmidt, calculating from the same observations, obtained, the former 4·95, the latter 4·84. Adopting a converse method from that of Carlini, Drobish, in 1826, measured the duration of the oscillations of the pendulum in a mining-shaft at Dolcoath, in Cornwall, and obtained 5·43.
Determination by Means of the Torsion Balance.—The torsion balance employed in measuring the density of the earth consists
Fig. 3.
of a straight rod a b (Fig. 3) of as uniform dimensions as possible, made of wood or metal, hanging by the cord c d, and supporting at its ends the balls a and b. A small mirror at d, in the middle of the ​rod, on which a perpendicular beam of light is made to fall, indicates, by means of a graduated circle engraved upon it, the most minute horizontal deflections of the balance. Two leaden balls. K and K' are brought within a suitable distance of the balls a and b, exercise an attractive force upon, them, and cause an horizontal deflection of the balance, in a direction opposed to the torsion force of the cord, the value of which may be ascertained by measurement. From this value is computed the force of the attraction which the leaden masses K and K' exercise upon a and b. Since the masses of the four balls, their relative distances from each other, and the amount of the attraction exerted upon them by the earth (which is given by the absolute weight of the balls), are all measurable, the ratio of the mass of the earth to the masses of the balls K and K' can be calculated, and from this, by the process already given, the mean density of the earth.
The results obtained by this method have a considerable degree of trustworthiness, for clear determinations are obtained in which errors are possible only in a small degree. The method was used by Cavendish in 1798, whose calculations gave 5·48, by Reich in Freiberg in 1837, who obtained 5·44, and by Baily in 1842, who obtained 5·6747. Reich repeated his experiments with improved apparatus between 1847 and 1850, using tin balls instead of leaden ones, and twisted copper wires or double iron wires instead of cord, and obtained 5·5756, a value which is often written briefly as 5·58. Hutton calculated the specific gravity of the earth from Cavendish's observations at 5·32, and E. Schmidt at 5.52.
Determination by Means of the Two-Armed Balance.—The idea of using the balance as means of measuring the mean density of the earth originated with the physicist Jolly, who suggested its application to this purpose in describing some improvements he had made in the instrument to increase its sensitiveness. The application was made by H. Poynting, in Manchester, who adopted the following method: Instead of a scale, he attached a weight (b Fig. 4), of 452·92 grammes to the end of a rod six feet in length, to which he opposed a counter-weight in the scale at the other end of the balance; a ball, C, weighing 154·220·6 grammes was brought to a position perpendicularly under b, when the mutual attraction of the two bodies occasioned a disturbance of the balance to the amount of 0·01 of a milligramme. The weight of the two mutually attracting bodies and the amount of attraction exerted upon them by the earth being known, and the distance apart of their centers of gravity having been carefully measured, Poynting calculated the mean density of our planet at 5·69, with a probable error of 0·15.
The approximate agreement of the results obtained by these four methods authorizes us to conclude that the masses of the interior of the earth possess a great density. If we consider, with Alexander von ​Humboldt, that the dry continental parts of the crust of the earth have a mean density of from 2·4 to 2·6, and the dry and oceanic parts of 1·5, and accept Reich's later estimate of the mean total density at 5·58, then the inaccessible, internal parts of the earth must have a
Fig. 4.
mean specific weight of 9·66. Only the metals among the bodies in the accessible parts of the earth possess so great a density; we have a right, then, to believe that the nucleus of the earth possesses a metallic constitution.
The density of the other heavenly bodies is deduced from that of the earth, by observing the amount of attraction which two bodies exert upon each other, and upon a third, and, having ascertained the distances apart of the three bodies, calculating their mutual densities by approved mathematical formulas.—Die Natur.