FIGURE OF THE EARTH.

"He took the golden compasses, prepared
In God's eternal store, to circumscribe
This universe, and all created things;
One foot he centred, and the other turned
Round through the vast profundity obscure,

In the earliest ages, the earth was regarded as one continued plane; but, at a comparatively remote period, as five hundred years before the Christian era, astronomers began to entertain the opinion that the earth is round. We are able now to adduce various arguments which severally prove this truth. First, when a ship is coming in from sea, we first observe only the very highest parts of the ship, while the lower portions come successively into view. Were the earth a continued plane, the lower parts of the ship would be visible as soon as the higher, as is evident from Fig. 10, page 70.

Fig. 10.

Since light comes to the eye in straight lines, by which objects become visible, it is evident, that no reason exists why the parts of the ship near the water should not be seen as soon as the upper parts. But if the earth be a sphere, then the line of sight would pass above the deck of the ship, as is represented in Fig. 11; and as the ship drew nearer to land, the lower parts would successively rise above this line and come into view exactly in the manner known to observation. Secondly, in a lunar eclipse, which is occasioned by the moon's passing through the earth's shadow, the figure of the shadow is seen to be spherical, which could not be the case unless the earth itself were round. Thirdly, navigators, by steering continually in one direction, as east or west, have in fact come round to the point from which they started, and thus confirmed the fact of the earth's rotundity beyond all question. One may also reach a given place on the earth, by taking directly opposite courses. Thus, he may reach Canton in China, by a westerly route around Cape Horn, or by an easterly route around the Cape of Good Hope. All these arguments severally prove that the earth is round.

Fig. 11.

But I propose, in this Letter, to give you some account of the unwearied labors which have been performed to ascertain the exact figure of the earth; for although the earth is properly described in general language as round, yet it is not an exact sphere. Were it so, all its diameters would be equal; but it is known that a diameter drawn through the equator exceeds one drawn from pole to pole, giving to the earth the form of a spheroid,—a figure resembling an orange, where the ends are flattened a little and the central parts are swelled out.

Although it would be a matter of very rational curiosity, to investigate the precise shape of the planet on which Heaven has fixed our abode, yet the immense pains which has been bestowed on this subject has not all arisen from mere curiosity. No accurate measurements can be taken of the distances and magnitudes of the heavenly bodies, nor any exact determinations made of their motions, without a knowledge of the exact figure of the earth; and hence is derived a powerful motive for ascertaining this element with all possible precision.

The first satisfactory evidence that was obtained of the exact figure of the earth was derived from reasoning on the effects of the earth's centrifugal force, occasioned by its rapid revolution on its own axis. When water is whirled in a pail, we see it recede from the centre and accumulate upon the sides of the vessel; and when a millstone is whirled rapidly, since the portions of the stone furthest from the centre revolve much more rapidly than those near to it, their greater tendency to recede sometimes makes them fly off with a violent explosion. A case, which comes still nearer to that of the earth, is exhibited by a mass of clay revolving on a potter's wheel, as seen in the process of making earthen vessels. The mass swells out in the middle, in consequence of the centrifugal force exerted upon it by a rapid motion. Now, in the diurnal revolution, the equatorial parts of the earth move at the rate of about one thousand miles per hour, while the poles do not move at all; and since, as we take points at successive distances from the equator towards the pole, the rate at which these points move grows constantly less and less; and since, in revolving bodies, the centrifugal force is proportioned to the velocity, consequently, those parts which move with the greatest rapidity will be more affected by this force than those which move more slowly. Hence, the equatorial regions must be higher from the centre than the polar regions; for, were not this the case, the waters on the surface of the earth would be thrown towards the equator, and be piled up there, just as water is accumulated on the sides of a pail when made to revolve rapidly.

Huyghens, an eminent astronomer of Holland, who investigated the laws of centrifugal forces, was the first to infer that such must be the actual shape of the earth; but to Sir Isaac Newton we owe the full developement of this doctrine. By combining the reasoning derived from the known laws of the centrifugal force with arguments derived from the principles of universal gravitation, he concluded that the distance through the earth, in the direction of the equator, is greater than that in the direction of the poles. He estimated the difference to be about thirty-four miles.

But it was soon afterwards determined by the astronomers of France, to ascertain the figure of the earth by actual measurements, specially instituted for that purpose. Let us see how this could be effected. If we set out at the equator and travel towards the pole, it is easy to see when we have advanced one degree of latitude, for this will be indicated by the rising of the north star, which appears in the horizon when the spectator stands on the equator, but rises in the same proportion as he recedes from the equator, until, on reaching the pole, the north star would be seen directly over head. Now, were the earth a perfect sphere, the meridian of the earth would be a perfect circle, and the distance between any two places, differing one degree in latitude, would be exactly equal to the distance between any other two places, differing in latitude to the same amount. But if the earth be a spheroid, flattened at the poles, then a line encompassing the earth from north to south, constituting the terrestrial meridian, would not be a perfect circle, but an ellipse or oval, having its longer diameter through the equator, and its shorter through the poles. The part of this curve included between two radii, drawn from the centre of the earth to the celestial meridian, at angles one degree asunder, would be greater in the polar than in the equatorial region; that is, the degrees of the meridian would lengthen towards the poles.

The French astronomers, therefore, undertook to ascertain by actual measurements of arcs of the meridian, in different latitudes, whether the degrees of the meridian are of uniform length, or, if not, in what manner they differ from each other. After several indecisive measurements of an arc of the meridian in France, it was determined to effect simultaneous measurements of arcs of the meridian near the equator, and as near as possible to the north pole, presuming that if degrees of the meridian, in different latitudes, are really of different lengths, they will differ most in points most distant from each other. Accordingly, in 1735, the French Academy, aided by the government, sent out two expeditions, one to Peru and the other to Lapland. Three distinguished mathematicians, Bouguer, La Condamine, and Godin, were despatched to the former place, and four others, Maupertius, Camus, Clairault, and Lemonier, were sent to the part of Swedish Lapland which lies at the head of the Gulf of Tornea, the northern arm of the Baltic. This commission completed its operations several years sooner than the other, which met with greater difficulties in the way of their enterprise. Still, the northern detachment had great obstacles to contend with, arising particularly from the extreme length and severity of their Winters. The measurements, however, were conducted with care and skill, and the result, when compared with that obtained for the length of a degree in France, plainly indicated, by its greater amount, a compression of the earth towards the poles.

Mean-while, Bouguer and his party were prosecuting a similar work in Peru, under extraordinary difficulties. These were caused, partly by the localities, and partly by the ill-will and indolence of the inhabitants. The place selected for their operations was in an elevated valley between two principal chains of the Andes. The lowest point of their arc was at an elevation of a mile and a half above the level of the sea; and, in some instances, the heights of two neighboring signals differed more than a mile. Encamped upon lofty mountains, they had to struggle against storms, cold, and privations of every description, while the invincible indifference of the Indians, they were forced to employ, was not to be shaken by the fear of punishment or the hope of reward. Yet, by patience and ingenuity, they overcame all obstacles, and executed with great accuracy one of the most important operations, of this nature, ever undertaken. To accomplish this, however, took them nine years; of which, three were occupied in determining the latitudes alone.[5]

I have recited the foregoing facts, in order to give you some idea of the unwearied pains which astronomers have taken to ascertain the exact figure of the earth. You will find, indeed, that all their labors are characterized by the same love of accuracy. Years of toilsome watchings, and incredible labor of computation, have been undergone, for the sake of arriving only a few seconds nearer to the truth.

The length of a degree of the meridian, as measured in Peru, was less than that before determined in France, and of course less than that of Lapland; so that the spheroidal figure of the earth appeared now to be ascertained by actual measurement. Still, these measures were too few in number, and covered too small a portion of the whole quadrant from the equator to the pole, to enable astronomers to ascertain the exact law of curvature of the meridian, and therefore similar measurements have since been prosecuted with great zeal by different nations, particularly by the French and English. In 1764, two English mathematicians of great eminence, Mason and Dixon, undertook the measurement of an arc in Pennsylvania, extending more than one hundred miles.

Fig. 12.

These operations are carried on by what is called a system of triangulation. Without some knowledge of trigonometry, you will not be able fully to understand this process; but, as it is in its nature somewhat curious, and is applied to various other geographical measurements, as well as to the determination of arcs of the meridian, I am desirous that you should understand its general principles. Let us reflect, then, that it must be a matter of the greatest difficulty, to execute with exactness the measurement of a line of any great length in one continued direction on the earth's surface. Even if we select a level and open country, more or less inequalities of surface will occur; rivers must be crossed, morasses must be traversed, thickets must be penetrated, and innumerable other obstacles must be surmounted; and finally, every time we apply an artificial measure, as a rod, for example, we obtain a result not absolutely perfect. Each error may indeed be very small, but small errors, often repeated, may produce a formidable aggregate. Now, one unacquainted with trigonometry can easily understand the fact, that, when we know certain parts of a triangle, we can find the other parts by calculation; as, in the rule of three in arithmetic, we can obtain the fourth term of a proportion, from having the first three terms given. Thus, in the triangle A B C, Fig. 12, if we know the side A B, and the angles at A and B, we can find by computation, the other sides, A C and B C, and the remaining angle at C. Suppose, then, that in measuring an arc of the meridian through any country, the line were to pass directly through A B, but the ground was so obstructed between A and B, that we could not possibly carry our measurement through it. We might then measure another line, as A C, which was accessible, and with a compass take the bearing of B from the points A and C, by which means we should learn the value of the angles at A and C. From these data we might calculate, by the rules of trigonometry, the exact length of the line A B. Perhaps the ground might be so situated, that we could not reach the point B, by any route; still, if it could be seen from A and C, it would be all we should want. Thus, in conducting a trigonometrical survey of any country, conspicuous signals are placed on elevated points, and the bearings of these are taken from the extremities of a known line, called the base, and thus the relative situation of various places is accurately determined. Were we to undertake to run an exact north and south line through any country, as New England, we should select, near one extremity, a spot of ground favorable for actual measurement, as a level, unobstructed plain; we should provide a measure whose length in feet and inches was determined with the greatest possible precision, and should apply it with the utmost care. We should thus obtain a base line. From the extremities of this line, we should take (with some appropriate instrument) the bearing of some signal at a greater or less distance, and thus we should obtain one side and two angles of a triangle, from which we could find, by the rules of trigonometry, either of the unknown sides. Taking this as a new base, we might take the bearing of another signal, still further on our way, and thus proceed to run the required north and south line, without actually measuring any thing more than the first, or base line.

Fig. 13.

Thus, in Fig. 13, we wish to measure the distance between the two points A and O, which are both on the same meridian, as is known by their having the same longitude; but, on account of various obstacles, it would be found very inconvenient to measure this line directly, with a rod or chain, and even if we could do it, we could not by this method obtain nearly so accurate a result, as we could by a series of triangles, where, after the base line was measured, we should have nothing else to measure except angles, which can be determined, by observation, to a greater degree of exactness, than lines. We therefore, in the first place, measure the base line, A B, with the utmost precision. Then, taking the bearing of some signal at C from A and B, we obtain the means of calculating the side B C, as has been already explained. Taking B C as a new base, we proceed, in like manner, to determine successively the sides C D, D E, and E F, and also A C, and C E. Although A C is not in the direction of the meridian, but considerably to the east of it, yet it is easy to find the corresponding distance on the meridian, A M; and in the same manner we can find the portions of the meridian M N and N O, corresponding respectively to C E and E F. Adding these several parts of the meridian together, we obtain the length of the arc from A to O, in miles; and by observations on the north star, at each extremity of the arc, namely, at A and at O, we could determine the difference of latitude between these two points. Suppose, for example, that the distance between A and O is exactly five degrees, and that the length of the intervening line is three hundred and forty-seven miles; then, dividing the latter by the former number, we find the length of a degree to be sixty-nine miles and four tenths. To take, however, a few of the results actually obtained, they are as follows:

Places of observation.Latitude.Length of a deg. in miles.
Peru,00° 00' 00"68.732
Pennsylvania,39 12 0068.896
France,46 12 0069.054
England,51 29 54½69.146
Sweden,66 20 1069.292

This comparison shows, that the length of a degree gradually increases, as we proceed from the equator towards the pole. Combining the results of various estimates, the dimensions of the terrestrial spheroid are found to be as follows:

Equatorial diameter,7925.648 miles.
Polar diameter,7899.170 "
Average diameter,7912.409 "

The difference between the greatest and the least is about twenty-six and one half miles, which is about one two hundred and ninety-ninth part of the greatest. This fraction is denominated the ellipticity of the earth,—being the excess of the equatorial over the polar diameter.

The operations, undertaken for the purpose of determining the figure of the earth, have been conducted with the most refined exactness. At any stage of the process, the length of the last side, as obtained by calculation, may be actually measured in the same manner as the base from which the series of triangles commenced. When thus measured, it is called the base of verification. In some surveys, the base of verification, when taken at a distance of four hundred miles from the starting point, has not differed more than one foot from the same line, as determined by calculation.

Another method of arriving at the exact figure of the earth is, by observations with the pendulum. If a pendulum, like that of a clock, be suspended, and the number of its vibrations per hour be counted, they will be found to be different in different latitudes. A pendulum that vibrates thirty-six hundred times per hour, at the equator, will vibrate thirty-six hundred and five and two thirds times, at London, and a still greater number of times nearer the north pole. Now, the vibrations of the pendulum are produced by the force of gravity. Hence their comparative number at different places is a measure of the relative forces of gravity at those places. But when we know the relative forces of gravity at different places, we know their relative distances from the centre of the earth; because the nearer a place is to the centre of the earth, the greater is the force of gravity. Suppose, for example, we should count the number of vibrations of a pendulum at the equator, and then carry it to the north pole, and count the number of vibrations made there in the same time,—we should be able, from these two observations, to estimate the relative forces of gravity at these two points; and, having the relative forces of gravity, we can thence deduce their relative distances from the centre of the earth, and thus obtain the polar and equatorial diameters. Observations of this kind have been taken with the greatest accuracy, in many places on the surface of the earth, at various distances from each other, and they lead to the same conclusions respecting the figure of the earth, as those derived from measuring arcs of the meridian. It is pleasing thus to see a great truth, and one apparently beyond the pale of human investigation, reached by two routes entirely independent of each other. Nor, indeed, are these the only proofs which have been discovered of the spheroidal figure of the earth. In consequence of the accumulation of matter above the equatorial regions of the earth, a body weighs less there than towards the poles, being further removed from the centre of the earth. The same accumulation of matter, by the force of attraction which it exerts, causes slight inequalities in the motions of the moon; and since the amount of these becomes a measure of the force which produces them, astronomers are able, from these inequalities, to calculate the exact quantity of the matter thus accumulated, and hence to determine the figure of the earth. The result is not essentially different from that obtained by the other methods. Finally, the shape of the earth's shadow is altered, by its spheroidal figure,—a circumstance which affects the time and duration of a lunar eclipse. All these different and independent phenomena afford a pleasing example of the harmony of truth. The known effects of the centrifugal force upon a body revolving on its axis, like the earth, lead us to infer that the earth is of a spheroidal figure; but if this be the fact, the pendulum ought to vibrate faster near the pole than at the equator, because it would there be nearer the centre of the earth. On trial, such is found to be the case. If, again, there be such an accumulation of matter about the equatorial regions, its effects ought to be visible in the motions of the moon, which it would influence by its gravity; and there, also, its effects are traced. At length, we apply our measures to the surface of the earth itself, and find the same fact, which had thus been searched out among the hidden things of Nature, here palpably exhibited before our eyes. Finally, on estimating from these different sources, what the exact amount of the compression at the poles must be, all bring out nearly one and the same result. This truth, so harmonious in itself, takes along with it, and establishes, a thousand other truths on which it rests.

LETTER VIII.