a degree, will give the actual distance in miles. Thus the direct distance between Paris and Buenos Aires is 99° 24′ 35", which is equal to 99.41° nearly. Multiplying this by 69.044, will give the direct distance in miles, which is equal to 6863.66404 miles, or 6863 miles 5 furlongs nearly. It has just been taken for granted that the shortest distance between two places on the globe is the arc of a great circle. This may be made evident by a few simple considerations. The plane of a great circle passes through the centre of the globe; that of a small circle does not: the radius of a great circle, therefore, is greater than that of a small one, and consequently the curvature or bending of the former is less than that of the latter. And as a straight line is the shortest distance between two points, so of two curved lines joining two points, that which is most like a straight line, i. e. the less curved line of the two, is the shorter; and therefore the arc of a great circle, lying between two places on the globe, is the shortest distance. : If we suppose, (as hitherto we have done,) that the earth is perfectly spherical, it is evident that a degree of latitude, being the 360th part of a meridian, which is always a great circle, must be everywhere of the same length. It is otherwise with degrees of longitude except for places upon the equator, a degree of longitude is an arc of a small circle, and is less than a degree of longitude measured upon the equator, which is a great circle. And the magnitude of a degree of longitude becomes gradually less in proportion as the distance from the equator increases, or as the latitude increases. The radii of the circles on which the longitude is measured (called the parallels of latitude) decrease from the equator to either pole; so that the circumferences of parallels of latitude decrease in like manner: but the circumference is always equal to the number of degrees into which it is divided, multiplied by the length of each degree; and as the number of degrees is the same in all circles, the length of each degree varies with the circumference, and must, therefore, decrease from the equator to the pole. The precise measure or law of this decrease may be proved in the following manner. Let (fig. 9.) the arc AE represent a degree of longitude upon the equator EQ; ea a corresponding arc of a de gree of longitude, measured upon the parallel of latitude e aq, in latitude A a. Fig. 9. Then, as the length of a degree varies as the circumference, and the circumference as the radius, we shall have the length of A E to that of e a at latitude Aa, as radius AO is to radius ao, where AO is the radius of the earth, and ao is the radius of the small circle ea q, the plane of which is parallel to the plane of the equator. In plane trigonometry, the radius ao is called the cosine of the arc A a, or the cosine of the latitude A a; hence the length of a degree of longitude at the equator is to the length of a degree at a given latitude A a, in the proportion of the radius of the earth to the cosine of the latitude; and as the degree at the equator and the radius of the earth are invariable, the length of a degree of longitude varies as the cosine of the latitude. CHAPTER VIII. Oblate-Spheroidal Figure of the Earth -Cause of this Figure-Centrifugal Force. THE various phenomena which indicate the nature of the earth's shape have been already described. They are sufficient to establish in a general way the roundness of the earth, but they are at the same time of that vague and indefinite character, as to be incapable of solving the more difficult problem of the earth's specific and exact shape. In order to determine this, it has been necessary to resort to experiments of an extremely delicate and tedious description, and to call in the aid of complex mathematical calculations founded upon the facts which the experiments have brought to light. The attention of some of the most eminent philosophers of Europe has for many years been given to this subject; and although the true figure of the earth cannot be considered as even yet determined with all the precision that is desirable, it is now conclusively proved that the earth is not a perfect sphere, but of an oblate-spheroidal form, bulging out at and about the equator, and flattened at the poles; and that the equatorial diameter is longer than the axis or polar diameter. The excess of the equatorial above the polar diameter represents, when compared with the whole diameter, the quantity by which the figure of the earth deviates from a perfect sphere; it is called the earth's ellipticity or compression. The discovery and proof of the earth's elliptical shape, and the laborious undertakings engaged in for determining the quantity of it, occupy some of the most interesting pages in the history of science. It is not perhaps to be much regretted that the person who first started the idea of the earth's spheroidal shape should be unknown. The first notion of it was in all probability nothing better than one of those happy conjectures which have been verified by subsequent proofs; but the name of Newton is as intimately allied with the discovery of the earth's true figure as with that of universal gravitation. In both cases the idea had been already entertained by several philosophers, but it was Newton who redeemed the truth from conjecture, and established it upon the basis of demonstration: with respect to the figure of the earth, he proved, from admitted principles and facts, that it must of necessity be an oblate-spheroid, and he assigned a ratio between the equatorial and polar diameters. The true figure of the earth is that which the particles composing it must assume, in order to be in a state of equilibrium or rest: the figure, therefore, depends essentially upon the forces which act upon these particles. The principal of these forces is the mutual attraction which subsists between the particles themselves. Any other force which acts in a different direction to this principal force, or with unequal intensity upon different particles, is a disturbing force; it disturbs and deranges that state or figure which the whole mass of An oblate spheroid is a solid body which may be conceived to be formed by the revolution of an ellipse or oval about its lesser axis. the earth would assume if affected only by the mutual attraction of the component particles. The method of conducting the investigation of the true figure of the earth, is one which is very usual in mechanical philosophy. The most simple and striking characteristics of the problem are singled out and considered alone, and the result obtained from them is afterwards varied and modified by the introduction of such minor and more complex conditions as are suggested by the problem in its true and practical form. In this manner, in the science of mechanics, the first principles and ground-work truths are ascertained, upon the supposition that the parts of machines are without weight, and that there is no such thing as friction, and the effect due in the practical result to these and other circumstances is estimated afterwards. In attempting to determine the true figure of the earth, it has accordingly been assumed in the first instance, that the earth is a body consisting only of fluid particles which move easily among themselves, and that the density of the particles, or the quantity of matter contained in the same space is the same throughout, and that the earth is in a state of rest. If this were the case, the only form in which equilibrium could take place among the particles which compose the earth, and which exert a mutual and equal attraction upon each other, is a perfect sphere. This may be exemplified by adverting to the familiar circumstance of drops or globules of water, or of quicksilver, upon a perfectly smooth and dry plain; by the force of mutual and equal attraction, the particles of which they consist, dispose themselves in a globular form. It is true, that the form of these drops is not perfectly spherical, but is rather flattened at the top; because the force of gravity which acts without an equal resistance upon the upper particles, makes them press down upon the lower, and thus deranges the effect which would be produced by their mutual attraction alone: but the fact is sufficient as a popular illustration of this conclusion, that fluid particles of an equal density, not affected by any external and partial force, would assume the form of a perfect sphere as that of a state of equilibrium. Now in this statement it is evident, that three conditions have been introduced which are not verified in the actual state and circumstances of the earth about its axis or a sidereal day), earth. These are, 1st. That the earth is a fluid body. 2d. That it is of the same density throughout. 3d. That it is in a state of rest. These conditions, therefore, not being fulfilled, it is necessary to inquire what alteration ensues in the form of equilibrium. Let us first take into consideration the fact of the earth not being in a state of rest, but having a double motion, one in its orbit, the other about its axis. The existence of motion in inanimate matter always indicates the impression of an external force. The force represented by this motion in the earth, is therefore in addition to that of the mutual attraction of its component particles. In order, however, that it should alter the form of equilibrium, it is clearly necessary that the additional force should be not merely external but partial. The earth's motion in her orbit has therefore no effect of this kind; because, acting equally upon all the particles, it can have no influence upon their mutual relation. The other motion of the earth about its axis gives rise to very different considerations. This motion is performed in a plane perpendicular to the axis; so that the particles of which the earth is composed move in circles, the planes of which being perpendicular to the axis, are either coincident with, or parallel to the plane of the equator. Thus (in fig. 10) the particle Q, at the it is evident that different parts of the earth will be differently affected by the motion of rotation :-immediately under the poles there exists (so far as this motion is concerned) a state of perfect rest; but in passing from the poles to the equator, the motion of rotation begins and becomes quicker and quicker (because in the same time a greater space or circle is described); till arriving at the equator, which is the largest circle, the plane of which is perpendicular to the axis, it has obtained its greatest velocity. This motion of rotation gives to all the particles affected by it, a tendency to fly off from the centre of rotation, and communicates to them what is therefore called a centrifugal force. The mutual attraction of the earth's particles, or their gravitating force, is directed towards the centre of the earth, this is called a centripetal or centre-seeking force. The combination of these two forces, the centrifugal and centripetal, will, as they are in different directions, and as the former acts in the partial manner we have described, produce a material alteration in the form, which, under the action of the latter alone, the particles of the earth would have assumed as a state of equilibrium. For it is evident that as the centrifugal force caused by the motion of rotation is in diminution of the centripetal force of gravity, the particles affected by this centrifugal force must give way to those which are not affected by it, and thus be pushed away from the centre of the earth; those also which are more affected by it, will give way to those which are less so, and will recede still farther from the centre of the earth; so that in order now to preserve equilibrium among themselves, there must be a decrease in the number of particles which have the greatest individual gravitating force, and a proportionate increase in the number of particles which have a diminished gravitating force. There are two reasons for the centrifugal force caused by the motion of the earth about its axis, being greatest at the equator, and gradually diminishing towards the poles. The motion of the earth on its axis being every where in a plane to which the axis is perpendicular, This is not strictly true: the direction of gravity being always perpendicular to the surface, and therefore, (the earth not being a sphere) it does not pass through the centre, but only near the centre, but this distinction may be disregarded in the present explanation. the particles lying under the equator move in the circle of the equator; all other particles between the equator and the poles move in circles parallel with the equator. Hence, particles moving in the equator have the greatest velocity and the greatest centrifugal force; those nearer the equator a greater velocity and a greater centrifugal force than those particles which are more distant from the equator. Moreover, the direction of the centrifugal force communicated by this motion is always opposite to the direction of the radius of the different circles in which the particles move; thus if Ld (fig.10), be the radius of the circle in which a particle at L moves, the direction of the centrifugal force at L will be La, and the direction of the centrifugal force at Q, in the equator will be Qq, QC being the radius of the equator and also of the earth. Now gravity or the centripetal force is always in a direction perpendicular to the surface, and therefore it is only at the equator, that this is exactly in an opposite direction to the centrifugal force. At L for instance, if La be taken as in the direction, and as also representing the quantity of centrifugal force there, it may be resolved in two directions a b, Lb, one of which only, a b, is directly opposed to the force of gravity, while the other is perpendicular to the direction of gravity, and is a tangent to the surface, and does not diminish the force of gravity. Hence it follows, that in the equatorial regions, where the directions of gravity and of the centrifugal force are both perpendicular to the surface, the two forces become altogether opposed to each other, and the whole amount of the centrifugal force of rotation takes effect in diminution of the force of gravity. But in parts distant from the equator (as at L), a portion only of the centrifugal force (viz. a b), and that in its diminished state, acts in opposition to the force of gravity; the rest of it (viz. L b) is in the direction of a tangent, and tends towards the equator. In order, therefore, that equilibrium may now be preserved among the component particles of the earth, a great accumulation of particles takes place in the equatorial regions, which by their number compensate their deficiency in gravitating force. And this effect is increased by that part of the centrifugal force acting between the poles and the equator, which is in a tangent direction to the earth's surface, and which tends to thrust down the particles on which it acts towards the equator. Hence the equatorial regions are elevated above the polar, and the height of all other intermediate parts is in some proportion of the distance of those parts from the equator. This is the alteration produced in the figure of equilibrium, by the diurnal rotation of the earth upon its axis. But the earth being neither altogether fluid, nor of the same density throughout, we must introduce some qualifications into the result we have just arrived at. The earth being partly solid, the particles of which it is so far composed do not move easily among themselves, but have an attraction of cohesion which opposes a certain resistance to the operation of the centrifugal force caused by its motion of rotation. This neutralizes and destroys part of the centrifugal force, and makes the earth's ellipticity to be less than it would be if the earth were altogether fluid. But the centrifugal force is not altogether destroyed by the attraction of cohesion; for it must have elevated even the solid parts of the equatorial regions; were it not so, the waters of the ocean, not being restrained in their motion by the same attraction of cohesion, would all have set towards the equator in order to restore the equilibrium which, by the diminution of the centripetal force of gravity there, had been disturbed, and would thus have overflowed the land at the equator and left the polar regions dry. It may here be remarked, that the two constant currents in the sea, which are observed to set from both poles towards the equator, may perhaps be accounted for, by the action of that part of the centrifugal force which is in the direction of a tangent to the earth's surface and towards the equator. The waters of the sea, having no attraction of cohesion, would obey the impulse of this force freely, as it is not opposed to the force of gravity. The westerly set of the same currents may be ascribed to their continually advancing into regions which have a greater easterly motion of rotation. The effect produced by the varying density of the earth, which increases towards the centre, has been proved by Clairaut, a celebrated French mathematician, to be a diminution of the oblateness of the earth, so that from this cause the height of the equatorial regions is somewhat less than on the supposition of an equal density; which is contrary to what Newton supposed would be the effect of an increase of density towards the centre. The result of Newton's inquiry into the figure of the earth was, that the equatorial diameter of the earth is to the polar as 230 229; from this ratio the ellipticity of the earth would be expressed by the fraction; or the polar diameter would be less than the equatorial by the 229th part of the whole, and the equatorial regions would be about 17 miles higher than the polar. A very simple mathematical process will enable us to exhibit the value of the centrifugal force at any point of the earth's surface. Let C (fig. 10), be the centre of the earth, Pp the polar axis, E Q the equator, La particle acted upon by the centrifugal force at any latitude LQ or a; from L let fall L d perpendicular to the polar axis, Ld will be the radius of the circle of rotation at latitude λ, and C Q will be that of the equatorial circle of rotation, and is the same with the radius of the earth. Now the whole amount of centrifugal force varies as the velocity; for motion or velocity is the producing cause of it, and the velocity of rotation of different parts of the earth's surface varies as the circle of rotation, and therefore varies as the radius of that circle:hence the centrifugal force at any point L, varies as Ld, which is the sine of P L, or the complement of the latitude, or the entire centrifugal force varies as the cosine of the latitude. Take La to represent the whole centrifugal force at L, resolve this in directions L b, a b, perpendicular and parallel to that of gravity: then a b is the only part of the centrifugal force which directly opposes the force of gravity. From L let fall L, perpendicular to CQ, then Lab and L C7, are similar triangles, and ab: La: el: C L, so that a b, or the centrifugal force opposed to gravity, is equal to La cl La CL cos. λ =whole centrifugal force at L x cos. 2. Now the whole centrifugal force we have shown to vary as the cos. of the latitude; therefore whole centrifugal force at L: centrifugal force at the equator:: cos. : 1, for at the equator the latitude=o and cos. o= radius; but the centrifugal force at the Oblate Figure of Jupiter, Saturn and Mars-Pendulum Experiments. IN the foregoing chapter we have explained the causes which have produced the earth's ellipticity. We now proceed to the various evidences of this fact which have been derived from observation and experiment. As the same causes must, under similar circumstances, produce similar effects, it was just to suppose that if the reasoning by which the earth's ellipticity is established be correct, the other planets of the solar system would exhibit the like appearance of a flattening at their poles and a bulging out of their equatorial parts; for their component particles are under similar circumstances of mutual attraction, of equilibrium, and of rotation about an axis*. This was the rather to be looked for, because of he traced in the principal features of the analogy or resemblance which is to the solar system. analogies, the orbits of all the planets Thus, among other revolving round the sun are elliptical; their periodic times, or the times of their revolutions, are in proportion to the cubes of their mean distances from gravitate towards the sun are as the the sun, and the forces by which they squares of those distances. The supposition of an ellipticity in the planets similar to that which is observed in the earth, was first verified in the planet Jupiter. This planet completes his We must except from the generality of this remark the planets Uranus, or Georgium Sidus, which is too distant, and Juno, Vesta, Ceres, and Pallas, which are too small to admit of any observations by which to ascertain whether they revolve about an axis or not. |