Geodesic

“ The point which we have now gained is one of very great importance. In our ordinary conceptions of space the geodesic surfaces are of course our ordinary planes, and the common curvature they possess is zero, but the condition that rigid bodies shall be capable of translation with unaltered features does not require that the curvatures shall be zero, it merely requires that the curvatures shall be constant. ”

“ In the geometry of two dimensions we can thus have either a plane or a sphere or a pseudosphere which are characterized by the property that a surface may be moved about in all directions without any change either in the lengths of its lines or in the magnitudes of its angles. The axiom which assumes that there is only one geodesic connecting two points marks off the plane and the pseudosphere from the sphere. The axiom that only one parallel can be drawn through a given point to a given line marks off the plane from the pseudosphere. ”

“ They take the place of straight lines in the intrinsic geometry of a surface. When we inquire whether the angles of a triangle add up to two right angles or not, we mean to speak of a triangle whose sides are geodesics. And when we speak of the distance between two points, we mean the distance along a geodesic.
The next step in our generalizing process is rather difficult: it is the transition to non-Euclidean geometry. We live in a world in which space has three dimensions, and our empirical knowledge of space is based upon measurement of small distances and of angles. ”

“ Being a law of space-time curvature or structure, we are able to derive from it the precise lay of the geodesics of space-time around matter. These geodesics, or paths of least resistance, are, we remember, the paths which planets, light rays and all free bodies will pursue. A knowledge of the lay of the geodesics around the sun will therefore allow us to determine the precise courses in space and the precise motions along these courses of all free bodies moving in a gravitational field. ”

“ To sum up—the interval between two (near) events is something quantitative which has an absolute significance in nature. The track between two (distant) events which has the longest interval-length must therefore have an absolute significance. Such tracks are called geodesics. Geodesics can be traced practically, because they are the tracks of particles undisturbed by material impacts. By the practical tracing of these geodesics we have the best means of studying the character of the natural geometry of the world. ”

“ What applies to distance applies equally to the straight line. There is nothing in the actual world having exactly the properties that straight lines were supposed to have; the nearest approach is the track of a light ray. Straight lines have to be replaced by geodesics, which are defined by what they do at each point, not all at once, like Euclidean straight lines. ”

“ When it came to determining the precise geometry of real space, various methods could be considered. We might proceed by appealing to rigid bodies, that is, to bodies which visually and tactually appeared to be the same wherever we might carry them. More generally, rigid bodies were assumed to be exemplified by material bodies maintained as far as possible under permanent conditions of temperature and pressure. We could also investigate space by means of light rays, assuming that their courses in vacuo would define geodesics. ”

“ If we assume that moving points describe geodesics in a non-Euclidean space which presents a high degree of non-Euclideanism, we must recognise that our material bodies (which behave Euclideanly) must diverge widely from standard when displaced. If, then, we consider the translational motion of a large non-rotating mass, all its molecules should describe geodesics. But this would be impossible, since, owing to the enormous variations in the shape of the body, as contrasted with the geometry of space, the majority of the molecules would be torn from their natural courses. ”

“ The basis of every extensive survey is an accurate triangulation, and the operations of geodesy consist in the measurement, by theodolites, of the angles of the triangles; the measurement of one or more sides of these triangles on the ground; the determination by astronomical observations of the azimuth of the whole network of triangles; the determination of the actual position of the same on the surface of the earth by observations, first for latitude at some of the stations, and secondly for longitude ”

“ Where the surface is spherical, the geodetic is always literally a “great circle,” a circle, that is to say, whose centre is the centre of the sphere. If instead of a sphere we be dealing with an ellipsoid, the geodetic becomes a variable figure, according to the position of our two points. {441} For obviously, if they lie in a line perpendicular to the long axis of the ellipsoid, the geodetic which connects them is a circle, also perpendicular to that axis ”

“ But suppose that the surface on which these beings live, instead of being an infinitely extended plane, is really the surface of an immense globe, like the earth on which we live. It needs no knowledge of geometry, but only an examination of any globular object—an apple, for example—to show that if we draw a line as straight as possible on a sphere, and parallel to it draw a small piece of a second line, and continue this in as straight a line as we can, the two lines will meet when we proceed in either direction one-quarter of the way around the sphere. ”

“ We know that the earth’s surface is divided into halves, called the northern and southern hemispheres, by an imaginary circle called the equator, whose plane passes through the center of the earth and is perpendicular to the earth’s axis. If the plane of the earth’s equator were infinitely extended it would describe upon the celestial sphere a great circle which would divide that sphere into two hemispheres, just as the plane of the terrestrial equator divides the earth into two hemispheres. ”

“ We see then that the effect of the uneven distribution of matter in the crust of the earth is to produce small elevations and depressions on the mathematical surface which would be otherwise spheroidal. No geodesist can proceed far in his work without encountering the irregularities of the mathematical surface, and it is necessary that he should know how they affect his astronomical observations. ”

“ When you have thus made a sphere for yourself, or, with an artificial globe before you, if you have access to one, proceed to point out on it the various arcs of azimuth and altitude, right ascension and declination, terrestrial and celestial latitude and longitude,—these last being referred to the equator on the earth, and to the ecliptic in the heavens.
When the circles of the sphere are well learned, we may advantageously employ projections of them in various illustrations. By the projection of the sphere is meant a representation of all its parts on a plane. ”

“ From experience, or possibly intuitively, we characterize existent space by certain fundamental qualities, termed axioms, which are insusceptible of proof; and these axioms, in conjunction with the mathematical entities of the point, straight line, curve, surface and solid, appropriately defined, are the premises from which the geometer draws conclusions. The geometrical axioms are merely conventions; on the one hand, the system may be based upon inductions from experience, in which case the deduced geometry may be regarded as a branch of physical science ”

“ The azimuth of a geodetic line cannot be observed, so that the line does not enter of necessity into practical geodesy, although many formulae connected with its use are of great simplicity and elegance. The geodetic line has always held a more important place in the science of geodesy among the mathematicians of France, Germany and Russia than has been assigned to it in the operations of the English and Indian triangulations. ”

“ By methods, the history and description of which we have not here sufficient space to give, it has been proved that the earth is, in form, a globe, or more strictly an ellipsoid, slightly drawn in at the poles and swollen round the equator. The polar diameter is 7899 miles, and the equatorial diameter 7926 miles, the difference amounting to only 27 miles. Thus, for ordinary purposes, we may regard the earth as being a true sphere. ”

“ B. A sphere you mean, my dear: a circle is a round line, every part of which is equally distant from the centre; a sphere or globe is a round body, the surface of which is every where equally distant from the centre.
Emily. If, then, the earth is not a perfect sphere, but prominent at the equator, and depressed at the poles, would not a body weigh heavier at the equator than at the poles? For the earth being thicker at the equator, the attraction of gravity perpendicularly downwards must be stronger. ”

“ Geddelics on Developable Surfaces.-On these the geodetics are the curves which become right lines when the surface is unrolled into a plane. From this property a first integral can be immediately deduced. - 3. Geodetics on Surfaces of Revolution.-In all such the geodetics are the curves given by the equation r sin ¢=const, 1' being the perpendicular on the axis of revolution, d> the angle at which the curve crosses the meridian. The general problem of the determination of geodetics on any surface may be advantageously treated in connexion with that of “ parallel " curves. ”

“ Let any one try the experiment of looking at a lighthouse, church spire, monument, gas-lamp, or other elevated object, from the distance of a few yards, and notice the angle at which it is observed: on going farther away, the angle will diminish and the object appear lower, until, if the distance be sufficiently great, the line-of-sight to the object, and the apparently [41] ascending surface of the Earth upon which it stands will converge to the angle which constitutes the vanishing point; at a single yard beyond which it will be invisible. ”

“ We may therefore, in all questions relating only to the directions of the stars, replace the angle between the directions of two stars by the arc joining the corresponding points on the celestial sphere, or, in other words, by the distance between these points on the celestial sphere. But such arcs on a sphere are easier both to estimate by eye and to treat geometrically than angles, and the use of the celestial sphere is therefore of great value, apart from its historical origin. ”

“ The shell thus formed is then cut along the line of the intended equator into two hemispheres, they are then again glued together and made to revolve round an axis the ends of which passed through the poles and entered a metal meridian circle. The sphere is then coated with plaster or whiting, and when it has been smoothed on a lathe and dried, the lines representing meridians and parallels are drawn upon it. Finally the globe is covered with the paper gores upon which the map is drawn. The adaption of these gores to the curvature of the sphere calls for great care. ”

“ Let the spheroid be so placed that its normal at A lies to the north of the normal to the earth’s surface by the small quantity ξ and to the east by the quantity η. Then in starting the calculation of geodetic latitudes, longitudes and azimuths from A, we must take, not the observed elements φ, α, but for φ, φ + ξ, and for α, α + η tan φ, and zero longitude must be replaced by η sec φ. At the same time suppose the elements of the spheroid to be altered from a, e to a + da, e + de. ”

“ Since Point and Plane represent in the realm of geometrical concepts what in outer nature we find in the form of the gravity-levity polarity, we may expect to meet Radius and Sphere as actual formative elements in nature, wherever gravity and levity interact in one way or another. ”

“ The science which inquires into matters of this kind, including besides the figure of the Earth, the length of the degree at different latitudes, and the distances of places from one another, alike in angular measure and in time, is called Geodesy; it is, in point of fact, land-surveying on a very large scale, in which instruments and processes of astronomical origin are brought into operation, and in which astronomers are more or less required to take the lead. ”

“ If the earth is a sphere then why could a map so draughted as truly to represent the surface of a sphere not be counted the most acceptable? This must have been the argument of those who especially applied themselves to the designing of maps suitable for a spherical surface, that is, for application to a globe ball. ”

“ If g is the intensity of gravity at some reference station, and Δg is the difference between intensities at two stations, the values of gravity in geodetic calculations enter as ratios (Δg) /g over the surface of the earth. Gravimetric investigations in conjunction with other forms of geophysical investigation, such as seismology, furnish data to test [304] hypotheses concerning the internal structure of the earth. ”

“ In fact, the sea horizon lies below the rational horizon by an amount varying with the elevation of the eye above the surface. Geometry enables us to determine just what the dip of the horizon must be for any given elevation of the eye. A rough and ready rule, which may serve for many purposes, is that the square root of the elevation of the eye in feet equals the dip of the horizon in minutes of arc, or of angular measure. ”

“ For the sake of brevity we shall presuppose a knowledge of Euclid’s Elements, although we shall use only a few of his propositions.
§ 1. Geometrical Elements. We consider space as filled with points, lines and planes, and these we call the elements out of which our figures are to be formed, calling any combination of these elements a “figure.”
By a line we mean a straight line in its entirety, extending both ways to infinity; and by a plane, a plane surface, extending in all directions to infinity. ”