Geodesy

“ The science of geodesy is part of mathematical geography, of which the arts of surveying and cartography are applications. The motions of the earth as a planet must be taken into account, as they render possible the determination of position and direction by observations of the heavenly bodies. The diurnal rotation of the earth furnishes two fixed points or poles, the axis joining which is fixed or nearly so in its direction in space. The rotation of the earth thus fixes the directions of north and south and defines those of east and west. ”

“ Geodif′erous, bearing or producing geodes. [Fr.,—Gr. geōdēs, earth-like, earthen—gē, earth, eidos, form.]
Geodesy, je-od′e-si, n. a science whose object is to measure the earth and its parts on a large scale.—ns. Geodē′sian, Geod′esist, one skilled in geodesy.—adjs. Geodes′ic, -al, Geodet′ic, -al, pertaining to or determined by geodesy. [Fr. géodésie—Gr. geōdaisia—gē, the earth, daiein, to divide.] ”

“ Wherefore the geodesy or measuring of right lines is thus in length, heighth, and breadth, from whence the Painter, the Architect, and Cosmographer, may view and gather of many famous place the windowes, the statues or imagery, pyramides, signes, and lastly, the length and heighth, either by a single or double: the breadth by a double dimension onely, that is, they may thus behold and take of all places the nature and symmetry ”

“ Natural irregularities in density and depths of the material making up the upper crust of the Earth also result in slight alterations of the direction of gravity. These alterations are reflected in the irregular shape of the geoid, the surface that is perpendicular to a plumb line.
Since the Earth is in fact flattened slightly at the poles and bulges somewhat at the equator, the geometric figure used in geodesy to most nearly approximate the shape of the Earth is the oblate spheroid or ellipsoid of revolution. ”

“ Geodesy of one distance is when the measurer for the finding of the desired dimension doth not change his place of standing. Geodesy of two distances is when the measurer by reason of some impediment lying in the way betweene him and the magnitude to be measured, is constrained to change his place, and make a double standing. ”

“ And also: "It is hoped that the beginner will be enabled to get a clear insight into the subject, and feel grateful that the discoveries and writings of many have been so condensed or elaborated as to make the study of geodesy pleasant."
The first chapter, which is an historic sketch of geodetic operations, is interesting and full of facts tending to show what progress has been made from the earliest times, and how the desire to know the real dimensions of the earth has kept man busy in devising and perfecting the methods of measurement. ”

“ 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. ”

“ In the case of the spheroid this equation becomes
which integrated gives ydx − xdy = Cds. This again may be put in the form r sin a = C, where a is the azimuth of the geodetic at any point—the angle between its direction and that of the meridian—and r the distance of the point from the axis of revolution. ”

“ 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. ”

“ To these two-dimensioned geometers geodesies would possess many of the attributes of straight lines in ordinary space. If the surface to which the beings were confined were actually a plane, then the geometry would be the same as our geometry. They would find that only one straight line could be drawn between two points, that through a point only one parallel to a given line could be drawn, and that the ends of a line would never meet even though the line be prolonged to infinity.
We might also suppose that intelligent beings could exist on the surface of a sphere. ”

“ 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. ”

“ 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 ”

“ A doubly infinite number of geodesic surfaces can be drawn through every point. If a rigid body be divided into two parts by a geodesic plane, then no matter how the body be displaced the plane of section will still be geodesic. The plane of section may be made to pass through any point, and the body may then be given such an aspect as shall cause the section to coincide with any geodesic surface through the point, but this necessarily involves that the section shall fit each geodesic surface, in other words, that all the geodesic surfaces shall have a constant curvature. ”

“ 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 ”

“ 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. ”

“ It is important to realize that geometry, as taught in schools since Greek times, ceases to exist as a separate science, and becomes merged in physics. The two fundamental notions in elementary geometry were the straight line and the circle. What appears to you as a straight road, whose parts all exist now, may appear to another observer to be like the flight of a rocket, some kind of curve whose parts come into existence successively. The circle depends upon measurement of distances, since it consists of all the points at a given distance from its center. ”

“ And if terrestrial physics includes terrestrial geometry, while celestial physics includes celestial geometry, then the geometrical part of terrestrial physics precedes the geometrical part of celestial physics; seeing that geometry gained its first ideas from surrounding objects. Until men had learnt geometrical relations from bodies on the earth, it was impossible for them to understand the geometrical relations of bodies in the heavens. So, too, with celestial mechanics, which had terrestrial mechanics for its parent. ”

“ Thus the geometer leaves to the man of science to decide, as best he may, what axioms are most nearly true in the actual world. The geometer takes any set of axioms that seem interesting, and deduces their consequences. What defines Geometry, in this sense, is that the axioms must give rise to a series of more than one dimension. ”

“ On their side, geometers, when measuring the size of the entire earth, avail themselves of the data furnished by the natural philosopher and astronomer; and the geographer on his part makes use of those of the geometer.
3. The heavens and the earth must be supposed to be divided each into five zones, and the celestial zones to possess the same names as those below. The motives for such a division into zones we have already detailed. These zones may be distinguished by circles drawn parallel to the equator, on either side of it. ”

“ If the sphere is translucent, our tiny rods displaced on the spherical surface will cast their shadows on the plane; and, of course, if we effect measurements on the plane by means of these shadows of our rods, we shall obtain exactly the same metrical relationships on the plane as we obtained formerly on the spherical surface. In particular, if a hundred tiny rods lying along a meridian span the distance from the south to the north pole on our sphere, the hundred shadows of these tiny rods will extend across the plane to infinity. ”

“ Arithmetic and Geometry have long been standard portions of the education of cultured persons throughout the civilized world; and hence all such persons have been able to accept and comprehend those portions of science which depend upon the idea of space: for instance, the doctrine of the globular form of the earth, with its consequences, such as the measures of latitude and longitude;—the heliocentric system of the universe in modern, or the geocentric in ancient times;—the explanation of the rainbow ”

“ But when we consider a curved surface, the straight lines of space cannot lie on its surface. The geodesics of the curved surface are then no longer straight lines in the Euclidean sense, but they are the nearest approach to such lines, i.e., the least curved of all lines traceable on the curved surface. In the case of the sphere, geodesics are great circles. All surfaces have their geodesics more or less capriciously distorted in shape according to the variation in curvature from place to place of the surface. ”

“ Consider, then, one such straight line or geodesic, and let it be that particular great circle which we call the equator. According to the axioms of Euclid’s geometry two perpendiculars drawn at two different points of the same straight line can never meet; they lie parallel to one another. On the sphere, on the other hand, two perpendiculars to the equator are two meridians and these meridians always meet at the pole. In fact it is impossible to draw any two great circles on the sphere which do not intersect. Hence we see that no parallel geodesics can exist in Riemann’s geometry. ”

“ 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. ”

“ The question whether the practical geometry of the universe is Euclidean or not has a clear meaning, and its answer can only be furnished by experience. All linear measurement in physics is practical geometry in this sense, so too is geodetic and astronomical linear measurement, if we call to our help the law of experience that light is propagated in a straight line, and indeed in a straight line in the sense of practical geometry. ”

“ 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 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. ”