Geode

“ The earth is only a larger form of a geode, and the law that created the geode in its hollow form undoubtedly fashioned the earth in the same way."
In presenting the theme of this almost incredible story, as told by Olaf Jansen, and supplemented by manuscript, maps and crude drawings entrusted to me, a fitting introduction is found in the following quotation:
"In the beginning God created the heaven and the earth, and the earth was without form and void." And also, "God created man in his own image." ”

“ If crystal growth continues until the geode is solid, it may bear superficial resemblance to a concretion, but the latter structure is one which has grown outward. The idea of “growth” in either case is that of mineral crystallization and enlargement, but does not in the least involve life like that of a plant or animal. Geodes have no value or use other than for ornamental purposes.
69 Fossils
Fossils are also found and collected by persons who are interested in rocks and minerals. ”

“ Quartz is the most common mineral deposited in geodes, but calcite, aragonite, dolomite, siderite, pyrite, galena, fluorite, and sphalerite also are found.
Geodes ranging in size from less than one inch to a foot or more in diameter can be gathered from streams where they have accumulated as residual boulders after the rock in which they were enclosed has been eroded.
Hollow geodes are the most desirable because they have better crystals. They can be distinguished from solid ones by their comparative lightness of weight. ”

“ This mineral occurs most frequently in the condition of amethystine quartz, in hexahedral prisms, lining the interior of geodes, in the bed of the River Desmoines, and on the Rock Rapids, in the channel of the Mississippi. The crystals which I have examined are generally limpid, with a high lustre, and of a pale violet color. Sometimes the tinge of color approaches to a full red, or is only apparent in the summit of the crystal. These geodes are sometimes eight or ten inches in diameter, with a rough and dark-colored exterior, often so nearly spherical as to resemble cannon balls. ”

“ The 19th century witnessed the culmination of the ellipsoidal era of geodesy, but the rapid accumulation of data made possible a better approximation to the figure of the earth by the geoid. The geoid is defined as the average level of the sea, which is thought of as extended through the continents. The basis of geodetic calculations, however, is an ellipsoid of reference for which a gravity formula expresses the value of normal gravity at a point on the ellipsoid as a function of gravity at sea level at the equator, and of latitude. ”

“ 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. ”

“ 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 ”

“ The geoid is above the ellipsoid of reference under mountains and it is below the ellipsoid on the oceans, where the geoid coincides with mean sea level. In physical geodesy, gravimetric data are used for the determination of the geoid and components of deflections of the vertical. For this purpose, one must reduce observed values of gravity to sea level by various reductions, such as free-air, Bouguer, isostatic reductions. If g0 is observed gravity reduced to sea level and γ is normal gravity obtained from the International Gravity Formula, then Δg = g0 - γ is the gravity anomaly. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ The science is concerned with the heavenly bodies. The earth on which we live is, to all intents and purposes, one of these bodies, and, so far as its relations to the heavens are concerned, must be included in astronomy. The processes of measuring great portions of the earth, and of determining geographical positions, require both astronomical observations proper, and determinations made with instruments similar to those of astronomy. Hence geodesy may be regarded as a branch of practical astronomy. (S. N.)
History of Astronomy. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ The zenith telescope is a particularly pleasant instrument to work with, and an observer has been known (a sergeant of Royal Engineers, on one occasion) to take every star in his list during eleven hours on a stretch, namely, from 6 o’clock p.m. until 5 a.m., and this on a very cold November night on one of the highest points of the Grampians. Observers accustomed to geodetic operations attain considerable powers of endurance. ”

“ But if, for some reason or other, the free fall of the body is interfered with, if, for instance, the rigidity of the earth’s crust prevents the body from falling farther along the geodesic, the geodesic will react and will strive to force the body through the earth’s crust. It was this force which we formerly ascribed to the pull of gravitation. ”

“ 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. ”

“ 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. ”

“ Again, in the geodetic systems which we meet with in morphology, it sometimes happens that we have two opposite systems of geodetic spirals separate and distinct from one another, as in Fig. 236, c; and it is also common to find the two systems interfering with one another, and forming a criss-cross, or reticulated arrangement. This is a very common source of reticulated patterns.
Among the ciliated Infusoria, we have in the spiral lines along which their cilia are arranged a great variety of beautiful geodetic curves ”

“ The communicator burped suddenly and the viewscreen came to life; a deep, dull red that brightened quickly. A tiny coal flared up, swelling in size and shifting from red to orange to yellow—the sun. Earth appeared as a hazy red dot that enlarged and resolved itself into a planet with distorted continents that trembled and changed, to resume their natural shapes and colors. Within a few seconds the sun was shining as ever, Earth loomed large and blue-green before them and the stars of Orion glittered unchanged beyond. Even their position in space was the same—they had not moved. ”

“ Having now described, and prepared the Planisphere for use, I shall next explain the motions of the Terraqueous Globe on which we live: and first, according to the true, or Copernican System of the Universe, this Globe is a Planet, revolving round the Sun in an orbit somewhat elliptical, and in the space of 365 days, 5 hours, and 49 minutes, being accompanied with the Moon, and hath the Sun at rest in the center of its orb.–Now, conceive at a vast distance, without this system, the sphere of the fixed Stars ”

“ Both masses had well-defined “peaks,” and I came to the conclusion that the instruction to “draw a line from one black rock to the other” would mean that a straight line must be drawn, or ranged, from one of these well-defined peaks to the other. So I temporarily removed the telescope from my sextant, and, levelling it upon the extreme peak, or highest point of the rock I occupied, brought it to bear upon the corresponding peak of the other rock. ”

“ The use of the Prime Vertical in the Planisphere is very obvious; for when, by the diurnal rotation of the Earth, any celestial object comes to this circle in the eastern Hemisphere, it appears due East: and if it be in the western Hemisphere, it is due West, and its Altitude, in both Cases, is seen by inspection; also, when any celestial object comes to the southern part of the Meridian, its meridian Altitude is seen on the scale of Altitude; and when any object is rising or setting, its Azimuth and Amplitude are seen on the Horizon, with the Point of the Compass on which it rises or sets. ”

“ The problem before us is the following:—A rod, or style, as it is called, being firmly fixed in a direction parallel to the earth’s axis, we have to find how and where points or lines of reference must be traced on some fixed surface behind the style, so that when the shadow of the style falls on a certain one of these lines, we may know that at that moment it is solar noon,—that is, that the plane through the style and through the sun then coincides with the meridian ”

“ The centre of Jupiter has the same distance: it lies between the sphere of Venus and that of Mercury, whilst the centre of Saturn lies between the spheres of Mars and Jupiter. Now, consider how improbable all this appears according to the laws of Natural Science. You will find it out when you consider the known distances and magnitudes of each sphere and each star, all expressed in terms of the radii of the earth. There is a uniform measure for all, and the excentricity of each sphere is not determined by units proportionate to its own magnitude. ”