Sphere

“ For in these the hollow convexity everywhere bends on itself, and leans upon the earth as its centre. Whereas the earth rises up solid and dense, like something that swells up and is protruded outwards. The heavens bend towards the centre, while the earth goes from the centre, the continual rolling of the heavens about it forcing its immense globe into the form of a 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. ”

“ If we had never known any other property of the sphere than its primitive character of having all its points equally distant from an interior point, how would we ever have been able to discover that the surface of the earth was spherical? ”

“ The answer seems to be this:—We have learnt by indirect methods that the Earth is a sphere; we have formed models approximately representing its shape and the distribution of its parts; generally when the Earth is referred to, we either think of an indefinitely extended mass beneath our feet, or else, leaving out the actual Earth, we think of a body like a terrestrial globe; but when we seek to imagine the Earth as it really is, we join these two ideas as well as we can—such perception as our eyes give us of the Earth’s surface we couple with the conception of a sphere. ”

“ It takes but little water just to touch At some one point the inside of a sphere, 100 And, as we turn the sphere, touch all the rest In due succession: but the finer air Which not so palpably nor obviously, Though no less universally, can touch The whole circumference of that emptied sphere, Fills it more fully than the water did ”

“ You know full well that the condition of the outer sphere is, as I have described it, a bleak, turbulent surface, the roof of the earth on which man exists, as a creeping parasite does on a rind of fruit, exposed to the fury of the ever-present earth storms.
"The central circle, or medial sphere, the shell, or layer of gravitation, lies conformably to the outer configuration of the globe, about seven hundred miles towards its center. It stretches beneath the outer circle (sphere) as a transparent sheet, a shell of energy, the center of gravitation. ”

“ What is the end of Nature? Where is the end of a sphere? The sphere balances at any and every point. So everything in Nature is at the top, and yet no one thing is at the top.
She works with reference to no measure of time, no limit of space, and with an abundance of material, not expressed by exhaustless. ”

“ When space is curved, and the cosmos is sphere within sphere, and the way from any one point to any other point is round the bend of the inevitable, that turns as the tips of the broad wings of the hawk turn upwards, leaning upon the air like the invisible half of the ellipse. ”

“ It follows that we are in the heart of a hollow sphere of loveliest blue, spangled with light. Now the sphere is the one perfect geometrical form. Over and round us then we have the one perfect shape. ”

“ You must understand what is meant by a great circle: it is a circle dividing the sphere into equal parts. Its plane therefore passes through the centre of the sphere. Now, our use of the celestial globe or sphere is founded on the assumption that it is a representation of the heavens, on the supposition that the eye of the observer is at the centre of the sphere. ”

“ Others thought the earth was a sphere because “the sphere is the most perfect form; it was the center of the universe because that is the place of honor; and it is motionless, because motion is less dignified than rest.” Some believed that the earth is round and rests upon the ocean. ”

“ The Sphere would willingly have continued his lessons by indoctrinating me in the conformation of all regular Solids, Cylinders, Cones, Pyramids, Pentahedrons, Hexahedrons, Dodecahedrons, and Spheres: but I ventured to interrupt him. Not that I was wearied of knowledge. On the contrary, I thirsted for yet deeper and fuller draughts than he was offering to me. ”

“ Yet a sphere with that radius of ten year’s flight would cover such an enormous quantity of space as to be beyond our comprehension and still be of no consequence in comparison with the spheres of one hundred, one thousand, one million, or one thousand millions of years radii. ”

“ Fill the whole dome, the entire extent of the surrounding sphere, bottom, sides, top, a sphere one hundred and eighty-five million miles in diameter, and everyone of these uncountable worlds would be touched with the same power as one; each would thrill with life. ”

“ 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. ”

“ That every visible figure will be represented by that part of the surface of the sphere, on which it might be projected, the eye being in the centre. And every such visible figure will bear the same ratio to the whole of visible space, as the part of the spherical surface which represents it, bears to the whole spherical surface. ”

“ For you have known that the universe is called a sphere by several astrologers, its center the center of the earth, and its radius equal to a line drawn from the center of the sun to the center of the earth. ”

“ Half a sphere is called a HEMISPHERE. 2. A body or the space bounded by a spherical surface. For most practical problems of navigation, the earth is considered a sphere, called the terrestrial sphere. ”

“ Unless the spheroid be in certain positions, the sphere possesses still higher beauty, as presenting the same circular and highly beautiful outline in every position; although a variety of positions is not essential to the perception of its peculiar beauty, whenever the observer has learned by different methods, and especially by different degrees of convergence of the two optic axes, to estimate the relative distances of the different points of the visible hemisphere, and thus to recognise the spherical form. ”

“ Thus the most perfect idea we can possibly acquire of a sphere, is by conceiving an infinite number of straight rays of equal lengths, issuing from the center, as from the eye, spreading every way alike; and circumscribed or wound about at their other extremities with close connected circular threads, or lines, forming a true spherical shell. ”

“ The sun’s rays, striking at all angles a sphere, that from moment to moment presents and withdraws different parts of its surface, and each of them for a different time daily throughout the year, would produce a considerable variety of changes even were the sphere uniform. ”

“ Seeing that the foregoing reasoning forces us to look upon the earth as a hollow sphere, or shell, in which the whole of the matter composing it is divided into two equal parts, attracted outwards and inwards by each other to a common plane, or region of meeting, we shall divide its whole volume into two equal parts radially, that is, one comprising a half from the surface inwards, and the other a half from the centre [170] outwards—that is to say, each one containing one-half of the whole volume of the earth. ”

“ Now, the surface of the earth seems to us (for this is another of the deceptive appearances which astronomy has to correct) to be a vast flat expanse, whose level is broken by hills and mountains, and the visible half of the celestial sphere seems to bend down on all sides and to rest 10upon the earth in a circle which extends all around us. This circle, where the heavens and the earth appear to meet, is called the horizon. ”

“ Every lesser circle of the sphere, will appear a circle to the eye, placed, as we have supposed all along, in the centre of the sphere. And, on the other hand, every visible circle will appear to coincide with some lesser circle of the sphere. ”

“ In many parts of the unfathomable ocean there may be communications or passages from the surface of the sphere on the outer side to the surface of the inner, at least all except the great mid-plane space, through which liquid apertures, light and heat may be communicated, perhaps, to the interior surface of the sphere. ”

“ A beautiful discovery, of which we shall speak further on, has already connected the geometry of spheres with that of straight lines and permits the introduction of the notion of coordinates of a sphere. ”

“ For instance, in order to form the conception of a sphere, I invent a cause at my pleasure - namely, a semicircle revolving round its center, and thus producing a sphere. (4) This is indisputably a true idea; and, although we know that no sphere in nature has ever actually been so formed, the perception remains true, and is the easiest manner of conceiving a sphere. ”

“ In our globe, which is a sphere, or rather an oblate spheroid, the centre of gravity will be the centre. Thus, if a plummet be suspended on the surface of the earth, it points directly to the centre of gravity, and, consequently, two plummet-lines suspended side by side cannot, strictly speaking, be parallel to each other. ”

“ A sphere with earth as its centre, and a radius of one minute of light’s flight, would contain 21 billions of earths; a sphere whose radius was nine minutes would contain 15,310 billions of earths, one of an hour’s radius 4,500 trillions, one of a day’s 62 quintillions, a week’s 20 sextillions, a month’s 1675 sextillions, and a year’s radius of the flight of light three octillions. ”

“ They would find that the space which they passed over from north to south on the earth, was proportional to the change of place of the horizon in the celestial sphere; and as the horizon is, at every place, in the direction of the earth’s apparently level surface, this observation would naturally suggest to them the opinion that the earth is placed within the celestial sphere, as a small globe in the middle of a much larger one. ”

“ The surface of a sphere cannot be regarded as a continuum, but as drops, or the fashion of the heavenly bodies, or as when a given quantity of air is displaced so many spheres are readily moved amidst other spheres. ”

“ This center in the great sphere of the universe is the earth. The poles of the sphere are the extremities of the axis on which it turns. The horizon cuts the sphere into two hemispheres, the one superior and the other inferior to the earth. The sky, which is continually turning, encircles all, one half of it being above, the other below the earth, which is as far removed from the superior part of the heavens as from the inferior.48
NOTES
30 Cicero’s allusion to Thales, cited p. ”

“ Wherever we are situated on the earth, we appear to be in the centre of a vast sphere, on the concave surface of which all celestial objects are inscribed. If we take any two points on the surface of the sphere, as two stars, for example, and imagine straight lines to be drawn to them from the eye, the angle included between these lines will be measured by the arc of the sky contained between the two points. ”

“ In perspective drawings of the sphere, the plane on which the representation is actually made may generally be any plane perpendicular to the line joining the centre of the sphere and the point of vision. ”

“ For in a right sphere the needle is parallel to the plane of the horizon; but when the cœlestial pole is vertically overhead, or when the pole of the earth is itself the place of the region, then the needle is perpendicular to the horizon. ”

“ But if I have got so far as to know that the earth is a sphere, and that its surface is spherical, I can cognize a priori and determine upon principles, from my knowledge of a small part of this surface—say to the extent of a degree—the diameter and circumference of the earth ”

“ The Apparent Concavity of the Earth as seen from a Balloon.—A perfectly-formed circle encompassed the visible planisphere beneath, or rather the concavo-sphere it might now be called, for I had attained a height from which the surface of the Earth assumed a regularly hollowed or concave appearance—an optical illusion which increases as you recede from it. ”

“ Not merely a hollow sphere, you understand. Everything—including empty space, solid matter and energy, is on the surface of this hypersphere. Thus our galaxy constitutes one point imbedded in the surface of the sphere ... ”

“ As we conceive of the sky, it does not consist of an entire sphere but only as a hemisphere bounded by the horizon. But we have no difficulty in extending the conception below the horizon, so that the earth with everything upon it is in the centre of a complete sphere. ”

“ The square of the number of times that each planet rotates during one revolution in its orbit is proportional to the cube of the diameter of its sphere of attraction.
The sphere of attraction is that within which the attraction of one planet is greater than that of the next interior or exterior planet.
PLANETARY ELEMENTS. Name of Body. Volume. Density. Light and Heat. Gravity. Time of Rotation. Distance from Sun. h. m. Sun 1415225 .250 ... ”

“ So you see, that except for the thickness of the blind rollers, the Cavorite exterior of the sphere will consist of windows or blinds, whichever you like to call them. Well, when all these windows or blinds are shut, no light, no heat, no gravitation, no radiant energy of any sort will get at the inside of the sphere, it will fly on through space in a straight line, as you say. ”