Armillary sphere

“ The armillary sphere [2] seems to me poorly designed and in bad proportion. Its confused circles and odd figures, giving it the look of a conjurer's apparatus, are enough to frighten a child. The earth is too small; the circles are too many and too large. ”

“ From India there are only two references, very closely related and appearing in the best known astronomical texts in connection with descriptions of the armillary sphere and celestial globe. These texts are both quite garbled, but so far as one may understand them, it seems that the types of spheres and globes mentioned95 are more akin to those current in China than in the West. The relevant portions of text are as follows (italics supplied) :
The circle of the horizon is midway of the sphere. ”

“ We know that like his contemporary, Fortin, he also constructed armillary spheres, one example of which it has been possible to locate. Fiorini thus refers to it, his citation being given in free translation.178 It is a Copernican sphere, that is, having a representation of the sun placed at the common191 center of the armillae instead of a representation of the earth as in the Ptolemaic sphere. It may be found in the palace of Sr. Scaramucci in S. Maria a Monte in the province of Florence. Attached to a base of wood about 20 cm. in height is an iron rod 35 cm. long. ”

“ A.D.MDLXIIII” and was a gift to the museum by the Grand Duke Leopold I. The diameter of its horizon circle, including the attached parts, is 41 cm., and its height, including its base, 76 cm. It has been described as follows: “An armillary sphere, the armillae of which are of gilded brass. The small globe within the circles representing the earth is of the clearest crystal. The horizon is of gilded brass and rests on a branched support ornamented with human heads in relief. ”

“ The different spheres constituting our planet, and the other orbs in creation, most probably do not revolve on axes, parallel to each other, nor perform their revolutions in the same periods of time; as is indicated by the spots on the belts of Jupiter, which move faster on one belt than another.
The atmosphere surrounding the sphere is probably more dense on the interior than the exterior surface, the increased pressure of which must increase the force of gravity ”

“ The application which the Dr. makes of this structure of the earth is this: that the concave sides of the spheres are made up of magnetic matter; that they revolve about their diurnal axes in about twenty-four hours; that the outer sphere moves either a little faster or a little slower than the internal ball; that the magnetic pole, both of the external shell and included globe, are distant from the poles of rotation; and that the variation arises from a change of the relative distances of the external and internal poles in consequence of the difference of their revolutions. ”

“ That if the earth be not a solid globe, but a hollow concentric sphere, the quantity of matter being diminished, the attraction of gravitation must be lessened so much that all moveable bodies resting on the earth would be thrown off by centrifugal force, in the line of a tangent from the surface of the sphere. ”

“ According to Ptolemy’s scheme, the eighth sphere formed the outermost boundary of the universe; but later astronomers added to this system two other spheres—a ninth, called the Crystalline, which caused Precession of the Equinoxes; and a tenth, called the Primum Mobile, or First Moved, which brought about the alternation of day and night, by carrying all the other spheres round the Earth once in every twenty-four hours. ”

“ Thus, the material molecules of the planetary surface are not all at an equal distance from the centre; and, consequently, the earth is not a perfect sphere. It is, as D'Alembert has demonstrated, the bulging, equatorial portion which experiences, owing to the solar attraction, a retrograde movement, carrying onward the rest of the globe in a general march, called the precession of the equinoxes. ”

“ The variation which has been observed in their number, shape, and dimensions, can in no way be better accounted for, than by concluding the planet to be constituted of a number of concentric spheres, of different breadths, revolving on different axes, and with different velocities, so as sometimes to present to our view the verge of one sphere, and sometimes that of another: and the rays of the sun falling on the parts of the verges presented to us, would occasion the diversified appearances which we discover. ”

“ He has now all the material for applying the method of exhaustion and so proves the main propositions about the surface and volume of the sphere. The rest of the book applies the same procedure to a segment of the sphere. Surfaces of revolution are inscribed and circumscribed to a segment less than a hemisphere, and the theorem about the surface of the segment is finally proved in Prop. 42. Prop. 43 deduces the surface of a segment greater than a hemisphere. Prop. 44 gives the volume of the sector of the sphere which includes any segment. ”

“ But as the bodies of the solar system are very nearly spherical, they will all act upon one another, and upon bodies placed on152 their surface, as if they were so many centres of attraction; and therefore we obtain the law of gravity which subsists between spherical bodies, namely, that one sphere will act upon another with a force directly proportional to the quantity of matter, and inversely as the square of the distance between the centres of the spheres. ”

“ The spheres, in many parts of the unfathomable ocean, is believed to be water quite through from the concave or convex surfaces to the great mid-plane-space, and probably the earthy or solid matter of the sphere, may in many places extend quite through from one surface to the [94] other, tending, like ribs or braces, to support the sphere in its proper form. Such a formation of spheres appears to be supported by various facts and phenomena; amongst the most prominent of which are Volcanoes and Earthquakes. ”

“ If it should appear, by such experiments, that the attraction of mercury was in the inverse ratio of the square of the distance, it would be clearly demonstrated that its constituent particles were spherical, because a sphere is the only figure which observes this law, and at whatever distance globes are placed the law of their attraction is always the same. ”

“ And although this spherical form possesses an intensity of power decreasing in proportion as the sphere expands, still it has everywhere the same property of giving existence to its centre, since it has everywhere an intrinsic spherical character essentially connected with a central point as its indispensable term. Whence we see that the substantial form, though virtually extending into an indefinite sphere, is everywhere terminated to its own matter. ”

“ 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. ”

“ Hence, a person standing, or trees growing, on the interior surface, would be in no more danger of being precipitated to the next sphere, between them and the centre, than those on the outer part of the sphere, when they should be turned (what is familiarly called) down.
The experiments made on the density of the globe, by observations with the plum-line, at the foot of a mountain, are very ingenious; but they must be subject to great uncertainty. ”

“ Then, after a second assumption that bodies which are forced upwards in a fluid are forced upwards along the perpendiculars to the surface which pass through their centres of gravity, Archimedes deals with the position of rest and stability of a segment of a sphere floating in a fluid with its base entirely above or entirely below the surface. ”

“ The mechanical method is used for finding (1) the area of any parabolic segment, (2) the volume of a sphere and a spheroid, (3) the volume of a segment of a sphere and the volume of a right segment of each of the three conicoids of revolution, (4) the centre of gravity (a) of a hemisphere, (b) of any segment of a sphere, (c) of any right segment of a spheroid and a paraboloid of revolution, and (d) of a half-cylinder, or, in other words, of a semicircle.
Archimedes then proceeds to find the volumes of two solid figures, which are the special subject of the treatise. ”

“ We next find that the three-dimensional space inside the fluid possesses a curvature at every point which is proportional to the density of the fluid at every point. Assuming this density to be the same throughout, we see that the curvature of space is constant, and we obtain a spherical space of three dimensions analogous to a cap on the two-dimensional surface of an ordinary sphere in three-dimensional Euclidean space. This curvature of space inside the sphere has nothing to do with the dimensions of our fluid sphere; the curvature is governed solely by the density of the fluid. ”

“ In fact a plane surface may be assimilated to the surface of a sphere of infinite radius; so that if our sphere were to grow indefinitely we should find that our measurements lost little by little their Riemannian characteristics, until it would be impossible to distinguish them from Euclidean measurements. ”

“ 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. ”

“ If the spheres were entirely removed from each other's influence, the state of stress would be uniform round each sphere, and the pulls on its surface-elements would balance, giving no resultant force on the sphere. ”

“ Since all the diurnal circles are parallel to the equator, consequently, they would all, like the equator [83] be perpendicular to the horizon. Such a view of the heavenly bodies is called a right sphere, which may be thus defined: a right sphere is one in which all the daily revolutions of the stars are in circles perpendicular to the horizon. ”

“ It is not easy to obtain any real conception of the vastness of this enormous sphere. Its diameter is 108.7 times that of the earth, and its circumference proportional, so that the traveler who could make the circuit of the world in 80 days would need nearly 24 years for his journey around the sun. Since the surfaces of spheres vary as the squares, and bulks as the cubes, of their diameters, it follows that the sun's surface is nearly 12,000 times, and its volume, or bulk, more than 1,280,000 times, greater than that of the earth. ”

“ We imagine a wave-front divided into elementary rings or zones—often named after Huygens, but better after Fresnel—by spheres described round P (the point at which the aggregate effect is to be estimated) , the first sphere, touching the plane at O, with a radius equal to PO, and the succeeding spheres with radii increasing at each step by 1/2λ. There are thus marked out a series of circles, whose radii x are given by x2 + r 2 = (r + 1/2nλ) 2, or x2 = nλr nearly; so that the rings are at first of nearly equal area. Now the effect upon P of each element of the plane is proportional to its area ”

“ Therefore, as the attractive force is the same on all particles at equal distances from the centre of a sphere, the equatorial particles would recede from the centre, till their increase in number balance the centrifugal force by their attraction. Consequently, the sphere would become an oblate or flattened spheroid, and a fluid, partially or entirely covering a solid, as the ocean and atmosphere cover the earth, must assume that form in order to remain in equilibrio. ”

“ Now, supposing the sun to be exactly of the same diameter as the earth, and placed directly over the equator, when the low side of the verge was turned towards the sun, the direct rays from his northern limb, independent of refraction, would pass the edge of the lower part of the verge, and fall on the inner part of the sphere, on the concave part of the high side opposite, as far as eighteen degrees, or upwards. ”

“ A feasible plan is to surround a delicately poised mass by a thick pair of hemispheres (which may be hollow for liquids) , and note with a refractometer any change of position, which, since the attraction of a sphere at a point within it is zero, will be due either to induction or to irregularities of the sphere. Errors due to irregularities may be readily eliminated by rotating the sphere. ”

“ Fig. 71.—Well-known model exhibiting the oblate spheroidal form as a consequence of spinning about a central axis. The brass strip a looks like a transparent globe when whirled, and bulges out equatorially.
No. 10. Applying the idea of centrifugal force to the earth considered as a rotating body, he perceived that it could not be a true sphere, and calculated its oblateness, obtaining 28 miles greater equatorial than polar diameter. ”

“ The material crust of the earth rests on this placid sphere of vigor, excepting in a few places, where, as in the crevice we have entered, gaps, or crevices, in matter exist, beginning from near the outer surface and extending diagonally through the medial and inner spheres into the intra-earth space beyond. This medial sphere is a form of pure force, a disturbance of motion, and although without weight it induces, or conserves, gravity. It is invisible to mortal eyes, and is frictionless, but really is the bone of the earth. ”