Sphere (geometry)

“ The doctrine of the sphere is Geometry; for all forms are contained in the sphere. All geometrical proofs admit of being conducted through the sphere. Geometry has originated directly from arithmetic, or is arithmetic itself, with this difference, that the latter regards series of numbers as individualities, the former, however, as a whole. Arithmetic is a geometry seriebus discretis; geometry is an arithmetic seriebus continuis, a solidified arithmetic. ”

“ Just as we arrived at the infinitely large sphere by letting a finite sphere grow, so must we consider any finite sphere as having grown from a sphere with infinitely small extension; that is, from a point. To travel from the point to the infinitely distant plane in the sense of projective geometry, therefore, means that we have first to identify ourselves with the point and 'become' the plane by a process of uniform expansion in all directions. ”

“ A plane perpendicular to a radius at its extremity is tangent to the sphere.
The student should always have his attention called to the analogue in plane geometry, where there is one. If here we pass a plane through the radius in question, the figure formed on the plane will be that of a line tangent to a circle. If we revolve this about the line of the radius in question, as an axis, the circle will generate the sphere again, and the tangent line will generate the tangent plane. ”

“ Just as the circle may be defined either as a plane surface or as the bounding line which is the locus of a point in a plane at a given distance from a fixed point, so a sphere may be defined either as a solid or as the bounding surface which is the locus of a point in space at a given distance from a fixed point. In higher mathematics the circle is defined as the bounding line and the sphere as the bounding surface; that is, each is defined as a locus. ”

“ Non-Euclidean geometry, however, showed other possibilities. The surface of a sphere has no boundary, yet it is not infinite. In [Pg 165] traveling round the earth, we never reach “the edge of the world,” and yet the earth is not infinite. The surface of the earth is contained in three-dimensional space, but there is no reason in logic why three-dimensional space should not be constructed on an analogous plan. What we imagine to be straight lines going on for ever will then be like great circles on a sphere: they will ultimately return to their starting point. ”

“ In plane geometry it is easily proved that if two circles intersect, the tangents from any point on their common chord produced are equal. For if the common chord is AB and the point P is taken on AB produced, then the square on any tangent from P is equal to PB × PA. The line PBA is sometimes called the radical axis.
Similarly in this proposition concerning spheres, if from any point in the plane of the circle formed by the intersection of the two spherical surfaces lines are drawn tangent to either sphere, these tangents are equal. ”

“ If the centre is at infinity the tangent cone becomes a right circular cylinder touching the sphere along a great circle, and if the projection is, as in our case, orthographic, then the section of this cone by a plane of projection will be a circle equal to the great circle of the sphere. We get such a circle in the plan and another in the elevation, their centres being plan and elevation of the centre of the sphere. ”

“ What are the differences between the geometry on a sphere and the geometry on a plane? If you make a triangle on the earth, whose sides are great circles, you will not find that the angles of the triangle add up to two right angles: they will add up to rather more. The amount by which they exceed two right angles is proportional to the size of the triangle. ”

“ If we draw through any point P in space a line to cut a sphere in two points, A and S, and then construct the fourth harmonic of P with respect to A and B, the locus of this fourth harmonic, for various lines through P, is a plane called the polar plane of P with respect to the sphere. With this definition and theorem one can easily find dual relations between points and planes in space analogous to those between points and lines in a plane. ”

“ From the elementary geometry of sphere and cylinder it is clear that each strip of the projection is equal in area to the zone on the model which it represents, and that each portion of a strip is equal in area to the corresponding portion of a zone. Thus, each small four-sided figure (on the model) bounded by meridians and parallels is represented on the projection by a rectangle which is of exactly the same area, and this applies to any such figure however small. ”

“ We do not now enter into the proof, but it is sufficiently obvious that a sphere of which the radius is the geometric mean between the greatest and least radii of curvature at each point will to a large extent osculate the surface, so that a portion of the surface in the neighbourhood of the point will, generally speaking, have the same curvature as the sphere. ”

“ 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. ”

“ For many purposes the mathematical surface is assumed to be a plane; in other cases a sphere of radius 6371 kilometres (20,900,000 ft.) . In the case of extensive operations the surface must be considered as a compressed ellipsoid of rotation, whose minor axis coincides with the earth’s axis, and whose compression, flattening, or ellipticity is about 1/298. ”

“ 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? ”

“ Just as beings which are confined within a narrow region surrounding a point on a spherical surface, may fall into the error that a sphere is a geometric figure in which one diameter is particularly distinguished from the rest. ”

“ 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. ”

“ He affirms therefore, that though it be true in the Theory, that a Circle, whose Plain is inclined to the Axis of the Sphere by an Angle, whereof half the Diameter is the Sine, and which touches the Sphere in its Pole, will touch in all its parts a spherical Surface, that shall turn upon that Axe. But that it is true also, that that must be but a Mathematical Circle, and without Breadth, and which precisely touches the Body in its middle: Whereas in the practice, a Circle capable to keep Sand and Putty, must be of some breadth ”

“ Two spheres intersect in a plane, and the equation to a system of spheres which intersect in a common circle is x2 + y2 + z2 + 2Ax + D = O, in which A varies from sphere to sphere, and D is constant for all the spheres, the plane yz being the plane of intersection, and the axis of x the line of centres. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ It must be remembered that the focus, or focal point of a concave mirror, is the centre of the curve presented by such mirror. To make this clear, let us recur to first principles. A sphere is generated by the revolution of a circle round its axis; we may see it by spinning a half-crown on the table. Suppose, then, a circle of three inches radius so rotated, the result would be a sphere of six inches in diameter. ”

“ If curvature of C map circle C' intersects the base, the coaxal system cutting C' orthogonally does not intersect in a real point, and C has accordingly no centre. It is of interest to examine in what way a pseudo sphere differs from a plane as regards the behaviour of parallel lines. ”

“ In two spheres mutually gravitating each towards the other, if the matter in places on all sides round about and equi-distant from the centres is similar, the weight of either sphere towards the other will be reciprocally as the square of the distance between their centres. ”

“ The solid angle subtended at a given point by a closed curve is measured by the area of a spherical surface whose centre is the given point and whose radius is unity, the outline of which is traced by the intersection of the radius vector with the sphere as it traces the closed curve. ”

“ This, like the sphere, is called a surface of positive curvature, in reference to the plane, which has no curvature.
Now, just as to the plane corresponds an uncurved or homaloidal space, so to a surface of positive curvature corresponds a space of positive curvature; and if the space in which we live can be proved to have the slightest positive curvature, it instantly follows that the universe is only finite in extent, and that every physical straight line, for example, every ray of light, if sufficiently produced, returns into itself. ”

“ For since the indicatrix at every point is a rectangular hyperbola, the angle between the elements of two intersecting curves-=angle between their conjugate tangents; but this =angle between conjugate tangents to representative curves on sphere=angle between these curves themselves.
The problem of finding a minimal surface to pass through a given curve in space, known as Plateau's problem, possesses an exceptional interest from the circumstance that it can be always exhibited to the eye in the following way by an actual physical experiment. ”

“ Many hollow cylinders and spherical figures, we know do exist on the surface of the earth; and notwithstanding their own gravity, which the different parts exert on each other, as well as the gravity of the earth, they retain their shape and position; and had the matter in the earth originally been thrown by a centrifugal force into the form of a hollow sphere, or had the first creating power originally given it that shape,—I can discover no good reason for a change; neither should I entertain any apprehensions of the particles of matter coalescing at the centre. ”

“ 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. ”

“ For the conception "sphere" requires that all diameters be equal, whereas we know that they are not so, since the distance from pole to pole has been proved to be smaller than that between opposite points on the equator. The earth is an ellipsoid of rotation, and this truth is absolute in the face of the errors which are expressed by the terms, plane and sphere. ”