Newton's rings

“ And the reason is, that the Rays which enter the Eye at several parts of the Pupil, have several Obliquities to the Glasses, and those which are most oblique, if consider'd apart, would represent the Rings bigger than those which are the least oblique. Whence the breadth of the Perimeter of every white Ring is expanded outwards by the obliquest Rays, and inwards by the least oblique. And this Expansion is so much the greater by how much the greater is the difference of the Obliquity ”

“ Farther, by carefully measuring the diameters of each ring it is found, that whereas the glasses touch where the dark spot appears in the center of the rings made by reflexion, where the air is of twice the thickness at which the light of the first ring is reflected, there the light by being again transmitted makes the first dark ring; where the plate has three times that thickness which exhibits the first lucid ring, it again reflects the light forming the second lucid ring; when the thickness is four times the first, the light is again transmitted so as to make the second dark ring ”

“ And if this Proportion be found true by experience it will amount to a demonstration that these Rings (like those formed in thin Plates) do depend on the thickness of the Glass. I procured therefore another concavo-convex Plate of Glass ground on both sides to the same Sphere with the former Plate. Its thickness was 5/62 Parts of an Inch; and the Diameters [Pg 306] of the three first bright Rings measured between the brightest Parts of their Orbits at the distance of six Feet from the Glass were 3·4-1/6·5-1/8· Inches. ”

“ We found that the stability of the motion of a solid ring depended on so delicate an adjustment, and at the same time so unsymmetrical a distribution of mass, that even if the exact conditions were fulfilled, it could scarcely last long, and, if it did, the immense preponderance of one side of the ring would be easily observed, contrary to experience. These considerations, with others derived from the mechanical structure of so vast a body, compel us to abandon any theory of solid rings. ”

“ The power of gravity is of a different nature from the power of magnetism. For the magnetic attraction is not as the matter attracted. Some bodies are attracted more by the magnet, others less; most bodies not at all. The power of magnetism, in one and the same body, may be increased and diminished; and is sometimes far stronger, for the quantity of matter, than the power of gravity; and in receding from the magnet, decreases not in the duplicate, but almost in the triplicate proportion of the distance, as nearly as I could, judge from some rude observations. ”

“ And if the perpendicular RG cut off the hyperbolic area PIGR, which shall be to the area PIEQ as any arc CD to the arc BC described in the whole descent; the resistance in any place D will be to the force os gravity, as the area (OR/OQ) IEF - IGH to the area PINM.
For since the forces arising from gravity with which the body is urged in the places Z, B, D, a, are as the arcs CZ, CB, CD, Ca, and those arcs are as the areas PINM, PIEQ, PIGR, PITS; let those areas be the exponents both of the arcs and of the forces respectively. Let Dd be a very small space described by the body in its descent ”

“ When we have completely mastered the geometry of the world of events, we shall have inevitably learnt the mechanics of it. That is why Einstein, studying the geometry of the world and discovering that it was strictly non-Euclidean, found that he was at the same time studying the mechanical force of gravitation. And when he had made up his mind which of the possible varieties of non-Euclidean geometry was obeyed, and so settled the laws of the new geometry, the same decision settled the law of gravitation—a law approximating to, but not identical with, the law which Newton had given. ”

“ The general theory leaves us at liberty to choose between an infinite universe of space-time in which the star-matter would be concentrated into a nucleus, or else between two major types of finite universe, that of de Sitter and that of Einstein. The stability of the sidereal distribution would suggest Einstein’s cylindrical universe. If this model be accepted, the relativity of rotation and the complete relativity of mass as due to interactions between bodies, an idea so cherished by Mach, can at last be vindicated. ”

“ Since the resistance of the thread is of less moment in greater globes, I tried the experiment also with a globe whose diameter was 181⁄4 inches. The length of the pendulum between the point of suspension and the centre of oscillation was 1211⁄2 inches, and between the point of suspension and the knot in the thread 1091⁄2 inches. The arc described by the knot at the first descent of the pendulum was 32 inches. The arc described by the same knot in the last ascent after five oscillations was 28 inches. The sum of the arcs or the whole arc described in one mean oscillation was 60 inches. ”

“ If the right lines AR, BR, with the arc ACB, the chord AB, and the tangent AD, constitute three triangles RAB, RACB, RAD, and the points A and B approach and meet: I say, that the ultimate form of these evanescent triangles is that of similitude, and their ultimate ratio that of equality. For while the point B approaches towards the point A, consider always AB, AD, AR, as produced to the remote points b, d, and r, and rbd as drawn parallel to RD, and let the arc Acb be always similar to the arc ACB. Then supposing the points A and B to coincide, the angle bAd will vanish ”

“ But when the nodes are in the quadratures, it disturbs them very much, and, attracting the body P perpetually out of the plane of its orbit, it diminishes the inclination of the plane in the passage of the body from the quadratures to the syzygies, and again increases the same in the passage from the syzygies to the quadratures. Hence it comes to pass that when the body is in the syzygies, the inclination is then least of all, and returns to the first magnitude nearly, when the body arrives at the next node. ”

“ For the globe is to its circumscribed cylinder as two to three; and therefore the force which can destroy all the motion of the cylinder while the same cylinder is describing the length of four of its diameters, will destroy all the motion of the globe while the globe is describing two thirds of this length, that is, eight third parts of its own diameter. ”

“ A Centripetal force is that by which bodies are drawn or impelled, or any way tend, towards a point as to a centre. Of this sort is gravity, by which bodies tend to the centre of the Earth; Magnetism, by which iron tends to the loadstone; and that force, whatever it is, by which the planets are continually drawn aside from the rectilinear motions, which otherwise they would pursue, and made to revolve in curvilinear orbits. A stone, whirled about in a sling, endeavours to recede from the hand that turns it ”

“ As Attraction is stronger in small Magnets than in great ones in proportion to their Bulk, and Gravity is greater in the Surfaces of small Planets than in [Pg 352] those of great ones in proportion to their bulk, and small Bodies are agitated much more by electric attraction than great ones; so the smallness of the Rays of Light may contribute very much to the power of the Agent by which they are refracted. ”

“ Furthermore, geodesics are the only lines that possess any absolute structural significance. If, then, bodies moving in a gravitational field do not follow them, we cannot attribute the courses of these bodies to the intrinsic geometry of the space. Once more we are compelled to appeal to a force of gravitation, just as Newton did, and our entire theory collapses. In order to succeed in an identification of the courses of bodies in a gravitational field with the geodesics of the continuum, a fourth non-spatial dimension is unavoidable. ”

“ These appearances seem to indicate the same slow progress of the rings towards separation which we found to be the result of theory, and the remark that the inner edge of the inner ring is more distinct seems to indicate that the approach towards the planet is less rapid near the edge, as we had reason to conjecture. As to the apparent unchangeableness of the exterior diameter of the outer ring, we must remember that the outer rings are certainly far more dense than the inner one, and that a small change in the outer rings must balance a great change in the inner one. ”

“ From the focus On those tangents let fall the perpendiculars ST, St, which produce to V, v, so that TV, tv may be equal to TS, tS. Bifect Vv in O, and erect the indefinite perpendicular OH, and cut the right Line VS infinitely produced in K and k, so that VK be to KS, and Vk to kS as the principal axe of the trajectory to be described is to the distance of it's foci. On the diameter Kk describe a circle cutting OH in H; and with the foci S, H, and principal axe equal to VH, describe a trajectory. ”

“ But we have all along supposed the angle of contact to be neither infinitely greater nor infinitely less than the angles of contact made by circles and their tangents; that is, that the curvature at the point A is neither infinitely small nor infinitely great, or that the interval AJ is of a finite magnitude. For DB may be taken as AD3: in which case no circle can be drawn through the point A, between the tangent AD and the curve AB, and therefore the angle of contact will be infinitely less than those of circles. ”

“ Wherefore the innumerable intersections of a right line with a spiral, since this is but one simple curve and not reducible to more curves, require equations infinite in number of dimensions and roots, by which they may be all exhibited together. For the law and calculus of all is the same. For if a perpendicular is let fall from the pole upon that intersecting right line, and that perpendicular together with the intersecting line revolves about the pole, the intersections of the spiral will mutually pass the one into the other ”

“ The equivalence postulate also appears to point to the cylindrical universe; for if we accept the hypothesis of an infinite quasi-Euclidean universe with its nucleus of stars, we shall have to assume that the rim of a rotating disk of sufficiently large proportions could be conceived of as extending well beyond the farthest stars of the nucleus. Yet, the larger the disk and the more remote the rim from the stars, the greater would become the centrifugal force acting on the rim. ”

“ To say, therefore, that a rotation of the firmament round a test-body would affect the field of force surrounding the body is equivalent to stating that a rotation of the stars would be capable of affecting the distribution of the metrical field round the body and of dragging the geodesics along. There is no reason to be surprised at some such effect taking place, seeing that the phenomenon of gravitation connotes that matter, whether at rest or in motion, must affect the curvature of space-time, hence the lay of the metrical field. ”

“ Hence it must follow that in a gravitational field near matter, space must also be non-Euclidean. But there is a marked difference between the distribution of forces of inertia and those of gravitation. Forces of inertia occurring far from matter can be cancelled by the observer’s changing his motion. Thus, centrifugal force on the disk can be made to vanish; all we have to do is to arrest the disk’s, hence the observer’s, rotation. On the other hand, we cannot get rid of the force of gravitation. ”

“ All these considerations apply solely when no external forces are acting on the moving body. The presence of a force tears the body away from the geodesic it would normally follow, and also accelerates its speed.
We now come to Newton’s law of gravitation. It was a remarkable fact that although, according to the law of inertia, the motions of free bodies should be directed along the straight geodesics of three-dimensional Euclidean space, yet the courses and motions of the planets round the sun and of projectiles near the earth’s surface deviated widely from these geodesic paths. ”

“ And hence if more lesser bodies revolve about the great one, it may easily be inferred that the orbits described will approach nearer to ellipses, and the descriptions of area's will be more nearly equable, if all the bodies mutually attract and agitate each other with accelerative forces that are as their absolute forces directly, and the squares of the distances inversely; and if the focus of each orbit be placed in the common centre of gravity of all the interior bodies ”

“ But being incapable of leaving the Earth, and viewing it at a distance; and it’s rotation being smooth and uniform, we can neither see it move on it’s Axis as we do the Planets, nor feel ourselves affected by it’s motion. Yet there is one effect of such a motion which will enable us to judge with certainty whether the Earth revolves on it’s Axis or not. All Globes which do not turn round their Axes will be perfect spheres, on account of the equality of the weight of bodies on their surfaces; especially of the fluid parts. But all Globes which turn on their Axes will be oblate spheroids ”

“ However, before discussing this aspect of Einstein’s theory, we must mention that even in classical science the relativity of accelerated and rotationary motion could be accepted in a restricted sense for mechanical phenomena.
Consider, for instance, the earth rotating among the stars. The space defined by a frame attached to the earth would be permeated with a field of centrifugal and Coriolis forces. Now we were always perfectly at liberty to regard the earth as non-rotating, and the stars as rotating round it, provided we retained the existence of this special field of force. ”