Physical plane

“ We may regard the foot of every perpendicular as the actual position of a moving point, and it can be proved that if the mass of each particle be multiplied into the area which the foot of its perpendicular describes in a second round any point in the plane, and then be added to the similar products from all the other particles, only observing the proper precautions as to sign, the sum will remain constant, i.e., in any other second the total quantity arrived at will be exactly the same. ”

“ If two similar mediums be separated from each other by a space terminated on both sides by parallel planes, and a body in its passge through that space be attracted or impelled perpendicular towards either of thos mediums, and not agitated or hindered by any other force; and the attraction be every where the same at equal diameter from either plane, taken towards the same hand of the plane; I say that the fine of incidence upon either plane will be to the due of emergence from the other plane, in a given ratio. ”

“ Here the radius, a straight line, is clearly the determining factor.
We now move to the other pole of the primary polarity, that is to the plane, and let the sphere arise by imagining the plane approaching an infinitely distant point evenly from all sides. We view the process realistically only by imagining ourselves in the plane, so that we surround the point from all sides, with the distance between us and
the point diminishing gradually. ”

“ In outer nature the all-embracing plane is as much the 'centre' of the earth's field of levity as the all-relating point is the centre of her field of gravity. All actions of dynamic entities, such as that of the ur-plant and its subordinate types, start from this plane. Seeds, eye-formations, etc., are nothing but individual all-relating points in respect of this plane. All that springs from such points does so because of the point's relation to the all-embracing plane. ”

“ If the motion is in a vertical plane, the action of gravity must be taken into the reckoning; if the motion is in a horizontal plane, the terms expressing variation of elevation of the filament will disappear. [3]
Let AB, CD (fig. 34) be two consecutive stream lines, at present assumed to be in a vertical plane, and PQ a normal to these lines making an angle φ with the vertical. Let P, Q be two particles moving along these lines at a distance PQ = ds, and let z be the height of Q above the horizontal plane with reference to which the energy is measured, v its velocity, and p its pressure. ”

“ I say, in the first place, that the whole force and efficacy of all the particles that are situate in AE, the circle of the equator, and disposed uniformly without the globe, encompassing the same after the manner of a ring, to wheel the earth about its centre, is to the whole force and efficacy of as many particles in that point A of the equator which is at the greatest distance from the plane QR, to wheel the earth about its centre with a like circular motion, as 1 to 2. ”

“ Let us now suppose a system of particles moving in space under their mutual actions. The projections of these particles on a plane will move as if they were the particles themselves subjected to the action of forces which are the projections of the actual forces on the same plane, and as the reactions between any two particles are equal and opposite, the projections of 379those reactions on the plane are equal and opposite. ”

“ If a heavy body, as for example, a sphere of metal be suspended by a string from a point A, Figure 66, vertically above N, the North Pole of the earth, and allowed to hang freely, the motion of the earth about its axis ANS will twist the string, and so cause the sphere to rotate about its vertical diameter. If, now, the sphere be drawn aside to a point B and allowed to drop gently, it will begin to vibrate in the plane NAB, and as the rotation communicated to the sphere does not tend to withdraw it from that plane, it will continue constantly to move in it. ”

“ 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. ”

“ The course of each planet lies throughout in one plane or flat surface, in which the sun is placed; but they do not all move in the same plane, though the different planes, in which they move, cross each other in very small angles. They all cross each other in lines, which pass through the sun; because the sun lies in the plane of each orbit. This inclination of the several orbits to each other is represented in fig. 90. The line, in which the plane of any orbit crosses the plane of the earth’s motion, is called the line of the nodes of that orbit. ”

“ The actual value of the constant will depend both on the system and on the plane. For a given system the constant will differ for the different planes which may be drawn, and there will be some planes in which that sum will be zero. In other words, in those planes the areas described by the feet of the perpendiculars, multiplied by the masses of the particles which are moving in one way, will be precisely equal to the similar sum obtained from the particles moving in the opposite direction. ”

“ The waves from a source surrounded by a uniform medium at rest spread out as spheres with the source as centre. If we take one of these spheres a distance from the source very great as compared with a single wave-length, and draw a radius to a point on the sphere, then for some little way round that point the sphere may be regarded as a plane perpendicular to the radius or the line of propagation. Every particle in the plane will have the same displacement and the same velocity, and these will be perpendicular to the plane and parallel to the line of propagation. ”

“ It touches the limite of absolute stability at the critical point . For we may take a pair of points in and having a common tangent plane as near to as we choose, and the line joining them upon the primitive surface made by a plane section perpendicular to the tangent plane, will pass through an area of instability. ”

“ The centre of gravity of the hemispherical base is, of course, in its axis; and therefore tends to approach the horizontal plane as much as possible, and this can never be accomplished, until the axis becomes perpendicular to the horizon. Whenever the curved surface is in any other position, the centre of gravity is not in the lowest place to which it can descend, as may be seen by the diagram which I have just sketched. ”

“ You can see at once that this condition could be obtained in a perfect convex surface fitting a perfect concave of the same radius. Fortunately we have a check to guard against this error. To produce a perfect plane, three surfaces must be worked together, unless we have a true plane to commence with; but to make this true plane by this method we must work three together, and if each one comes up to the demands of this most rigorous test, we may rest assured that we have attained a degree of accuracy almost beyond human conception. Let me illustrate. Suppose we have plates 1, 2, and 3, Fig. ”

“ Since then the area described by the line Cp in an immovable plane is proportional to the time, it is manifest that a body, being acted upon by a just quantity of centripetal force, may revolve with the point p in the curve line which the same point p, by the method just now explained, may be made to describe in an immovable plane. ”

“ Homogeneal Rays which flow from several Points of any Object, and fall perpendicularly or almost perpendicularly on any reflecting or refracting Plane or spherical Surface, shall afterwards diverge from so many other Points, or be parallel to so many other Lines, or converge to so many other Points, either accurately or without any sensible Error. ”

“ If the ball is projected upwards, its motion is gradually retarded; if downwards, it is continually accelerated. This is true if the plane is fairly smooth—like a well-planed plank—and the inclination of the plane not very small. If we imagine the experiment performed on an ideal plane, which is supposed perfectly smooth, we should expect the same results to follow, however small the inclination of the plane. ”

“ Let us think of a sphere rotating round a fixed axis. Every particle of that sphere will of course describe a circle around the axis, and all these circles will lie in parallel planes. We may for our present purpose regard each atom of the body as a little planet revolving in a circular orbit, and therefore the moment of momentum of the entire sphere will be found by simply adding together the moments of momentum of all the different atoms of which the sphere is composed. ”

“ Imagine any particle projected along a horizontal plane without friction; then we know, from what has been more fully explained in the preceding pages, that this particle will move along this same plane with a motion which is uniform and perpetual, provided the plane has no limits. But if the plane is limited and elevated, then the moving particle, which we imagine to be a heavy one, will on passing over the edge of the plane acquire, in addition to its previous uniform and perpetual motion, a downward propensity due to its own weight ”

“ Now if the globes are not homogeneous, the space described by the denser globe must be augmented in the ratio of the density. For the motion, with an equal velocity, is greater in the ratio of the density, and the time (by this Prop.) is augmented in the ratio of motion directly, and the space described in the ratio of the time.
Cor. 5. And if the globes move in different mediums, the space, in a medium which, cæteris paribus, resists the most, must be diminished in the ratio of the greater resistance. ”

“ The fact that the force acting on each planet is directed to the sun, necessitates the equable description of areas.
2. The fact that the force varies as the inverse square of the distance, necessitates motion in an ellipse, or some other conic section, with the sun in one focus.
3. The fact that one attracting body acts on all the planets with an inverse square law, causes the cubes of their [Pg 207] mean distances to be proportional to the squares of their periodic times. ”

“ 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. ”

“ If a rare medium consist of very small quiescent particles of equal magnitudes, and freely disposed at equal distances from one another: to find the resistance of a globe moving uniformly forward in this medium. Case 1. Let a cylinder described with the same diameter and altitude be conceived to go forward with the same velocity in the direction of its axis through the same medium; and let us suppose that the particles of the medium, on which the globe or cylinder falls, fly back with as great a force of reflexion as possible. ”

“ If by any distribution of imaginary sources within the spherical surface we can effect this, the value of the corresponding potential outside the sphere is the true and only one. The potential inside the sphere must really be constant and equal to that at the surface. ”

“ In looking at the properties and motions of a planet, therefore, as our earth for example, we find that a planet is a sphere, or more correctly an oblate spheroid; that the earth or planet is a magnet possessing polarity, having a north and south pole; that it has rotation on an axis, in addition to translation in an orbit, and that it is subject to the universal Law of Gravitation. ”

“ If we now assume that all rocks flow tinder pressure, that the critical point is variable and that the modulus of compression is also variable, being greater for the lighter rocks and less for the heavier, and that this modulus is greatly accelerated at the critical point, we have a law which will regiment the facts of geonomy as the facts of astronomy are marshaled by the law of gravity. ”

“ Let us now see how these act on bodies moving in large and small curves, and how the waters on the earth's surface are driven by centrifugal force toward a line tangent to her orbit. Since the length of the orbital curve of the earth is very great, and therefore not much deflected from a straight line, the waters are driven very little above the usual surface, no matter how rapidly the earth herself may move in this curve. ”

“ When this system is inverted back again into the form of the two spheres in contact, we have a corresponding series of negative images, the distances of which from the point of contact are of the form , where is positive for the sphere and negative for the sphere . The charge of each image, when the potential of the spheres is unity, is numerically equal to its distance from the point of contact, and is always negative. ”

“ We shall simplify the matter by regarding the orbits of the planets or other bodies as circles The fact that these orbits are ellipses, which are, however, very nearly circles, will not appreciably affect the argument.
Let us, then, suppose a single planet revolving round a fixed sun, in the centre. The energy of this system has two parts. There is first the energy due to the velocity of the planet, and this is found by taking half the product of the mass of the planet and the square of its velocity. ”