Range of motion

“ When the sphere is in motion it produces the same effect as a uniformly-charged line whose length bears to the diameter of the sphere the same ratio as the velocity of the sphere bears to the velocity of light. ”

“ The surface itself is moving by the diurnal rotation of the earth upon its axis, as well as by the annual motion in its orbit round the sun. These motions, and others to which the earth is subject, must be all compounded by the theorem of the parallelogram of forces before we can obtain the absolute state of the body with respect to motion or rest. ”

“ And therefore the same law takes place in a system consisting of many bodies as in one single body, with regard to their persevering in their state of motion or of rest. For the progressive motion, whether of one single body, or of a whole system of bodies, is always to be estimated from the motion of the centre of gravity. ”

“ The motion of a heavy body down a curve differs in an important respect from the motion down an inclined plane. Every part of the plane being equally inclined to the vertical direction, the effect of gravity in the direction of the plane is uniform; and, consequently, the phenomena obey all the established laws of uniformly-accelerated motion. ”

“ If a state of motion exists in a system of bodies, in which the components of the light vector are certain functions of relative coordinates and time t, thus also the functions that arise when we replace t by -t, represents the values of the components that correspond to a possible motion. ”

“ The science of the mathematics of motion deals with the speed of one and its trajectory, the speed of another and its trajectory, and of their collisions, and for this purpose it has to deal with the measure of their relations, and forever relation is considered and thus an illusion is sometimes produced, when motion itself seems to be wholly relation. ”

“ Any kind of centripetal force being supposed, and the centre of force, and any plane whatsoever in which the body revolves, being given, and the quadratures of curvilinear figures being allowed; it is required to determine the motion of a body going of from a given place, with a given velocity, in the direction of a given right line in that plane. ”

“ I understand that, besides the law of universal gravitation, there are two fundamental laws of motion. By one of these laws, if a body be set in motion and be acted on by no other than the projectile force, it will move forward in a straight line and with a uniform velocity forever. ”

“ Any length being given, which is passed through in a given time with uniform motion, to find out what length shall be passed through in the same time with motion uniformly accelerated, that is, with such motion that the proportion of the lengths passed through be continually duplicate to that of their times, and that the line of the impetus last acquired be equal to the line of the whole time of the motion. ”

“ 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 Proposition) is augmented in the ratio of motion directly, and the space described in the ratio of the time.
Corollary 5. And if the globes move in different mediums, the space, in a medium which, caeteris paribus, resists the most, must be diminished in the ratio of the greater resistance. ”

“ And when a body is thrown upwards, its uniform gravity impresses forces and takes off velocities proportional to the times; and the times of ascending to the greatest heights are as the velocities to be taken off, and those heights are as the velocities and the times conjunctly, or in the duplicate ratio of the velocities. And if a body be projected in any direction, the motion arising from its projection is compounded with the motion arising from its gravity. ”

“ B. You perplex yourself by confounding the idea of circular motion, with that of motion in a right line; you must think only of the motion of a body round a fixed line, and you will find, that if the parts farthest from the centre had not the greatest velocity, those parts would not be able to keep up with the rest of the body, and would be left behind. Do not the extremities of the vanes of a windmill move over a much greater space, than the parts nearest the axis of motion? (plate 3. fig. 1.) ”

“ Many modern physicists mythologize in this manner about motion, being unable to distinguish motion as an abstract property, because motion is formulated in terms of space and force in terms of parallelograms. ”

“ It is obvious that the motions of a pair of points may be varied in any manner, whether by direct or by lateral deviation, and yet that their comparative motion may remain constant, in consequence of the deviations taking place in the same proportions, in the same directions and at the same instants for both points. ”

“ The laws of motion as established by observation and experiment at the surface of the earth must be considered as mathematically certain. Now, it is an observed fact that the planets in their motion deviate from straight lines in a certain way. ”

“ The astronomy of the ancients admitted none but uniform circular motions, and could therefore be completely cultivated by the aid of their elementary geometry. But the pure science of motion might be extended to all motions, however varied as to the speed or the path of the moving body. ”

“ The manner in which you compound these different momenta to obtain their resultant strikes me as so novel that my mind is left in no small confusion. I do not refer to the composition of two uniform motions, even when unequal, and when one takes place along a horizontal, the other along a vertical direction; because in this case I am thoroughly convinced that the resultant is a motion whose square is equal to the sum of the squares of the two components. The confusion arises when one undertakes to compound a uniform horizontal motion with a vertical one which is naturally accelerated. ”

“ More important astronomically—and probably intrinsically also—is what he calls the science of local motion,80 which deals with the motion of bodies. He builds up on the basis of his early experiments (§ 116) a theory of falling bodies, in which occurs for the first time the important idea of uniformly accelerated motion, or uniform acceleration, i.e. motion in which the moving body receives in every equal interval of time an equal increase of velocity. ”

“ That Bodies refract Light by acting upon its Rays in Lines perpendicular to their Surfaces. But in order to this Demonstration, I must distinguish the Motion of every Ray into two Motions, the one perpendicular to the refracting Surface, the other parallel to it, and concerning the perpendicular Motion lay down the following Proposition. ”

“ The law of the complex effect is explained, by being resolved into the separate laws of the causes which contribute to it. Thus, the law of the motion of a planet is resolved into the law of the acquired force, which tends to produce a uniform motion in the tangent, and the law of the centripetal force, which tends to produce an accelerating motion toward the sun; the real motion being a compound of the two. ”

“ Furthermore, change of motion never takes place abruptly; if its direction is changed, it is always through a curve without angles; if its velocity alters, it is by degrees. The gradual changes which are constantly taking place are conceived by geometers to be compounded together according to the rules of the parallelogram of forces. ”

“ Applying these general considerations to the case of motion, we find that, even within the sphere of immediate sense-data, it is necessary, or at any rate more consonant with the facts than any other equally simple view, to distinguish instantaneous states of objects, and to regard such states as forming a compact series. Let us consider a body which is moving swiftly enough for its motion to be perceptible, and long enough for its motion to be not wholly comprised in one sensation. ”

“ When a body in a state of uniform motion receives an impulse in a direction not coinciding with that of its motion, it will move uniformly after the impulse in an intermediate direction, which may be determined by the principles established for the composition of motion in the last chapter. ”

“ Motion. A continued and successive change of place, either of a whole body, or of the particles of [Pg 211] which a body is composed; the earth in revolving upon its axis only, would not change its place as a body, but all the particles of which it is composed, would revolve round a common axis of motion. In revolving in its orbit, its whole mass is constantly occupying a new portion of space. ”

“ The general law that the change of motion is proportional to the moving force and takes place in its direction, provided we take care always to regard the reaction of curves, surfaces, obstacles, &c., as so many real moving forces of (for a time) unknown magnitude, will always help us out of any dynamical scrape we may get into. ”

“ Hence the work which any moving mass whatever is competent to perform, in virtue of the motion which it at any moment possesses, is jointly proportional to its weight and the square of its velocity. Here, then, we have a second measure of work, in which we simply translate the idea of height into its equivalent idea of motion. ”

“ It was thus stated, “The motion impressed on each particle of any system by the forces which act upon it, may be resolved into two, the effective motion, and the motion gained or lost: the effective motions will be the real motions of the parts, and the motions gained and lost will be such as would keep the system at rest.” The distinction of statics, the doctrine of equilibrium, and dynamics, the doctrine of motion, was, as we have seen, fundamental; and the difference of difficulty and complexity in the two subjects was well understood, and generally recognized by mathematicians. ”

“ In revising our successive judgments concerning a body's course or velocity, we take for granted that there is an actual course or an actual velocity—we take for granted that there are fixed points in space with respect to which all motions are absolute; and we find it impossible to rid ourselves of this idea. Nevertheless, absolute motion cannot even be imagined, much less known. ”

“ They comprised myriads of infinitesimal points, ostensibly motionless but actually moving very lazily: an individual point moved at most by a two-minute arc in an hour, and that not in a straight line but as it were oscillating about its own center of motion. ”

“ The analysis is more complex than that of Laplace, where the mutual attraction of the ocean was neglected, but this was perhaps inevitable. Our first task is to form the equations of motion and continuity, which will be equally applicable to all forms of the theory.
§ 15. Equations of Motion.—Let be the radius vector, colatitude and east longitude of a point with reference to an origin, a polar axis and a zero-meridian rotating with a uniform angular velocity from west to east. ”

“ And, by composition, the sum of the particles of time, in which all the particles PQ of the velocity AP are generated, will be as the sum of the particles of the sector ATD; that is, the whole time will be as the whole sector. ”

“ For those authors who treat of the motion of heavy bodies use to consider the ascent and descent of such bodies, not only in a perpendicular direction, but at all degrees obliquity upon any given planes; and for the same reason we are to consider in this place the motions of bodies tending to centres by means of any forces whatsoever. when those bodies move in eccentrical planes. These planes are supposed to be perfectly smooth and polished so as not to retard the motion of the bodies in the least. ”

“ And universally, if equally swift bodies are resisted in the ratio of any power of the diameters: the spaces, in which homogeneous globes, moving with any velocity whatsoever, will lose parts of their motions proportional to the wholes, will be as the cubes of the diameters applied to that power. ”

“ The motion of a sphere, rolling and spinning in the interior of a spherical bowl, or on the top of a sphere, is found to be of the same character as the motion of the axis of a spinning top about a fixed point. ”

“ By the same reasoning if a body, acted on by forces tending to two or more centres in any the same right line CO, should describe in a free space any curve line ST; the area AOP would always proportional to the time. ”