Superposition

“ The same things being supposed, I say, that the space described in the ascent or descent is as the difference of the area by which the time is expressed, and of some other area which is augmented or diminished in an arithmetical progression; if the forces compounded of the resistance and the gravity be taken, in a geometrical progression. Take AC (in these three figures) proportional to the gravity, and AK to the resistance; but take them on the same side of the point A, if the
body is descending, otherwise on the contrary. ”

“ Wherefore the points B, C, D lie in a conic section described about the focus S, in such manner that all the right lines drawn from the focus S to the several points of the section, and the perpendicular: let fall from the same points on the right line GF are in that given ratio. ”

“ For division (which is all that a mill, or pestle, or any other body, does upon another, in reducing it to insensible parts) can never take away either solidity, extension, figure, or mobility from any body, but only makes two or more distinct separate masses of matter, of that which was but one before ; all which distinct masses, reckoned as so many distinct bodies, after division, make a certain number. These I call original or primary qualities of body, which I think we may observe to produce simple ideas in us, viz. solidity, extension, figure, motion or rest, and number. ”

“ This division is taken out of the 4 d j. of Euclide, where rectitude or straightnes is attributed to a line, as if from it both surfaces and bodies were to have it. And even so the rectitude of a solid figure, here-after shall be understood by a right line perpendicular from the toppe unto the center of the base. Wherefore rectitude is propper unto a line: And therefore also obliquity or crookednesse, from whence a surface is judged to be right or oblique, and a body right or oblique.
5. A right line is that which lyeth equally betweene his owne bounds: A crooked line lieth contrariwise. 4. ”

“ I say that the attractive force of the whole solid, in the recess from its plane superficies, will decrease in the ratio of a power whose side is the distance of the corpuscle from the plane, and its index less by 3 than the index of the power of the distances. ”

“ The spaces which a body describes by any finite force urging it, whether that force is determined and immutable, or is continually augmented or continually diminished, are in the very beginning of the motion one to the other in the duplicate ratio of the times. ”

“ For the distances of the bodies from their common centre of gravity are reciprocally as the bodies; and therefore in a given ratio to each other; and thence by composition of ratio's, in a given ratio the whole distance between the bodies. ”

“ If any plane figure revolves about an external axis in its plane, the volume of the solid generated by the revolution is equal to the product of the area of the figure and the distance travelled by the centroid of the figure. ”

“ Is it possible, by continued subdivision, to reach a particle the division of which would put an end to the existence of the substance—not in the sense of annihilation, but, in other words, to reach a particle whose division compels a substance to suffer death by yielding up those identifying properties by which alone we distinguish it from other kinds of matter? ”

“ But if their quantities of matter are unequal, then, at an average of all the chances, each will intercept a number of particles proportional to its quantity of matter, and so the forces with which the bodies are impelled toward one another, will be as the quantity of matter directly, and the square of the distance inversely. ”

“ The total work done by two concurrent forces acting on a particle, or on a rigid body, in any infinitely small displacement, is equal to the work of their resultant. Let AB, AC (fig. 46) represent the forces, AD their resultant, and let AH be the direction of the displacement δs of the point A. The proposition follows at once from the fact that the sum of orthogonal projections of AB→, AC→ on AH is equal to the projection of AD→. ”

“ And, even could a central force be truly conceived as acting in lines not straight, the average distribution of its effects upon the inner surface of the surrounding sphere would still follow the same law. ”

“ But the errors that are generated by proportional forces, similarly applied to the bodies at similar parts of the similar figures, are as the forces and the squares of the times conjunctly.
Cor. 3. The same thing is to be understood of any spaces whatsoever described by bodies urged with different forces; all which, in the very beginning of the motion, are as the forces and the squares of the times conjunctly. ”

“ A line having no balance or symmetry in itself may become balanced. The line may be regarded as if it were a series of dots close together. The line is then a relation of positions indicated by dots. It is a composition of attractions corresponding and equal. It is only necessary, then, to find what I have called the center of equilibrium, the balance-center of the attractions, and to indicate that center by a symmetrical inclosure. The line will then become balanced.
Fig. 93
Here is a line. ”

“ The philosophy of the fact is simply this--while a body revolves on its axis, the component particles of its mass move in circles, the centres of which are placed in the axis; a centrifugal force therefore is generated, which is resisted by the cohesion of the parts of the mass, and this tendency of each particle to fly off is expended in exciting a pressure upon the axis ”

“ In a system of bodies whose forces decrease in the duplicate ratio of the distances, if the lesser revolve about one very great one in ellipses, having their common focus in the centre of that great body, and of a figure exceedingly accurate; and moreover by radii drawn to that great body describe areas proportional to the times exactly; the absolute forces of those bodies to each other will be either accurately or very nearly in the ratio of the bodies. ”

“ We suppose as before that the mass especially considered is bounded by a surface of which all that lies in the region of non-homogeneity is such as may be traced by a moving normal to the dividing surface. ”

“ A sphere is the figure of equilibrium which any aggregate of units tends to assume, under the influence of simple mutual attraction. Where the number of units is small and their mutual polarities are decided, this proclivity towards spherical grouping will be overcome by the tendency towards some more special form, determined by their mutual polarities. ”

“ Imagine curve lines to be described on any plane, and to revolve about any given axes passing through the centre of force, and by that revolution to describe curve superficies; and that the bodies move in such sort that their centres may be always found in those superficies. If those bodies reciprocate to and fro with an oblique ascent and descent, their motions will be performed in planes passing through the axis, and therefore in the curve lines, by whose revolution those curve superficies were generated. ”

“ Let us note, in the first place, that the appearance which our plane diagrams suggest of inequality of the several cells is apt to be deceptive; for the differences of magnitude apparent in one plane may well be, and probably generally are, balanced by equal and opposite differences in another. ”

“ The inference is so simple a one as naturally to be recognized by those whose thoughts go a little beyond their lessons in geometry.38 If the reviewer means to ask, whence I get the implied assumption that central forces act only in straight lines, I reply that this assumption has a warrant akin to that of Newton’s first axiom, that a moving body will continue moving in a straight line unless interfered with. For that the force exerted by one centre on another should act in a curved line, implies the conception of some second force, complicating the direct effect of the first. ”

“ For if the bodies are divided into particles proportional to the wholes, and alike situated in them, it will be, as the attraction towards any particle of one of the bodies to the attraction towards the correspondent particle in the other body, so are the attractions towards the several particles of the first body, to the attractions towards the several correspondent particles of the other body; and, by composition, so is the attraction towards the first whole body to the attraction towards the second whole body. ”

“ 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 be always proportional to the time. ”

“ These proportions take place also, when the attraction arises from the attractive virtue of both spheres mutually exerted upon each other. For the attraction is only doubled by the conjunction of the forces, the proportions remaining as before. ”

“ If we draw a line upon the primitive surface, dividing it into parts which represent respectively stable and unstable equilibrium, in respect to continuous changes, i.e., dividing the surface which is concave upward in both its principal curvatures from that which is concave downward in one or both, this line, which may be called the limit of essential instability, must have a form somewhat like that represented by in figure 2. ”

“ In general, therefore, if by the positive side of a surface for which is constant we understand the side on which has the greater value, we may say that denotes the component parallel to of the force exerted by the matter on the positive side of a surface for which is constant upon the matter on the negative side of that surface per unit of the surface measured in the state of reference. ”

“ The Proposition, That the Superficies of a Sphere is equal to the Area's of Four of its greatest Circles, which is one of the most wonderful Inventions that was ever found in Geometry, shews him to have been a much greater Man, than all that is said of him by the Roman, or Greek Historians. ”

“ Supposing that the centripetal force is reciprocally proportional to the square of the distance of the places from the centre; it is required to define the spaces which a body, falling directly, describes in given times. ”

“ The forces (mentioned at the end of section II) acting upon the center of the sides of length , are parallel to and remain parallel to at the transformation, as well as the connecting line of their contact points. ”

“ Thus much of the Perpendicle, and parallell equality of plaine right lines: Their Proportion is the last thing to be considered of them.
The truth of this element dependeth upon the nature of the parallells: And that throughout all kindes of equality and inequality, both greater and lesser. For if the lines thus cut be perpendiculars, the portions [71] intercepted betweene the two parallels shall be equall: for common perpendiculars doe make parallell equality, as before hath beene taught, and here thou seest. ”

“ We have a division of Concrete Science which generalizes those concomitants of this re-dis­tri­bu­tion that hold good among all orders of concrete objects—a division which explains why, along with a {92} predominating integration of matter and dissipation of motion, there goes a change from an indefinite, incoherent homogeneity, to a definite, coherent heterogeneity ”

“ 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. ”

“ I say, that those two figures AacE, PprT, are to one another in that same ratio. For as the parallelograms in the one are severally to the parallelograms in the other, so (by composition) is the sum of all in the one to the sum of all in the other; and so is the one figure to the other; because (by Lem. III) the former figure to the former sum, and the latter figure to the latter sum, are both in the ratio of equality. ”

“ The whole problem of the solid, up to a certain point, is contained in our plane diagram of Fig. 138. And when our particular solid is a solid of revolution, then it is obvious that a study of its plane of symmetry (that is to say any plane passing through its axis of rotation) gives us the solution of the whole problem. The right cone is a case in point, for here the in­ves­ti­ga­tion of its modes of symmetrical subdivision is completely met by an examination of the isosceles triangle which constitutes its plane of symmetry. ”

“ An attracting body being given, it is required to find the ratio of the decrease of the centripetal forces tending to its several points. The body given must be formed into a sphere, a cylinder, or some regular figure, whose law of attraction answering to any ratio of decrease may be found by Prop. ”

“ The expression for the superficial tension is most readily found with the aid of the idea of superficial energy, introduced into the subject by Gauss. Since the tension is constant, the work that must be done to extend the surface by one unit of area measures the tension, and the work required for the generation of any surface is the product of the tension and the area. ”

“ Rule 2a. If the primary mass is divided into two horizontal divisions, the dominance should be either in the upper or the lower section. Plate 7 shows this division of the primary mass—the simplest division of the space. A space divided just half way from top to bottom would be monotonous and expressive of the ratio of one to one. This arrangement as we have already discovered in the second chapter is not conducive to good design. ”

“ To fix our ideas, let us suppose that there are four homogeneous masses , and , which meet one another in four surfaces, which we may call , and , these surfaces all meeting along a line . This is indicated in figure 11 by a section of the surfaces cutting the line at right angles at a point . In an infinitesimal variation of the state of the system, we may conceive of a small surface being formed between and (to be called ) , so that the section of the surfaces of discontinuity by the same plane takes the form indicated in figure 12. ”

“ Supossing the sine of incidence upon any superficies to be in a given ratio to the sine of emergence; and that the inflection of the passs of those bodies near that superficies is performed in a very short space which may be confirmed as a point, it is required to determine such a superficies as may cause all the corpuscles issuing from any one given place to converge to another given place. ”