Compactness

“ It has, moreover, the great advantage of compactness, forming a single solid mass, with no straggling or outlying portions; and it is strongly defended on almost every side by natural barriers offering great difficulties to an invader. ”

“ The plurality’s position seems to be that race does not predominate in the creation of a districting map so long as the map does not violate other traditional districting criteria such as compactness, contiguity, equally populated districts, minimizing county splits, etc. ”

“ It is self-evident that any proportion of space must be less than infinite space. And if infinite space can be conceived as having proportions, and we could conceive all these proportions, then these proportions would be equal to the whole! ”

“ And this is how and why we construct the qualities of our ideal geometrical space. We make it one and identical by correlating our sense-spaces, by fusing the multitude of fields of vision and by refusing to recognise the spaces of our dream experiences, in order that we may have a common standard to which we can refer all our space-perceptions. We make it empty and invariable by abstracting from that which fills it and changes in it, in order that nothing may distract us from the contemplation of its pure form. ”

“ With cubic existence (and its correlative three-dimensional space) , with weight and texture we have therefore got from the contemplated shape to a thought alien to that shape and its contemplation. The thought, to which life and its needs and dangers has given precedence over every other: What Thing is behind this shape, what qualities must be inferred from this aspect? After the possibility of occupying so much space, the most important quality which things can have for our hopes and fears, is the possibility of altering their occupation of space; not our locomotion, but theirs. ”

“ Its powers take in objects of all dimensions; yet they are not situated as bodies in a material place, where the greater occupy more space than the less: for the thought of a mile, or ten thousand miles, does no more fill or stretch the soul, than that of a foot, an inch, or a mathematical point. And whereas all matter has its parts, and those extended, one without another, into length, breadth, and thickness, and so is measurable by inches, yards, or solid measures; there is nothing of measurable extension in any thing belonging to the soul, neither length, breadth, nor thickness ”

“ If, therefore, they mean by body and extension the same that other people do, viz. by BODY something that is solid and extended, whose parts are separable and movable different ways; and by EXTENSION, only the space that lies between the extremities of those solid coherent parts, and which is possessed by them,—they confound very different ideas one with another; for I appeal to every man’s own thoughts, whether the idea of space be not as distinct from that of solidity, as it is from the idea of scarlet colour? ”

“ For the sake of clearness and precision, let us designate the primitive concrete data, the result of perception, as extension, and the concept, the result of abstraction, as space—properly so called.
I. At the outset what is given us by intuition is extension under a concrete form. What first becomes known to us is not space but a limited and determined extension—what the child can hold in its hand, reach by a movement of its arms, later on the room which it crosses with uncertain steps ”

“ It is certain, in fact, that when we wish to consider separately a certain portion of indefinite space, conceived as gaseous, we mentally solidify its exterior envelope, so that a line and a surface are habitually, to our minds, just as solid as a volume. ”

“ That the Nature of extension or dilation strives or endeavours to get space, ground, or compass, I mean those corporeal motions endeavour to make place and space by their extensions, that is, to spread their parts of matter into a larger compass or body. And when I say, That Contractions endeavour to cast or thrust out space, place, ground, or compass, My meaning is, That those corporeal motions endeavour to draw their parts of matter into a more close and solid body, for there is no place nor space without body. ”

“ When we see an object we have the sensation and intuition of extension. The space perceived or sensed is, in this case, the extension itself perceived. We imagine a multitude of extended objects, and a capacity which contains them all. We imagine this capacity as the immensity of the ethereal regions, a boundless abyss, a dark region beyond the limits of creation. ”

“ So that with them the whole extent of space ir, the common and natural measure of every thing that hath length and breadth, and the magnitude of every body and of every figure is expressed by its being such a part of the universe. In like manner, the common and natural measure of length, is an infinite right line, which, as hath been before observed, returns into itself, and hath no limits, but bears a finite ratio to every otherline. ”

“ Again, must this quasi-perpendicularity be regarded as a mere convenient metaphor, or may it be an actual perpendicularity, the spiritual forces pulling in a space of their own, more or less like our known space? In the latter case, are the two spaces coincident [1] ; or does the spiritual include the other as that includes a given plane or line; or is every line and every plane of one space perpendicular to every line and plane of the other? The third alternative seems untenable; for, admitting it, how could a spiritual force produce even a deflective effect in physical space? ”

“ For otherwise, if the mind be limited, there must be a definite limit to its comprehensive faculty, and it is easy to conceive that such a limit would soon become apparent to every student; as apparent as it is that a being, confined within three dimensions of space, cannot, without altering his nature, escape from these three dimensions, nor from the laws which govern matter having length, breadth and thickness alone, without the external fourth dimension, with its interchangeability of exterior and interior angles. ”

“ Finity is that property by virtue of which geometric space is limited to the scope of the individual consciousness of a unodim, a duodim or a tridim and by virtue of which it is finite in extent. Flexity is that property by virtue of which geometric space is regarded as possessing curvature, and in consequence of which progress through it is made in a curved, rather than a geodetic line, also by virtue of which it may be flexed without disruption or dilatation. ”

“ We have seen that space vanishes internally into relations between units which never can exist. But, on the other side, when taken itself as a unit, it passes away into the search for an illusory whole. ”

“ But what, in the case supposed, is the fact or predicament that is asserted, when we speak or think of endless space, or of space that transcends all our powers of measurement? Is it correct to say that the law of thought fails, because we can not express endless space in feet or miles? Is it true that we have only "symbolized" the product of endless space out of the data or elements of measurable space? ”

“ We have here a process similar to that observed in the different sciences where we find, for example, geometry exclusively adopts space as its object, jurisprudence law, philosophy the explication of the eternal Idea and its determinate existence and self-identity in the facts of experience, wherein each of the above mentioned sciences develops these objects by differentiation out of their differences, without one of them actually presenting to consciousness in its completeness that which we are accustomed in ordinary modes of thought to call concrete real existence. ”

“ Furthermore, quantitative distinctions in the material world, i.e. distinctions of position, shape, and the like, are all of them primarily known to us in universal form, — not as individual but as specific characters of the object that we have before us. Quantity is no more and no less individuated for our reason than is any other object of thought. This place, this shape, this size, and this definite matter, are just as hard to define in an individual way as this angelic nature, or this immaterial soul. ”

“ What do we wish to express when we say that our space is infinite? Nothing more than that we might lay any number whatever of bodies of equal sizes side by side without ever filling space. Suppose that we are provided with a great many wooden cubes all of the same size. ”

“ Hence, throughout the picture, the expression of space and size is dependent upon obscurity, united with, or rather resultant from, exceeding fulness. We destroy both space and size, § 16. Space and size are destroyed alike by distinctness and by vacancy.either by the vacancy, which affords us no measure of space, or by the distinctness, which gives us a false one. The distance of Poussin, having no indication of trees, nor of meadows, nor of character of any kind, may be fifty miles off, or may be five; we cannot tell—we have no measure, and in consequence, no vivid impression. ”

“ Therefore it is that all sensible beings, even those which have not reason, have been endowed with this intuition of extension or space: which is like an unlimited field on which the different parts of the universe are represented. ”

“ Again, if a body be considered as having superficies, and be understood to be so moved, that all the several parts of it describe several lines, then the way of every part of that body is called thickness or depth, and the space which is made is called solid, consisting of three dimensions, any two whereof are applied whole to every several part of the third. ”

“ When we have expressed in figures the greatest measurable idea of space that can be so expressed, what do we "symbolize," when we say that beyond that measured space there stretches a space that we can not measure, and to which there is of necessity no limit? ”

“ So much of law and order, of symmetry and beauty in every part, implies, it may be thought, that every part has been framed with a view to some use;—that its symmetry and its beauty are the marks of some noble purpose. ”

“ Solidity, extension, space — the foundation properties of the material world — mean, as has been said above, certain movements and energies of our own bodies, and exist in our minds in the shape of feelings of force, allied with visible and tactile, and other sensible impressions. ”

“ The arguments with which I have fought against the opinion that space is a pure nothing, are not founded on our imaginations, but on the impossibility of nothing being extended, or having any properties. This is the principal argument which I have used against those who, while they hold space to be a pure nothing, maintain the possibility of the conception or existence of the properties which they attribute to space. ”

“ Supposing you knew-not by sight or by instinct, but by sheer intellectual knowledge, as I know the truth of a mathematical proposition—that what we call empty space was full, crammed. Not with lumps of what we call matter like hills and houses, but with things as real—as real to the mind. ”

“ And if the three or more dimensions as attributes of space prove to be unthinkable, and also the point without extension, the line without superficies, and the superficies without solidity—so too in consequence are all the concepts derived from them, such as those of geometrical figures, none of which has, or can have, reality. ”

“ Conformity to rule and symmetry, which are the abstract unity and definition of all that is essentially spatial and its configuration no less than of all which is external under temporal condition, are mainly the co-ordinating principles of quantity and size, as we have already noticed in our consideration of the beauty of Nature. ”

“ If the second part of my system be true, that the idea of space or extension is nothing but the idea of visible or tangible points distributed in a certain order, it follows, that we can form no idea of a vacuum, or space, where there is nothing visible or tangible. ”

“ Since space is only the extension of bodies, and there is no space where there are no bodies, it follows that the extension which we conceive distinct from bodies, with fixed points and dimensions, immovable in itself, and the receptacle of all that is movable, is a pure illusion, and there is nothing in reality corresponding to it. ”

“ There is no proportion between equal things. They can have symmetry only, and symmetry without proportion is not composition. It is necessary to perfect beauty, but it is the least necessary of its elements, nor of course is there any difficulty in obtaining it. ”

“ Measure a corner of the Creation, and multiply that space in proportional progression, and the entire Infinite will multiply its circles filled with universes, which will pass in proportional segments between the ideal and elongating branches of your Compass. ”

“ Even the state of the question makes us suppose space to be a substance, that is, a being subsisting by itself without requiring another being in which to exist. The difficulty once overcome on this supposition, it is solved in its most essential and inaccessible point, and therefore in all others. ”

“ If, abstracting this relation from the world of which it is the uniform but most elementary determination, we regard it as a relation between objects having no other determination, these become spaces and nothing but spaces—space pure and simple, vacuum. But we have known the world in confused fulness before we detach its constituent relations in the clearness of unreal abstraction. ”

“ When we say, the whole is greater than its part, we announce a general fact, which rests, it is true, on our ideas of whole and part; but, in abstracting these notions, we begin by considering them as subsisting in space, and time, and body, and again, in linear, and superficial, and solid space. ”