Fourteen Points

“ In any circle, the nearer the points of the circumference are to the centre, the closer they will necessarily be to one another. As we draw nearer, each for himself, to our Centre, we shall feel that we have approximated to all those who stand round the same centre, and draw from it the same life. ”

“ I like to think of all the points in a circle really feeling a placid satisfaction in the thought that they are equidistant from the centre! They are infinite in number, and so can well afford to despise the arrogance of a point within, and to ignore the envious murmurs of a point without! ”

“ The human being is then divided into four parts and represent the four cardinal points or directions of the existence. However, there is a fifth point, a fifth direction: the TOP and BOTTOM. What rises transcending the human plain if it is possible to balance in equilibrium the complementing pair of opposites. ”

“ For my soul, being a real being as well as my body, is certainly as capable of changing distance with any other body, or being, as body itself ; and so is capable of motion. And if a mathematician can consider a certain distance, or a change of that distance between two points, one may certainly conceive a distance, and a change of distance, between two spirits ; and so conceive their motion, their approach or removal, one from another. ”

“ Indeed, it thus became the second principle, as a circle which, by developing, becomes a figure and a a surface, whose circumference, centre, and rays are distinct, occupying different points. The origin of things is better than their goal. ”

“ Since the sun revolves round the earth it describes a circle, and every circle must have a centre; that we know already. This centre is invisible, it is in the middle of the earth, but we can mark out two opposite points on the earth's surface which correspond to it. ”

“ The position of a point is made to depend on the length of three lines which are always drawn in determinate directions, and the line joining two points is in like manner considered as the resultant of three lines. ”

“ In a similar manner, by placing two singular points of degree, unity and strength, , , at a distance along an axis , and at the origin respectively, when is indefinitely increased, and diminished so that is finite and , we obtain a singular point of degree 2, strength at the origin, the axes being . ”

“ In addition to the private spaces belonging to the private worlds of different percipients, there is, however, another space, in which one whole private world counts as a point, or at least as a spatial unit. This might be described as the space of points of view, since each private world may be regarded as the appearance which the universe presents from a certain point of view. ”

“ I observed the latitude 13 degrees 32 minutes south; longitude made 35 degrees 19 minutes west; course north 89 degrees west, distance 108 miles.
(*Footnote. One person turns his back on the object that is to be divided: another then points separately to the portions, and each of them asking aloud, "Who shall have this?" to which the first answers by naming somebody. This impartial method of division gives every man an equal chance of the best share.) ”

“ It follows that the number of parts of which the line is made up is at least as great as the number of points less one, if we refuse to say that the points are on the line; and is as great as the number of points, if we are willing to say that they are on the line. To move over the whole line, then, a point must come within one term of the end of an endless series, or it must pass over an endless number of small pieces of line until it comes to the very end. ”

“ It is written that no man hath seen God at any time; but has any man seen a Point or a Line or a Plane Surface at any time? A Point has neither parts nor magnitude—I seem to remember—but only position. ”

“ This succession, which stimulates the curiosity and so arouses the attention required by every object in turn, is the order followed by most men, and it is the right order for all children. To take our bearings so as to make our maps we must find meridians. Two points of intersection between the equal shadows morning and evening supply an excellent meridian for a thirteen-year-old astronomer. But these meridians disappear, it takes time to trace them, and you are obliged to work in one place. So much trouble and attention will at last become irksome. ”

“ Nothing is that which fills no space.
If one single point placed in a circle may be the starting point of an infinite number of lines, and the termination of an infinite number of lines, there must be an infinite number of points separable from this point, and these when reunited become one again; whence it follows that the part may be equal to the whole. ”

“ If the two resistances over which the points A and B move are mounted in the same position with respect to the cardinal points of the compass, then the points themselves will always be in the same position with respect to the cardinal points when a balance is obtained. ”

“ To the question, where and what is the fourth dimension, the answer must be, it is here—in us, and all about us—in a direction toward which we can never point because at right angles to all the directions that we know. ”

“ As soon, however, as we accept the view that points and straight lines are complicated constructions by means of classes of physical entities, the hypothesis that we have an a priori intuition enabling us to know what happens to straight lines when they are produced indefinitely becomes extremely strained and harsh; nor do I think that such an hypothesis would ever have arisen in the mind of a philosopher who had grasped the nature of physical space. ”

“ As to subject-matter, there are points which stand strongly out; the quadrature of the circle, for instance. But there are others which cannot be drawn out so as to be conspicuous in a review of writers: as one instance, I may take the centrifugal force.
When I was about nine years old I was taken to hear a course of lectures, given by an itinerant lecturer in a country town, to get as much as I could of the second half of a good, sound, philosophical omniscience. The first half (and sometimes more) comes by nature. ”

“ A series of concentric circles offer points of external contact more numerous in some and less numerous in others, according as their circumference is nearer or further from the common centre, but the centre itself is unaffected by such chances; it remains always the same, one and indivisible. ”

“ Is there any motion in a straight line? A straight line infinitely projected becomes a circle, it returns to the starting point. You must end where you begin; and as you began in God, you must go back to God. ”

“ I mean that every Point in you for you are a Square and will serve the purpose of my illustration every Point in you, that is to say in what you call your inside, is to pass upwards through Space in such a way that no Point shall pass through the position previously occupied by any other Point ”

“ He points out that we make such judgments. When we say that the three angles of a triangle equal two right angles, or that every event has a cause, we are saying something universal and necessary, something not founded on experience. ”

“ But when we seek in imagination to follow out this curved surface as it actually exists, slowly bending round until all its meridians meet in a point eight thousand miles below our feet, we find ourselves utterly baffled. ”

“ And what does it matter after all, under the great Dome, a few hordes of factmongers more or less, glimmering and wonderless, crawlers on the bottom of the sea of time, lovers of the ooze of knowledge, feeling with slow, myopic mouths at Infinite Truth?
But when I see my four faces—the faces of my four special boys, when I hear the college bells ringing to them, it matters a great deal. My soul will not wait. What is the ridge-pole of the world? The distance of a ridge-pole does not count. The extent of a universe does not seem to make very much difference. ”

“ Hence space is employed as the type even of the concept of time itself, representing it by a line, and its limits—moments—by points. Time, on the other hand, approaches more to a universal and rational concept, comprising under its relations all things whatsoever, to wit, space itself, and besides, those accidents which are not comprehended in the relations of space, such as the thoughts of the soul. ”

“ The point cannot move entirely in a straight direction, making turns and angles at intervals, otherwise we should obtain, as we have seen, and as is, indeed, quite obvious, a polygon, and not a circle: neither can the point move entirely out of the straight, otherwise the direction which is continually changing would be altogether lost. ”

“ By experience I know what figure a man, or any other known object, makes to my eye, at the distance of ten feet: I perceive the gradual and proportional diminution of this visible figure, at the distance of twenty, forty, a hundred feet, and at greater distances, until it vanish altogether. Hence a certain visible magnitude of a known object, becomes the sign of a certain determinate distance, and carries along with it the conception and belief of that distance. ”

“ As the centre of a circle is not the point which shows the distance of one part from another but really that focus from which proceed the countless lines comprised within and constituting the complete circuit so the Lord is the centre as well as the circumference of the full circle of life. The Lord in His sanctified Self is present in the inner soul, aye, seated in the very heart of every created being. ”

“ The young philosopher was unable to answer this question, and his father, therefore, informed him that, if a perpendicular line were drawn from the centre of gravity of a body to the centre of the earth, such a line would be termed the line of direction; along which every body, not supported, endeavours to fall ”

“ When a circle is made, and from its centre a number of lines are drawn to the circumference, all these lines, though ever so distant in the circumference, meet together in the point, which is in the middle. ”

“ The circle represents the Infinite, without beginning or end; the straight line represents the finite, which has both beginning and end. The sphere represents self-existent and eternal individuality and Mind; the straight line, a belief in a self-existent and temporary material existence. ”

“ SHORT CUTS.
Did you ever know a path across a level field to be straight, when formed by the feet of wayfarers alone? There is the opposite stile which you seek, there is nothing to turn you to the right hand or to the left, but your own swerving fancy; and that makes the field-path crooked, invariably. It seems as if no one could walk straight alone, nor indeed correct himself, once for all, when wrong. The moment he becomes conscious of a deviation from the true track, he leaves it again in the other direction. ”

“ You seem to mark out for yourself a certain sphere, and if you go beyond it, you think you do yourself an injury. Thus, while you have an apparent movement, you are only describing a circle, whose centre and circumference is self. ”

“ The point by reason of its indivisibility is immeasurable, and the circle by reason of its arc cannot be exactly squared, and therefore cannot be measured with precision." Notes by Miss Hillard: "This is why the Deity is represented by a point. Paradiso, xxviii. 16: 'A point beheld I,' 'Heaven and all nature, hangs upon that point,'" etc. Bk. IV. xvi. 6, quoting Aristotle's Physics: "The circle can he called perfect when it is a true circle. ”

“ Every line may be, and by the ancient Geometricians was, considered as a point produced, the two extremes being its poles, while the point itself remains in, or is at least represented by, the midpoint, the indifference of the two poles or correlative opposites. ”

“ Our idea of a point, I apprehend to be simply our idea of the minimum visibile, the smallest portion of surface which we can see. A line, as defined by geometers, is wholly inconceivable. We can reason about a line as if it had no breadth; because we have a power, which is the foundation of all the control we can exercise over the operations of our minds; the power, when a perception is present to our senses, or a conception to our intellects, of attending to a part only of that perception or conception, instead of the whole. ”

“ The simplest symbol of it is the circle, the line which returns into itself, not the straight line produced indefinitely; and the readiest way to find it is to consider the satisfaction of desire. There we have myself and its opposite, and the return from the opposite, the finding in the other nothing but self. ”

“ After all, there is some logic—a peculiar logic of today—in this state of being a point. A point has more unknowns than any other entity. If a point should start to move, it might become thousands of curves, or hundreds of solids. ”

“ For if a circumference is described from the central point, which seems to be a circle,—for it is impossible for a circle to be described without a point, and it is impossible for a circle to be without a point,—surely the earth consisted before all, they say, in a state of chaos and disorganization. ”

“ A straight line, for example, is supposed to be a certain track in space whose parts all exist simultaneously. We shall now find that what is a straight line for one observer is not a straight line for another. ”

“ Distance or space, in its simple abstract conception, to avoid confusion, I call expansion, to distinguish it from extension, which by some is used to express this distance only as it is in the solid parts of matter, and so includes, or at least intimates, the idea of body : whereas the idea of pure distance includes no such thing. ”