Circlet

“ A circle is in itself a consummate wonder of geometrical symmetry. It is the line in which the omnipotent energy delights to move. There is no fault in it to be amended. The first drawn circle and the last both embody the same complete fulfillment of a perfect design. Then look at the rays which pass from the inner to the outer circle. How beautifully they bring the greater and lesser circles into connection with each other! ”

“ Have your circus on top of a large table, or on the floor, or out of doors on the bare ground. Almost any place will do where there is a good-sized level surface.
Make the circus ring about forty inches in diameter, outline it with chalk and cover the marked line with either sawdust or dry corn meal built into a narrow ridge similar to that of earth around the large rings; then your circle will look just like the ring of a real circus. ”

“ Under the flaming yellow stars it stood out luminous, marking the exact center of an enormous circle; a circle roofed by the radiantly flecked heavens, bounded by mountains which rose against the sky-line, abrupt as a wall, black as ink. ”

“ A body of men walking in a circle remain together as a body within the circumference of the circle as it moves with the earth; change their paths to a cycloid directed to a distant point, and the body of men will move away in that direction; change their paths to parallel right lines, and as a body they may soon be a mile away and still in a circle. ”

“ A circle may be a great circle of either the celestial sphere, the earth, or even of a baseball. The poles of a great circle are the points on the surface of its sphere, penetrated by the diameter perpendicular to the plane of the great circle. As for example, the poles of the earth are connected by the diameter that is perpendicular to the plane of the equator. An angle at any pole is measured on the great circle which subtends it. For instance, angles at the poles of the earth are measured on the equator; angles at the zenith on the horizon. ”

“ If a circle be defined as a figure, such that all straight lines drawn from the center to the circumference are equal, every one can see that such a definition does not in the least explain the essence of a circle, but solely one of its properties. (5) Though, as I have said, this is of no importance in the case of figures and other abstractions, it is of great importance in the case of physical beings and realities: for the properties of things are not understood so long as their essences are unknown. ”

“ You must understand what is meant by a great circle: it is a circle dividing the sphere into equal parts. Its plane therefore passes through the centre of the sphere. Now, our use of the celestial globe or sphere is founded on the assumption that it is a representation of the heavens, on the supposition that the eye of the observer is at the centre of the sphere. ”

“ The circle was undoubtedly known to the early civilizations, its simplicity specially recommending it as an object for study. Euclid defines it (Book I. def. 15) as a “plane figure enclosed by one line, all the straight lines drawn to which from one point within the figure are equal to one another.” In the succeeding three definitions the centre, diameter and the semicircle are defined, while the third postulate of the same book demands the possibility of describing a circle for every “centre” and “distance.” ”

“ We have seen that the stereographic projection of any circle of the sphere is itself a circle. But in the case in which the circle to be projected passes through V, the projection becomes, for a great circle, a line through the centre of the sphere ”

“ I have no doubt myself that it found its way into this Appendix in the fifth (? or fourth) century B.C. from a Tâoist source. Kû Hsî, in his 'Lessons on the Yî for the Young,' gives for it the figure of a circle,—thus, ○; observing that he does so from the philosopher Kâu (A.D. 1017–1073) [2] , and cautioning his readers against thinking that such a representation came from Fû-hsî himself. To me the circular symbol appears very unsuccessful. 'The Great Extreme,' it is said, 'divided and produced two lines,—a whole line and a divided line.' But I do not understand how this could be. ”

“ A person to whom the double circular motion is familiar in the lathe naturally looks upon one circle moving upon another as in simple motion, if the second circle be fixed to the revolving radius, so that one and the same point of the moving circle travels upon the fixed circle. ”

“ On strong paper describe a circle a little larger than the surface of your sun. On this trace two spirals, one-twelfth of an inch apart, and open the intervening space with a knife, cutting the paper from the circumference, decreasing in breadth to a certain distance from the centre. Then cut the remainder of the circle into similar spirals, alternately open and close. Paste it on an iron hoop, supported by an iron cross, and set all on a stand which will let it freely turn round its centre. ”

“ The circle, therefore, is the cause of ornament to all things, even to the last participations, and leaves nothing destitute of itself, since it supplies beauty, similitude, formation, and perfection to the universe. ”

“ It is a very important proposition, theoretically and practically, for it enables us to find the center of a circle if we know any part of its arc. A civil engineer, for example, who wishes to find the center of the circle of which some curve (like that on a running track, on a railroad, or in a park) is an arc, takes two chords, say of one hundred feet each, and erects perpendicular bisectors. It is well to ask a class why, in practice, it is better to take these chords some distance apart. ”

“ The use of a rude form of circle to set forth the human head reminds one of the employment by living savage tribes of the same form as the symbol of a house (hut?) , a wreath, and so forth. Yet there is a measure of resemblance even in this abstract symbolism: the circle does roughly resemble the contour of the head: as, indeed, the square or rectangle may Fig. 3. be said less obviously to do when hair and whiskers and the horizontal line of the hat break the curved line.
But it is not the mere contour which represents the face: it is a circle picked out with features. ”

“ The circle being drawn, a line A, 179, is drawn through its centre, and from its intersection with the circle as at b, here a step on each side is marked as c, d, then lines c to d, and c and d to e, where A meets, the circle will describe a trigon. If the figure is to stand vertical, all that is necessary is to draw the line a vertical, as in Figure 180. A ready method of getting the dimension across corners, across the flats, or the length of a side of a given polygon, is by means of diagrams, such as shown in the following figures, which form excellent examples for practice. ”

“ My neighbour snatched his cap and pulled it down on to his nose.
'What do I find so awful?' he shouted. 'Why, this: the circle is the destruction of all independent development; the circle is a hideous substitute for society, woman, life; the circle... oh, wait a bit, I'll tell you what a circle is! A circle is a slothful, dull living side by side in common, to which is attached a serious significance and a show of rational activity ”

“ Four times during the revolution of the great circle each large clock emits for a couple of minutes a species of chime, the nature of which my ignorance of music renders me unable to describe:—viz., when the line dividing the green and black semicircles is horizontal at noon and midnight, and an hour before, at average sunrise and sunset, it becomes perpendicular. ”

“ A ring of 44 feet outer and 30 feet inner dimensions, has been formed by constructing a sunken circle about 12 inches below the level of the original floor, which is rendered capable of removal to just such an extent as will disclose the ring, the boarding being replaced to form the ordinary promenade, at such times as the circus performances are not going on. ”

“ 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. ”

“ For God there is no circle, there can be none. But we outline a portion of an immense circle, of which we can touch but one point. We indicate only the inner periphery of a circle it is impossible to finish, and then retrace our steps.
When the circle is made small, we make it with one, two, three or four fingers, with the hand, with the arm. If the circle is vast as can be made with the arms, it is homogeneous. ”

“ In every framework of thought which man had ever devised, the circle was prominent, vital. It played its part in every creed of every race, of every time. It was as essential to the ancient arts of magic as to the current methods of science. It played its part in the movement of planets, the shape of stars, perhaps the essence of the total universe.
Man might be too didactic in requiring that awareness develop a physical science comparable to his own, but surely awareness, whatever form it took, would know the circle. ”

“ Suppose the circle whose centre is O and one of the points of intersection A to remain fixed, while the second circle turns round that point in such a manner that the second point of intersection B becomes ultimately consecutive to A; then, since the line OO′ always bisects AB, we see that when B ultimately becomes consecutive to A, the line OO′ passes through A. In consequence of the motion, the common chord will become in the limit a tangent to each circle, as in the second diagram.—Comberousse, Géométrie Plane, page 57. ”

“ We may take any point along the major axis, as shown, for instance, in Fig. 149. Let B be this point, taken at random. Then describe the half circle (C) . We may, also, arbitrarily, take any point, as, for instance, D on the minor axis E, and by drawing the diagonal lines (F) we find marks on the circle (C) , which are the meeting lines for the large curve (H) , with the small curve (C) . In this case we have formed an ovate or an oval form. Experience will soon make perfect in following out these directions.
Focal Points.—The focal point of a circle is its center, and is called the focus. ”

“ The value of the circle as a unifying and therefore as a simplifying agent cannot be overestimated, especially in solving the problems which occur in composition where the circle has not been a part of the original scheme, but where, when applied, it seems to bring a relief to confusion and disorder. ”

“ Symbol. An enormous circle, or transparent sphere, floats in the air and men are putting their heads into this circle.
Interpretation. In this symbol everything that I was thinking of is expressed. The validity of the transsubjective concerns all men without exception; the circle goes through all the heads. This validity must have its cause in something common to all. The heads all belong to the same apparently homogeneous sphere. Not all judgments are transsubjective; with their bodies and limbs men are outside of and under the sphere and stand on the earth as separate individuals. ”

“ A circle may be described with any given point as a center and any given line as a radius. This involves the use of the second of the two geometric instruments, the compasses.
5. Any figure may be moved from one place to another without altering the size or shape. This is the postulate of the homogeneity of space, and asserts that space is such that we may move a figure as we please without deformation of any kind. It is the basis of all cases of superposition. ”

“ It is, then, possible to call the circle perfect when it is truly a circle, that is, when it is joined with its own proper or relative virtue, it is then complete in its nature, and it may then be called a noble circle. ”

“ A system coaxal with the two given circles is readily constructed by describing circles through the common points on the radical axis and any third point; the minimum circle of the system is obviously that which has the common chord of intersection for diameter, the maximum is the radical axis—considered as a circle of infinite radius. ”

“ A circle is a round straight line with a hole in the middle. ”

“ The boys form a circle at the right, girls form a circle at the left, hold cat-tails high; all circle toward the right, all circle toward the left, go to center and back in each circle; each circle forms a line, stand with sides to audience, face each other, march forward and back twice; march forward, pass between and back to places, repeat and stay on opposite sides; march again, meet, hold cat-tails touching, end couples pass under the next, and next, until all have passed under and returned to place in lines. ”