Spiral

“ The spiral makes fewer revolutions, and the hem or moulding which forms it is flat, as is also the inter-spiral or general surface of the volute, which has never any secondary spirals upon it, though that and the intervolute are sometimes enriched with foliage. ”

“ Almond.—Here the oblique sinus, as already described, encloses an almond-shaped figure, blunt above, pointed below, and formed of concentric furrows.
6. Spiral.—When the transverse flexures described in 1 do not pass gradually from straight lines into curves, but assume that form suddenly with a more rapid divergence, a semicircular space is necessarily created, which stands upon the straight and horizontal lines below, as it were upon a base. This space is filled by a spiral either of a simple or composite form. The term ‘simple’ spiral is to be understood in the usual geometric sense. ”

“ Now, motion under resistance takes a spiral direction, as may be seen by the motion of a body rising or falling through water. A bubble rising rapidly in water describes a spiral closely resembling a corkscrew, and a body of moderate specific gravity dropped into water may be seen to fall in a curved direction, the spiral tendency of which may be distinctly observed. ”

“ For the semidiameters of circles, inasmuch as circles are understood to be made up of the aggregate of them, are so many sectors; and therefore in the description of a spiral, the sector which describes it is diminished in duplicate proportions to the diminutions of the circumference of the circle in which it is inscribed; so that the complement of the spiral, that is, that space in the circle which is without the spiral line, is double to the space within the spiral line. ”

“ No effort of consciousness seemed capable of changing the degree of curvature of the spiral or circle and keep it smooth, though one could of course at any time break into the spiral or circle and walk off in another direction. ”

“ In the main, the logarithmic spiral is char­ac­ter­is­tic, not of the living tissues, but of the dead. And for the same reason, it will always or nearly always be accompanied, and adorned, by a pattern formed of “lines of growth,” the lasting record of earlier and successive stages of form and magnitude.
It is evident that the spiral curve of the shell is, in a sense, a vector diagram of its own growth; for it shews at each instant of time, the direction, radial and tangential, of growth, and the unchanging ratio of velocities in these directions. ”

“ In projecting, the lines in one view are used to mark those in other views, and to find their shapes or curvature as they will appear in other views. Thus, in Figure 225a we have a spiral, wound around a cylinder whose end is cut off at an angle. The pitch of the spiral is the distance A B, and we may delineate the curve of the spiral looking at the cylinder from two positions (one at a right-angle to the other, as is shown in the figure) , by means of a circle having a circumference equal to that of the cylinder. ”

“ We are not going round in circles, we are mounting higher, the circle is a spiral, we have already climbed up many steps.” ”

“ The essentials are a fixed slide of some bold design—star-shaped, circular, octagonal, or what not—and a moving shade that is of spiral pattern. The spiral pattern is the secret of the whole thing. Let the above be spun behind a star, so as to be only visible through its rays, and the star will seem to be alive. Turn it one way, the waves flow outwards; turn it the other, the waves flow inwards. The [258] standing pattern must have a centre, and the spiral must be worked on the same centre. That done, all is well. Sometimes a double spiral is desirable, and then the pattern in Fig. ”

“ Let us next regard our logarithmic spiral from the dynamical point of view, as when we consider the forces concerned in the growth of a material, concrete spiral. In a growing structure, let the forces of growth exerted at any point P be a force F acting along the line joining P to a pole O and a force T acting in a direction perpendicular to OP; and let the magnitude of these forces be in the same constant ratio at all points. ”

“ If now we suppose a consciousness connected with the film in such a way that the intersection of the spiral with the film gives rise to a conscious experience, we see that we shall have in the film a point moving in a circle, conscious of its motion, knowing nothing of that real spiral the record of the successive intersections of which by the film is the motion of the point. ”

“ When we say here, for short, that the shell is a logarithmic spiral, we merely mean that it is essentially related to one: that it can be inscribed in such a spiral, or that cor­re­spon­ding points (such, for instance, as the centres of gravity of successive chambers, or the extremities of successive septa) wall always be found to lie upon such a spiral. ”

“ Instead of being like a succession of globular skins, each completely covered by its successor, the growth is often spiral and the construction is as if the nucleus had been rolled one, two, or three complete revolutions in a continuous plate of nacre, and the spiral envelope then finally merged into another plate and the process repeated. ”

“ Next put the point of the pencil in the loop and move it around the pin just as you did in making the circle and you will find that you have drawn a very pretty geometrical spiral which is known as the spiral [121] of Archimedes. It is so called because Archimedes was the first to explain that it was caused by a point moving with uniform angular speed and receding from the center at a constant rate.
Fig. 50b. how to draw a spiral with a thread
How to Draw an Ellipse. ”

“ A spiral curve is a [45] harmonious line, says the designer: repeat it, reversed, and you prolong the harmony; repeat it again, with variations, and you complete the harmony. Or, harmonious effect is produced by recurring form and line. Here is a circular form; here is a meandering line: combine and repeat them, and you get a logical and harmonious border motive. ”

“ Algebra, Part 2 (Edinburgh, 1879) , p. 209.
922. In the year 1692, James Bernoulli, discussing the logarithmic spiral [or equiangular spiral, ρ = αθ] ... shows that it reproduces itself in its evolute, its involute, and its caustics of both reflection and refraction, and then adds: "But since this marvellous spiral, by such a singular and wonderful peculiarity, pleases me so much that I can scarce be satisfied with thinking about it, I have thought that it might not be inelegantly used for a symbolic representation of various matters. ”

“ If the sun, for example, have from six to twelve inches diameter, the width of the strokes that form the spiral need not be more than one-twentieth part of an inch, and the spaces between them that form the transparent parts, about two-tenths of an inch. ”

“ The light wave makes its way, roughly speaking, spirally through ether; objects of large surface and slow descent, such as certain suitably shaped pieces of paper, descend through the atmosphere in a spiral path; a bubble of air ascends spirally through water; the plant climbs a tree by spiral windings; a horse mounting a steep ascent with a heavy load takes a zigzag or spiriform course; men ascend and descend by spiral stair-ways ”

“ To have done with this part of our subject, here is another property of the logarithmic spiral. Roll the curve along an indefinite straight line. Its pole will become displaced while still keeping on one straight line. The endless scroll leads to rectilinear progression; the perpetually varied begets uniformity. ”

“ When he finds a simple geometrical construction, for instance in the honey-comb, he would fain refer it to psychical instinct or design rather than to the operation of physical forces; when he sees in snail, or nautilus, or tiny foraminiferal or radiolarian shell, a close approach to the perfect sphere or spiral, he is prone, of old habit, to believe that it is after all something more than a spiral or a sphere, and that in this “something more” there lies what neither physics nor mathematics can explain. ”

“ The beautiful form of the spiral structure of the tongue should be viewed under a high magnifying power, when it will be seen that no continuing spiral structure really exists; each ring, apparently detached, does not extend quite round ”

“ The disruptive energy of internal repulsion, as above stated, increases in force as the convolutions of the spiral become more and more compressed and the spiral becomes more and more circular in form. ”

“ We have no means, other than observation, of determining that one leaf is vertically above another, and spiral series such as we have been dealing with will appear, whether such orthostichies exist, whether they be near or remote, or whether the angle of divergence be such that no precise vertical superposition ever occurs. ”

“ Let us now place our (elliptical) spiral in such a position that it casts no shadow except an ellipse, and, for convenience of reference, let us agree that the longer axis points north and south. Let us picture to ourselves a tribe of microscopic creatures, whose true destiny should be to proceed upwards in the direction of the coil as a whole, and who have a blind but irresistible impulse to do so. They have no mode of ascending except along the wire ”

“ Besides the rosettes of which we have had occasion to speak so often we encounter at every step a spiral ornament the design of which remains without much modification, while a certain variety is given to its general effect by changing the arrangement of its colours. In the example reproduced in Fig. 126 large black disks, like eyes, are embraced by a double spiral in which blue and yellow alternate. [382]
Fig. ”

“ It mounts aloft in a rapidly increasing spiral, surveying the near and minute objects first, then the larger and more distant, till having circled about the spot five or six times and taken all its bearings it darts away for home. ”

“ The great pillars are ornamented in various ways—some with a great spiral groove running from bottom to top; others with two spirals, ascending in different directions, so as to cross over one another; some are fluted or channeled straight up and down ”

“ In other words, the form of the spiral will determine the eccentricity of the ellipse, subject to perturbations, however, of various sorts. During this stage of coalescence from an open ring into a sphere, these bodies will take on, by cooling and condensation, their planetary forms; and as the forming spheres, by the retreat of their masses down the lines of approach to the sun, advance, their forward and nearer extremities will be more powerfully acted upon by gravity than those parts in the rear, and a forward plunge or axial movement of rotation will be set up. ”

“ This seems to be the law of progress in everything we do; it moves along a spiral rather than a perpendicular; we seem to be actually going out of the way, and yet it [34] turns out that we were really moving upward all the time. ”

“ On another large button there are only two globular projections at each corner, and the lozenge has a broad border filled with an uninterrupted row of small circles; in the centre we see a double circle filled with spirals, of the form we have so frequently passed in review; the remaining space in each acute angle being filled up with an omega-like spiral and three small figures similar to those in No. ”

“ As the small nebulae and the spirals inhabit the same regions of the sky, it is probable that the order of distance of the two classes is the same. Classes e, f, may even turn out to be spirals themselves. ”

“ We see, accordingly, that in plane spirals whose constant angle lies, say, between 65° and 70°, we can only obtain contact between consecutive whorls if the rate of growth of the inner border of the tube be a small fraction,—a tenth or a twentieth—of that of the outer border. In spirals whose constant angle is 80°, contact is attained when the respective rates of growth are, ap­prox­i­mate­ly, as 3 to 1; while in spirals of constant angle from about 85° to 89°, contact is attained when the rates of growth are in the ratio of from about 3 ⁄ 5 to 9 ⁄ 10. ”

“ And as we lifted higher and higher the circle changed before our eyes into a sphere—a tremendous, faintly glowing sphere of size inconceivable, filling the heavens with its vast bulk, feebly luminous like the ghost of some mighty sun, rushing through space to meet us as we sped up and over it. ”

“ As the curve is the true line of beauty, and as the circle in Egyptian hieroglyphics is ever used as the symbol of renewed life—the type or sign of the generative principle—so the motion produced by the centripetal and centrifugal forces, seems to be that of nature. ”

“ And when a Man by a stroke upon his Eye sees a flash of Light, are not the like Motions excited in the Retina by the stroke? And when a Coal of Fire moved nimbly in the circumference of a Circle, makes the whole circumference appear like a Circle of Fire; is it not because the Motions excited in the bottom of the Eye by the Rays of Light are of a lasting nature, and continue till the Coal of Fire in going round returns to its former place? ”

“ You know full well that the condition of the outer sphere is, as I have described it, a bleak, turbulent surface, the roof of the earth on which man exists, as a creeping parasite does on a rind of fruit, exposed to the fury of the ever-present earth storms.
"The central circle, or medial sphere, the shell, or layer of gravitation, lies conformably to the outer configuration of the globe, about seven hundred miles towards its center. It stretches beneath the outer circle (sphere) as a transparent sheet, a shell of energy, the center of gravitation. ”