Cube root

“ There, before my ravished eye, a Cube, moving in some altogether new direction, but strictly according to Analogy, so as to make every particle of his interior pass through a new kind of Space with a wake of its own—shall create a still more perfect perfection than himself, with sixteen terminal Extra-solid angles, and Eight solid Cubes for his Perimeter. And once there, shall we stay our upward course? In that blessed region of Four Dimensions, shall we linger on the threshold of the Fifth, and not enter therein? ”

“ The interior of each is a Light-buff (the colour of the substance of the cube) , the sides are of the colours of the vertical faces of the cube, and the points of the colours of the vertical lines of the cube, viz., Dark-blue, Blue-green, Light-yellow, Vermilion lines, and Brown, French-grey, Dark-slate, Green points. Thus, the square, in moving in the unknown direction, traces out a succession of squares, the assemblage of which makes the cube in layers. So also the cube, moving in the unknown direction, will at any point of its motion, still be a cube ”

“ The occasional notice of what appear to be perfect cubes, added to the certainty that cleavages are practicable parallel with every plane of that solid, at first tempted me to presume that it ought to be considered as the primitive form; but the consideration of some circumstances connected with the crystallization of this substance, induced me wholly to abandon the idea. The inspection of the annexed drawings will evince that, of the five modifications described, the first alone is compatible with the cube. ”

“ Cubes of pyrites (fig. 89) are striated parallel to one edge, the striae on adjacent faces being at right angles, and due to oscillatory combination of the cube and the pentagonal dodecahedron (compare fig. 36) ; whilst cubes of blende (fig. 90) are striated parallel to one diagonal of each face, i.e. parallel to the tetrahedron faces (compare fig. 31) . These striated cubes thus possess different degrees of symmetry and belong to different symmetry-classes. Oscillatory combination of faces gives rise also to curved surfaces. ”

“ The cube is the most stable of them because resting on a quadrangular plane surface, and composed of isosceles triangles. To the earth then, which is the most stable of bodies and the most easily modelled of them, may be assigned the form of a cube; and the remaining forms to the other elements,—to fire the pyramid, to air the octahedron, and to water the icosahedron,—according to their degrees of lightness or heaviness or power, or want of power, of penetration. The single particles of any of the elements are not seen by reason of their smallness; they only become visible when collected. ”

“ Illustrating Cube Shading.—In Fig. 137 I show merely nine lines joined together, all lines being of equal thickness.
As thus drawn it may represent, for instance, [Pg 79] a cube, or it may show simply a square base (A) with two sides (B, B) of equal dimensions.
Shading Effects.—Now, to examine it properly so as to observe what the draughtsman wishes to express, look at Fig. 138, in which the three diverging lines (A, B, C) are increased in thickness, and the cube appears plainly. On the other hand, in Fig. 139, the thickening of the lines (D, E, F) shows an entirely different structure. ”

“ Six of these squares joined together formed eight solid angles, each produced by three plane right angles: and the shape of the body thus formed was cubical, having six square planes for its surfaces.” In his cosmology Plato assigned this solid to “earth,” for “‘earth’ is the least mobile of the four (elements—‘fire,’ ‘water,’ ‘air’ and ‘earth’) and most plastic of bodies: and that substance must possess this nature in the highest degree which has its bases most stable.” The mensuration of the cube, and its relations to other geometrical solids are treated in the article Polyhedron ”

“ Aristæus, probably a little older than Euclid, wrote a book upon these solids.
As an interesting amplification of this proposition, the centers of the faces (squares) of a cube may be connected, an inscribed octahedron being thereby formed. Furthermore, if the vertices of the cube are A, B, C, D, A', B', C', D', then by drawing AC, CD', D'A, D'B', B'A, and B'C, a regular tetrahedron will be formed. Since the construction of the cube is a simple matter, this shows how three of the five regular solids may be constructed. The actual construction of the solids is not suited to elementary geometry. ”

“ Examples of simple forms amongst crystallized substances are octahedra of alum and spinel and cubes of salt and fluorspar. More usually, however, two or more forms are present on a crystal, and we then have a combination of forms, or simply a “combination.” Figs. 6, 7 and 8 represent combinations of the octahedron and the cube; in the first the faces of the cube predominate, and in the third those of the octahedron; fig. 7 with the two forms equally developed is called a cubo-octahedron. Each of these combined forms has all the elements of symmetry proper to the simple forms. ”

“ After several months a third complication was introduced, by imagining that each color represented a lineal bed of flowers, the flowers having been previously gathered by the child and their colors compared. At this time solid figures would be placed in the center of the innermost plane figure outlined by the sticks, thus bringing out clearly the relations of the sides of such solids to certain planes. Thus a cube would stand in a square, a tetrahedron or pyramid in the center of a triangle. ”

“ The light proceeding from the cube was now circularly analysed, by means of the interposition of a lamina of mica of a proper thickness between the plate of Iceland spar and the analysing prism. The axis of this lamina lay so that the segments of the arcs were removed from the central point to the first and third quadrants. When the cube was yet unheated, its action was thus in direct opposition to its action in the first degree of its heating. ”

“ This modification consists in a decrease on the six solid angles of the primitive crystal, so that each is replaced by an equilateral quadrangular plane, perpendicular to the axis that passes through the angle, and forms the passage of the octahedron into the cube. The angle caused by the meeting of the faces P and 1, Fig. 16, is, according to Haüy, 125°, 15′, 52′′ Fig. 24 shews the direction of the laminæ of the cube. Fig. 25 may rather be considered an accidental circumstance, than as forming an important part of this series. ”

“ Let us call the figure which corresponds to a square in a plane and a cube in our space, a tessaract. Model 1 is a cube. Let us assume a tessaract generated from it. Let us call the tessaract Urna. The generating cube may then be aptly called Urna Mala. We may use cubes to represent parts of four-space, but we must always remember that they are to us, in our study, only what squares are to a plane-being with respect to a cube.
Let us again examine the mode in which a plane-being represents a Block of cubes with slabs. Take Block 1 of the 81 Set of cubes. ”

“ If we suppose the plane being to have a general idea of the existence of a higher solid—our solid—we must next trace out in detail the method, the discipline, by which he would acquire a working familiarity with our space existence. The process begins with an adequate realisation of a simple solid figure. For this purpose we will suppose eight cubes forming a larger cube, and first we will suppose each cube to be coloured throughout uniformly. [Pg 151] Let the cubes in fig. 91 be the eight making a larger cube. ”

“ Clinographic drawings are most frequently used for representing crystals. In representing, for example, a cubic crystal (fig. 11) a cube face a5 is first placed parallel to the plane on which the crystal is to be projected and with one set of edges vertical; the crystal is then turned through a small angle about a vertical axis until a second cube face a2 comes into view, 574 and the eye is then raised so that a third cube face a1 may be seen. ”

“ If we do not inquire about one of the cubes itself, we are all right; that being granted we can know the block.
But if we look into what each of these unit cubes, or what each of these unit positions is, we find quite an infinity opening before us. There is nothing definitely of which we can say that the whole unit cube is built up, and each of the positions has a perfectly endless number of positions in it, if we come to examine it closely. ”

“ If the eight cubes drawn, in fig. 82, are put close together, as on the right hand of the diagram, they form a cube, and in them, as thus arranged, a going up is represented by adding red to the zero, or null colour, a going away by adding yellow, a going to the right by adding white. White is used as a colour, as a pigment, which produces a colour change in the pigments with which it is mixed. From whatever cube of the lower set we start, a motion up brings us to a cube showing a change to red, thus light yellow becomes light yellow red, or light orange, which is called ochre. ”

“ The second solid is composed of the same triangles, which unite as eight equilateral triangles, and make one solid angle out of four plane angles—six of these angles form a regular octahedron. The third solid is a regular icosahedron, having twenty triangular equilateral bases, and therefore 120 rectangular scalene triangles. The fourth regular solid, or cube, is formed by the combination of four isosceles triangles into one square and of six squares into a cube. ”

“ Now as each half inch cube has six faces measuring half an inch by half an inch, it has a superficial surface of three square inches, and as there are eight of the half inch cubes, there are twenty-four superficial inches to be colored against six in the inch cubes. It will be seen, therefore, that by increasing the fineness of a pigment a greater surface is presented to be colored, and hence more color is required. ”

“ These “bordered” pits are very characteristic of the wood of all conifers.
The structure of the root is best studied in the seedling plant, or in a rootlet of an older one. The general plan of the root is much like that of the pteridophytes. The fibro-vascular bundle (Fig. 75, M, fb.) is of the so-called radial type, there being three xylem masses (x) alternating with as many phloem masses (ph.) in the root of the seedling. This regularity becomes destroyed as the root grows older by the formation of a cambium ring, something like that in the stem. ”

“ A cube is a body with sides all of equal breadth and their surfaces perfectly square. When thrown down, it stands firm and steady so long as it is untouched, no matter on which of its sides it has fallen, like the dice which players throw on the board. The Pythagoreans appear to have drawn their analogy from the cube, because the number of lines mentioned will be fixed firmly and steadily in the memory when they have once settled down, like a cube, upon a man's understanding. ”

“ A cube one inch in the side has six square inches of surface to one cubic inch of bulk, while a cube ten inches in the side has 600 square inches of surface to 1000 cubic inches of bulk, so that the surface of the small cube is ten times greater in proportion to its contents than that of the large one. Now, supposing a child to be one-tenth of the size of its mother, besides its feebler powers of generating heat, it will have just ten times as much surface in proportion to its size by which to lose heat as that mother has. ”

“ Take the example of an ordinary cube. The child looks at it, and there is a visual impression formed of its color, of the length of each side, of the area of a surface, of the combination of the surfaces so as to give rise to the idea of solidity. The simple ideas are combined into the more complex idea—the visual one of a cube. But by the aid of touch other qualities can be ascertained. The hardness, weight, sharpness of edges and angles, smoothness or roughness of surfaces, form the tactual idea of a cube. ”

“ They are to be rolled about in play, and thus develop ideas of color, material, form, motion, direction, and muscular sensibility. A sphere, cube, and cylinder of hard wood compose second, the second ‘gift.’ Here, therefore, are found a known factor in the sphere and an unknown one in the cube. A comparison is made of the stability of the cube with the movability of the sphere, and the two are harmonized in the cylinder, which possesses the characteristics third, of each. ”

“ Stems or branches which remain under ground give out roots as freely as roots themselves give off branches. Stems which creep on the ground most commonly root at the joints; so will most branches when bent to the ground, as in propagation by layering; and propagation by cuttings equally depends upon the tendency of the cut end of a shoot to produce roots. Thus, a piece of a plant which has stem and leaves, either developed or in the bud, may be made to produce roots, and so become an independent plant. ”

“ The full symmetry of the cubic system is not realised, however, by a study of the cube alone; 39we only appreciate it when we come to examine the general form of the cubic system, that which is produced by starting with a face oblique to all three axes, and with different amounts of obliquity to each, and seeing how many repetitions of the face the symmetry demands. ”

“ Which being multiplied Cubik wise, will sone shew it selfe, whether it be dubble or no, to the Cubik number of your Fundamentall Cube. By lines, thus: As your lesse and first line, (in your hollow Pyramis or Cone,) is to the second or greater, so let the Radical side of your Fundamẽtall Cube, be to a fourth proportionall line, by the 12. proposition, of the sixth boke of Euclide. Which fourth line, shall be the Rote Cubik, or Radicall side of the Cube, dubble to your Fundamentall Cube: which is the thing we desired. God be thanked for this Inuention, & the fruite ensuing. ”