Combe

“ Measure an angle of creation, make a proportionally progressive multiplication, and all infinity shall multiply its circles, peopled by universes, passing in proportional segments between the extending symbolical arms of your compass. Suppose now that, from whatever point of the infinite above you, a hand holds another compass or square, then the lines of the celestial triangle will meet of necessity those of the compass of science and will form therewith the mysterious star of Solomon. ”

“ Others, again, are in the murky depths of vast arithmetical solutions extracting, with heavy breathings, the cube root from some formidable quantity, and bringing it to the surface exhausted and far from certain as to the ultimate utility of their discoveries. ”

“ What sense besides vision suggests to us the idea of her figure? And even in objects which we can reach, what is that circumstance in the sense of vision which suggests to us that the colour belongs to the sphere, except that we see the colour where we see the sphere? If we do not see figure, we do not see position; for figure is the relative position of the parts of a boundary. If we do not see position, why do we ascribe the yellow colour to the sphere on our left, rather than to the cube on our right? ”

“ If he got the set of right appearances which a cube would present to him when, turning about in a systematic way, it came at intervals into his plane, and if, moreover, he fixed his mind on these appearances, he might at last, if it was in him, rise to the conception of a cube as we know it.
Now, the parts by which a higher solid comes into our space are solids, and what we have to form is a set of solids coming and going in a systematic way, as the higher figure is moved about in a systematic way. ”

“ If the ivory be not a cube, it must have a spherical or some other form. And the cube, in order to be a cube, must be of some material;—if not of ivory, of wood, or stone, for instance, A figure without matter is merely a geometrical conception;—a modification of the idea of space. Matter without figure is a mere abstract term;—a supposed union of certain sensible qualities which, so insulated 39 from others, cannot exist. ”

“ It follows that we are in the heart of a hollow sphere of loveliest blue, spangled with light. Now the sphere is the one perfect geometrical form. Over and round us then we have the one perfect shape. ”

“ 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. ”

“ For fluid and solid, hot and cold, form the material of all composite bodies; and all other differences are secondary to these, such differences, that is, as heaviness or lightness, density or rarity, roughness or smoothness, and any other such properties of matter as there may be. second degree of composition is that by which the homogeneous parts of animals, such as bone, flesh, and the like, are constituted out of the primary substances. ”

“ The light, that converges on our eye, from all the stars of heaven, within what we term the field of our vision, is collected, in a space, that cannot be larger than the retina on which it falls. A cube or a sphere is represented to us, by the two dimensions of a coloured plane, variously shaded, as truly, as by the object itself with its triple dimensions ”

“ It will of course be understood that by the term Symmetry I here intend, not that more indefinite attribute of form which belongs to the domain of the fine arts, as when we speak of the ‘symmetry’ of an edifice 68 or of a sculptured figure, but a certain definite relation or property, no less rigorous and precise than other relations of number and position, which is thus one of the sure guides of the scientific faculty, and one of the bases of our exact science. ”

“ What is a solid? It is the union of three dimensions. If sight can give us the idea of surfaces which consist of two, why not also of solids which consist of three dimensions? This one reflection is enough to show that it has been denied without reason; but I shall not stop with this, but shall prove, by the most rigorous observation, and the analysis of its phenomena, that sight can give us the idea of a solid. ”

“ For purposes of study, however, you must regard objects of three dimensions as bounded by lines, just as they appear in photographs, drawings, or other flat representations, geometric or perspective. In regarding objects from the point of view of decoration there is still another element to be considered; that is, the element of material, the substance of which objects consist, for it is evident that the ornament which would be appropriate to wood, for instance, Fig. 4. might not be appropriate to metal or to stone. ”

“ Again, just as solid figures can be represented on paper by perspective, four-dimensional figures can be represented perspectively by solids. If there are two squares, one lying over the other, and the underneath one be pushed away, its sides remaining parallel with the one that was over it, then if each point of the one be joined to the corresponding point of the other, we have a fair representation on paper of a cube. ”

“ We may avoid all such confusion by using and to represent a strain. Let us consider an element of the body strained which in the state is a cube with its edges parallel to the axes of , and call the edges according to the axes to which they are parallel, and consider the ends of the edges as positive for which the values of or are the greater. ”

“ 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. ”

“ If from every point of a material body a new direction goes off, the matter which fills up the space produced by the solid moving in this new direction will have the solid it started from as its surface, and will be to it as a solid cube is to the square which bounds it on the top. ”

“ If we draw black outlines, such as represent the edges of a cube seen in perspective, certain of the lines will cross each other; and we may make this cube appear to assume two different positions, by determining in our own mind that the lines which belong to one end of the cube shall be understood to be before or to be behind those which they cross. ”

“ Living shapes. Every visible thing has a shape. A rock, a pyramid, a scaffolding round a house has a shape; but you cannot say that they are alive. Now this thing was alive. This thing bore perhaps no closer relation to the shape of a live being than to the shape of a rock, a pyramid or a scaffolding. ”

“ In a word the whole of Geometry is an appeal to the Imagination in which the geometer says to us, "I know that my propositions are not exactly true except with respect to invisible, ideal, and imaginary figures, planes, and solids. These ideas, therefore, you must endeavour to imagine. In order to relieve the strain on your imagination, I will place before you material and visible figures about which my reasoning will be approximately true. ”

“ The Proposition, That the Superficies of a Sphere is equal to the Area's of Four of its greatest Circles, which is one of the most wonderful Inventions that was ever found in Geometry, shews him to have been a much greater Man, than all that is said of him by the Roman, or Greek Historians. ”

“ Of things said to have fallen from the sky, we make, in the usual scientific way, two divisions: miscellaneous objects and substances, and symmetric objects attributable to beings like human beings, sub-dividing into—wedges, spheres, and disks. ”

“ Surfaces and Curves in which the Composition of the Body represented is Variable and its Temperature and Pressure are Constant.
When there are three components, the position of a point in the plane may indicate the composition of a body most simply, perhaps, as follows. The body is supposed to be composed of the quantities of the substances , the value of being unity. ”

“ As white light is seen to be composed of but three primary colours, as Music is based on the Triad, as Space is known to us in three dimensions only, and Geometry, "the Head of all Learning," is based upon the Circle, Square, and Triangle, so may we see life in its three primary aspects: the Animal, Vegetable, and Material. ”

“ If each visible solid owes its form to superficies, and each superficies its form to lines, and each line its form to a point—and a point has no form, because it has no parts—then, who shall stone the man that cries out, "Ex nihilo geometria fit?" ”

“ The movement has been barren of any modification perceptible to the senses and appreciable to the mind. There has been no lesson unless you have, by words speaking to the mind, succeeded in making the child comprehend the idea of a cube derived from its intrinsic properties; a body with six equal sides and eight equal angles. ”

“ The right-angled isosceles triangle produces from itself a square, andthe square generates from itself the cube, which is the figure peculiar to earth. But the figure of a beautiful and perfect sphere was imparted to the most beautiful and perfect world, that it might be indigent of nothing, but contain all things, embracing and comprehending them in itself, and thus might be excellent and admirable, similar to and in concord with itself, ever moving musically and melodiously. ”

“ FLOWERS and other forms as LINE-MOTIVES. The rectangular space may be subdivided, as was the square, by a simple line-motif,—flower, fruit, still life, animal or figure,—following some Principle of Composition. In chapter III, under Subordination, an exercise was suggested and illustrated; it could be taken up again at this point, with new subjects, for a study of Variation. As rectangular compositions will be found under Notan and Color, it is not necessary to consider them further here as pure line, except in the case of Landscape, to which a special chapter is given. ”

“ And as geometricians, in the case of boys who cannot yet be initiated into the perception of incorporeal and impassive substance, convey their ideas through the medium of spheres, cubes, and dodecahedrons, so celestial Love has contrived beautiful mirrors of beautiful things, and exhibits them to us glittering in the shapes colours and appearances of youths in all their flower, and calmly stirs the memory which is inflamed first by these. ”

“ Every existent thing, that is, every individual object, is a compound of matter and form. We may compare them in this respect to the material and the shape of a thing, though we must be careful not to think that form is merely shape. Geometry considers shapes as if they existed by themselves. ”

“ There came a day when, for simple exhaustion, if symmetry should really demand just one more, he was ready to take symmetry into account. Symmetry was harmony, and the idea of harmony began to haunt him; he said to himself that harmony was of course every thing. ”

“ If by the soul you mean the Form or Essence of the living animal, by the circle, the Form of the circle, by the right angle, the Form or Essence thereof, — then this Form is posterior in regard to the notional parts of which it is constituted, but prior in regard to the particular circle or right angle. ”

“ A third child, excessively trained in form studies by the constant use of geometric solids, and much practice in looking for the fundamental geometric forms lying at the basis of the multifarious objects that exist in the world, will, as a matter of course, apperceive geometric forms, ignoring the other phases of objects. ”

“ We get it when we pass from a short vertical line to a long horizontal one, from a small square to a large circle, as well as when we pass between those figures whose shapes are congruous. But when the shapes are congruous our consciousness of the relation is a good deal more distinct, and it is most distinct of all when, in the exercise of our analytic attention, we notice, first, a part, and then the whole, of a single line or shape. ”

“ For the movement of composite bodies is, as we said, determined by that simple body which preponderates in the composition. These premises clearly give the conclusion that there is in nature some bodily substance other than the formations we know, prior to them all and more divine than they. ”

“ Now the first peculiarity distinguishing things from shapes is that they can occupy more or less cubic space: we can hit up against them, displace them or be displaced by them, and in such process of displacing or resisting displacement, we become aware of two other peculiarities distinguishing things from shapes: they have weight in varying degrees and texture of various sorts. Otherwise expressed, things have body, they exist in three dimensional space ”

“ The different positions of the several parts of the body with regard to the eye, when put together, make a real figure, which is truly extended in the length and breadth, and which represents a figure that is extended in length, breadth, and thickness. In like manner, a projection of the sphere is a real figure, and hath length and breadth, but represents the sphere, which hath three dimensions. ”

“ One mistake in the pattern does not necessarily do away with the efficacy of a comb, as a Panggang man once said, “It is like a gap or hole in a bird-trap: the bird can hop through it, but it is always a question whether it sees the gap.” ”

“ Similarly, all the movements of space which transform a given solid into itself form a group. For instance, the cube is transformed into itself by twenty-four distinct movements. Nine around the lines which join the middle points of opposite faces, six around those which join the middle points of opposite edges, eight around the diagonals, and the identity. ”