Diagram

“ The diagram was, however, so destitute of all credibility, that neither these easily deceived women, nor the most rustic class of men, nor those who were ready to be led away by any plausible pretender whatever, ever gave their assent to the diagram. Nor, indeed, have we ever met any individual, although we have visited many parts of the earth, and have sought out all those who anywhere made profession of knowledge, that placed any faith in this diagram. ”

“ The assertion that correct diagrams can be made mechanically is not borne out by the facts. It is easier to avoid precision in oral analysis than in written. The diagram drives the pupil to a most searching examination of the sentence, brings him face to face with every difficulty, and compels a decision on every point. ”

“ Yet all that he had asserted respecting the function to which the axioms are confined in the demonstrations of geometry, holds equally true of the definitions. Every demonstration in Euclid might be carried on without them. This is apparent from the ordinary process of proving a proposition of geometry by means of a diagram. What assumption, in fact, do we set out from, to demonstrate by a diagram any of the properties of the circle? ”

“ The different diagrams which we obtain by different laws of association are all such as may be obtained from one another by a process of deformation, and this consideration is sufficient to demonstrate their properties from the well-known properties of the diagram in which the volume and pressure are represented by rectangular co-ordinates. ”

“ If we make a symbolic diagram on a blackboard, of the laws of association between ideas, we are inevitably led to draw circles, or closed figures of some kind, and to connect them by lines. When we hear that the nerve-centres contain cells which send off fibres, we say that Nature has realized our diagram for us, and that the mechanical substratum of thought is plain. In some way, it is true, our diagram must be realized in the brain ”

“ The correctness or the truth of a map consists in the correspondence between the reality and the diagram, which is an arbitrary sign of it. Throughout the whole of our ordinary life we find that we make use of symbols and signs that are not themselves either parts of or copies of the things for which they stand. ”

“ When the number of words in each group is increased there is, of course, closer agreement of their diagrams, and this became so evident in the earlier stages of the investigation that the conclusion was soon reached that if a diagram be made representing a very large number of words from a given author, it will not differ sensibly from any other diagram representing an equally large number of words from the same author. ”

“ The study of the motion of a material system is much assisted by the use of a series of diagrams representing the configuration, displacement and acceleration of the parts of the system.
Diagram of Configuration.—In considering a material system it is often convenient to suppose that we have a record of its position at any given instant in the form of a diagram of configuration. The position of any particle of the system is defined by drawing a straight line or vector from the origin, or point of reference, to the given particle. ”

“ The diagrams illustrating different cases under the law of combination can easily be perfected into neural diagrams, though, to be sure, any diagram is ultra-simple as compared with the great number of neurones that take part in even a simple reaction. ”

“ In geometry, for instance, he learns to distinguish the relations one part of his diagram has to another. He reasons from known relations to those which are unknown, and thus discovers many new truths. ”

“ When the geometrician draws a diagram with the most perfect accuracy; when he keeps his eye fixed upon it, while he goes through a long process of reasoning, and demonstrates the relations of the several parts of his figure; he does not consider, that the visible figure presented to his eye, is only the representative of a tangible figure, upon which all his attention is fixed ”

“ Upon which the same disorder, confusion, and error follow, as would if a man, going to demonstrate something of an heptagon, should, in the diagram he took to do it, leave out one of the angles, or by oversight make the figure with an angle more than the name ordinarily imported, or he intended it should when at first he thought of his demonstration. ”

“ We draw with a continuous line, after having looked at, or thought of, our model. How shall we explain such a faculty, except by our habit of discovering at once the organization of the outlines of common objects, that is to say, by a motor tendency to draft their diagram in one continuous line? ”

“ The method of representation, in which the co-ordinates of the point in the diagram are made equal to the volume and entropy of the body, presents certain characteristics which entitle it to a somewhat detailed consideration, and for some purposes give it substantial advantages over any other method. ”

“ And like a diagram or map, it is in a way artificial since it gives weight to one element at the expense of the others. But again like the diagram or map, the thing it shows is a fact, a fact which is more readily grasped by this artificial device than by bald statement. Maps do not take the place of photographs, nevertheless they have their own peculiar place in making intelligible the make-up of the physical world. ”

“ Readers will recognize the diagram as a familiar friend of their youth. A man possessed a square-shaped estate. He bequeathed to his widow the quarter of it that is shaded off. The remainder was to be divided equitably amongst his four sons, so that each should receive land of exactly the same area and exactly similar in shape. ”

“ If a branching diagram had not been used, and only the names of the groups had been written in a linear series, it would have been still less possible to have given a natural arrangement; and it is notoriously not possible to represent in a series, on a flat surface, the affinities which we discover in nature amongst the beings of the same group. ”

“ The experiment may be made as follows:—The percipient, being blindfolded, is seated at a table with his back to the operator, without contact and in perfect silence. A diagram—for instance, a circle with a cross in the centre—is distinctly drawn by a third person and so held as to be in full view of the operator, who looks at it in silence, steadily and with concentrated attention. ”

“ By making things mathematical they make them thin. Take the living lines out of that landscape, simplify it to a right angle, and you flatten it out to a mere diagram on paper. Diagrams have their own beauty ”

“ Even so elementary an operation as that of abstracting from the particular configuration of the triangles or other figures, and the relative situation of the particular lines or points, in the diagram which aids the apprehension of a common geometrical demonstration, is a very useful, and far from being always an easy, exercise of the faculty of generalization so strangely imagined to have no place or part in the processes of mathematics.—Mill, John Stuart. ”

“ Then if we took a photograph in yellow light we should get only the E's as in the second diagram; if we took a photograph in red light we should get only the A's. The two photographs would be quite different although we had actually photographed the same picture. ”

“ The plans and designs drawn by architects and engineers are used to determine the value of certain real magnitudes by measuring certain distances on the diagram. For such purposes it is essential that the drawing be as accurate as possible. ”

“ The diagram shows all the irregularities plotted in the light of our present knowledge; and, to compare with their amounts, a few standard things are placed on the same scale, such as the smallest interval capable of being detected with the unaided eye, the distance of the component stars in ε Lyræ, the constants of aberration, of nutation, and of stellar parallax. ”

“ But, that unpractised learners, even in making use of one theorem to demonstrate another, reason rather from particular to particular than from the general proposition, is manifest from the difficulty they find in applying a theorem to a case in which the configuration of the diagram is extremely unlike that of the diagram by which the original theorem was demonstrated. A difficulty which, except in cases of unusual mental power, long practice can alone remove, and removes chiefly by rendering us familiar with all the configurations consistent with the general conditions of the theorem. ”

“ If we superimpose upon one another (or even set side by side) pictures, or plane projections, of an organism, drawn at successive intervals of time, we have such a three-dimensional diagram, which is a partial representation (limited to two dimensions of space) of the organism’s gradual change of form, or course of development; and in such a case our contour-lines may, for the purposes of the embryologist, be separated by intervals representing a few hours or days, or, for the purposes of the palaeontologist, by interspaces of unnumbered and innumerable years79. ”

“ Very neat and serviceable diagrams may be produced even by children with no natural taste for drawing, but they need to be shown how to work, and perhaps to have the lines of a diagram suggested to them at first by a rough blackboard sketch, or it may not occur to them that a few simple lines will show all that is necessary better than a would-be realistic sketch of apparatus, with impossible perspective and smudgy shading. ”

“ The trains of visible ideas, which occur in dreams, are deducible, partly from the sympathy here mentioned, partly from that of the last paragraph.
Eleventhly, Our stock of visible ideas may be considered as a key to a great part in our knowledge, and a principal source of invention in poetry, painting, mathematics, mechanics, and almost every other branch of the arts and sciences. In mathematics and mechanics the invention of the diagram is, in effect, the solution of the problem. ”

“ We need not confuse the reader by mathematics, but to establish our theory we produce the Diagram IA, and it will be here seen that the relative proportions existing in the segments of the circles have been observed in the triangles. ”

“ All you need is to make a simple plan or diagram on a sheet of paper or cardboard and number eight counters, 1 to 8. Then a whole family can enter into an amusing competition to find the best possible solution of the difficulty. ”

“ Now the acts of mutual giving in due proportion may be represented by the diameters of a parallelogram, at the four angles of which the parties and their wares are so placed that the side connecting the parties be opposite to that connecting the wares, and each party be connected by one side with his own ware, as in the accompanying diagram. ”

“ They symbolize inconceivable things, whereas, to answer his purpose, they should symbolize conceivable ones; seeing that the ordinary acceptation of lineal diagrams, although sufficiently true for the conclusions of mathematics, does not cut deep enough for a fourth dimension, which, if diagrams are to facilitate our comprehension of it, needs the exact truth of what they postulate. ”

“ In the diagram given of one of the possible solutions it will be seen that there are eight places at which we may start this particular tour; but there is only one route in each case, because we must complete the compartment in which we find ourself before passing into another. In any solution we shall find that the squares distinguished by stars must be entering or exit points, but the law of reversals leaves us the option of making the other connections either at the diamonds or at the circles. ”

“ And though mathematical demonstration depends not upon sense, yet the examining them by diagrams gives great credit to the evidence of our sight, and seems to give it a certainty approaching to that of demonstration itself. For, it would be very strange, that a man should allow it for an undeniable truth, that two angles of a figure, which he measures by lines and angles of a diagram, should be bigger one than the other, and yet doubt of the existence of those lines and angles, which by looking on he makes use of to measure that by. ”

“ For suppose we limit ourselves to the 27 positions numbered in Diagram I. Two of these positions form one shape, three of them will form another shape, and so on. And in going over each of these arrangements we gradually get to know the whole set of them which form the block.
Having given up for the time any question as to the possible subdivisions of the cube, and looking on each cube as a unit position, we have 27 positions. These positions can be taken in different selections, and each selection is a shape. ”

“ There is danger of requiring too much written analysis. When the ordinary constructions have been made clear, diagrams should be used only for the more difficult sentences, or, if the sentences are long, only for the more difficult parts of them. In both oral and written analysis there is danger of repeating what needs no repetition. When the diagram has served its purpose, it should be dropped. ”

“ In the Grasset diagram, No. 3, A B C D correspond to a b c d in skeleton diagram No. 4, while the dotted forms, E and F, No. 3, correspond to e and f, No. 4.
In the Herkomer, No. 2, w x y z is the rectangle W X Y Z of diagram No. 1; 1 2 3 4 5 correspond to A, 6 7 8 9 10 to B, 11 12 13 14 to C; while the d e and f g equal D E, F G; p is our plumb-line, P.
It must be distinctly understood that any number of objects may be contained in a rectangle; let a child scribble upon this page, anywhere, a dozen or more {64} DIAGRAM NO. 3. DIAGRAM NO. 4. ”

“ It invariably requires something of the nature of a diagram; that is, an "Icon," or Sign that represents its Object in resembling it. It usually, too, needs "Indices," or Signs that represent their Objects by being actually connected with them. But it is mainly composed of "Symbols," or Signs that represent their Objects essentially because they will be so interpreted. Demonstration should be Corollarial when it can. An accurate definition of Corollarial Demonstration would require a long explanation ”