Lattice work

“ In the simplest example of the three-dimensional graph we have to solve the system
and a system for the general lattice constructed upon the same principle. The system has been discussed by MacMahon, Phil. Trans. vol. clxxxvii. A, 1896, pp. 619-673, with the conclusion that if the numbers of nodes along the axes of x, y, z be limited not to exceed the numbers m, n, l respectively, then writing for brevity 1 – xs = (s) , the generating function is given by the product of the factors
one factor appearing at each point of the lattice. ”

“ For it is not unknown that Lactantius, a writer of prominence in other lines although but little versed in mathematics, spoke very childishly about the form of the earth when he ridiculed those who declared that it was spherical. Hence it should not seem strange to the learned if some shall look upon us in the same way. Mathematics is written for mathematicians, to whom these labors of ours will seem, if I mistake not, to add something even to the republic of the Church.... ”

“ It is not otherwise with this quest after a philosophical method. After committing so many errors one would like to contrive a logical calculus which would put us beyond the need of taking trouble, and without putting any strain on our personal acuteness, would present us with the correct results in every case with the certainty of a machine. Those labours have been constantly fruitless. However much rough work machinery saves us, no machine has as yet been invented which will undertake for us the whole work of life at once. ”

“ We have already seen how strenuously Lotze insists upon the fact that the given subject-matter of thought is to be regarded wholly as the work of a physical mechanism, "without any action of thought." ”

“ But as it has been found practicable to frame a technical form, by conforming to which we can make sure of finding the conclusion of the ratiocination, our end can be completely attained without our ever thinking of any thing but the symbols. Being thus intended to work merely as mechanism, they have the qualities which mechanism ought to have. They are of the least possible bulk, so that they take up scarcely any room, and waste no time in their manipulation ”

“ Strictly speaking, these are not lathes, for in order to do the things before mentioned it is necessary to use after attachments in connection with them, so that the combination of them produces the results spoken of. There is, absolutely, an unlimited field for the genius of workmen to exert itself in designing patterns and executing work of an ornamental character. ”

“ The mechanic who earns his living by working ten hours a day in a workshop, does not care to go home and pursue the same calling in the evening; but he can institute an agreeable change in his life, beautify his dwelling, and cultivate his taste, by the use of the lathe, and thus obtain ornaments that would cost large sums if purchased at the stores; or he may, indeed, make the lathe a source of revenue, and sell the product of his skill and ingenuity at high prices to those who admire, but have not the ability to construct. ”

“ Many a mechanic is working hard at fluxions in his garret, ignorant of the improved notation of modern days. He finds his great reward in the delight which the exercise of his intellectual faculties affords, and, with no thought of scientific celebrity, he revels in the profundities of the higher calculus. Such pure and disinterested love of science is one of the most hopeful features of the labour question. ”

“ We see only a few of the spokes of a revolving wheel, and it accords more with the modesty of true science, to ascribe the apparent imperfection to our point of view, than to conclude that the circle is defective. The more we enlarge our view the more complete does the circle become; and it is but the doctrine of inductive philosophy to conclude, that if our vision were entirely unobstructed, the machinery would appear in all its perfection. ”

“ Yet even here, if we look beneath the apparent freedom and informality, we find certain laws at work which seem to differ only in degree from the more definite and constructive control of line which we have been considering. In the first place, there are our direct impressions from nature; and, secondly, our conscious aims and efforts to express an idea in our minds. We have the same restricted and definite forms of language and materials in each case—line, form, space, brushes, pencil, colour, paper, canvas, or clay. ”

“ It is a specification that emerges, correspondently, in both datum and ideatum. That which is left behind in the evolution of accepted meaning is still characterized as real, but real now in relation only to a way of experiencing—to a peculiarity of the organism. That which is moved toward is regarded as real in a cosmic or extra-organic sense.
1. The data of thought.—When we turn to Lotze, we find that he makes a clear distinction between the presented material of thought, its datum, and the typical characteristic modes of thinking in virtue of which the datum gets organization or system. ”

“ Imagine yourselves before a living model, man or woman, with a pencil, and a piece of paper twice the dimensions of your hand, on which to copy it. Certainly, you cannot be expected to reproduce the magnitude of the limbs, for your paper is too small; nor can you be expected to reproduce their color, for you have only black and white to work with. What you have to do is to reproduce their relationships, and first the proportions, that is to say, the relationships of magnitude. ”

“ Working by the methods of graphic statics, the engineer’s task is, in theory, one of great simplicity. He begins by drawing in outline the structure which he desires to erect; he calculates the stresses and bending-moments necessitated by the dimensions and load on the structure; he draws a new diagram representing these forces, and he designs and builds his fabric on the lines of this statical diagram. He does, in short, precisely what we have seen nature doing in the case of the bone. ”

“ But I invite you today to step into a little dramatic workshop, instead of a scientific library; and to see an humble workman in the craft, trying, with repeated experiments—not to elucidate the laws of dramatic construction, but to obey them, exactly as an inventor (deficient, it may be, in all scientific knowledge) tries to apply the general laws of mechanics to the immediate necessities of the machine he is working out in his mind. The moment a professor of chemistry has expressed a scientific truth, he must illustrate it at once by an experiment, or the truth will evaporate. ”

“ It embraces, within a comparatively small compass, all the rules and formulas for mathematical computation, and all the practical results of mechanical philosophy. It is, indeed, a collection of the useful results of science and the interesting facts which have been developed by experience. It may safely be said, that no work, of the same extent, contains so much information, with the rules for applying it to practical purposes. ”

“ Whenever we perceive a case of what we call operation, we invariably find that the result which issues from the operation assumes different forms, when the same so-called cause enters into relation with different objects, and that it is consequently as much determined by the nature of that member in the relation which we regard as passive or receptive as by the nature of the other member, which appears to us to be more especially active. This is a thought which is never left out of sight in the investigations of natural science. ”

“ Passing over the most abstract science, Logic, on the due guidance by which, however, the large producer or distributor depends, knowingly or unknowingly, for success in his business-forecasts, we come first to Mathematics. Of this, the most general division, dealing with number, guides all industrial activities; be they those by which processes are adjusted, or estimates framed, or commodities bought and sold, or accounts kept. No one needs to have the value of this division of abstract science insisted upon. ”

“ 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. ”

“ It is a ratiocination; and when our knowledge of the causes is so perfect, as to extend to the exact [pg 470] numerical laws which they observe in producing their effects, the ratiocination may reckon among its premisses the theorems of the science of number, in the whole immense extent of that science. Not only are the highest truths of mathematics often required to enable us to compute an effect, the numerical law of which we already know; but, even by the aid of those highest truths, we can go but a little way. ”

“ Since such is the purport of all propositions which convey any real knowledge, and since ratiocination is a mode of acquiring real knowledge, any theory of ratiocination which does not recognize this import of propositions, can not, we may be sure, be the true one.
Applying this view of propositions to the two premises of a syllogism, we obtain the following results. The major premise, which, as already remarked, is always universal, asserts, that all things which have a certain attribute (or attributes) have or have not along with it, a certain other attribute (or attributes) . ”

“ Lotze vibrates between the notion of thought as a tool in the external sense, a mere scaffolding to a finished building in which it has no part nor lot, and the notion of thought as an immanent tool, as a scaffolding which is an integral part of the very operation of building, and which is set up for the sake of the building-activity which is carried on effectively only with and through a scaffolding. ”

“ Again, it is a mistake to plan many small projects consisting of small parts in the hope of effecting economy. The awkwardness of the average grammar school boy will make it necessary to discard much of such stock. Where the parts are of some size, it is possible in most every instance to give him a new but smaller set of dimensions and require him to continue to work on the piece originally given. ”

“ A collection of traits left as the common residuum, the caput mortuum, of a million objects, would be merely a collection, an inventory or aggregate, not a general idea; a striking trait emphasized in any one experience which then served to help understand some one other experience, would become, in virtue of that service of application, in so far general. Synthesis is not a matter of mechanical addition, but of application of something discovered in one case to bring other cases into line. ”

“ The work of assembling the factors of production in any productive establishment, whether it be a shop, farm, factory, or transportation system, calls for a degree of knowledge and care comparable with that of the chemist in the assembling of chemical elements, though, as stated before, the chemist must follow definite formulae with mathematical precision, because of the law of definite proportions. ”

“ 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? ”

“ In this respect the organized and animated body agrees with a common machine, the work of human hands. It agrees with, but it also surpasses it. It has an internal power of self-adjustment. No amount of work would convert a thin line into a strong rope, or a light framework into a strong one. ”

“ But as we become better used to the large rôle which these phenomena play,—not that the phenomena are new, but that their present importance is new, and hence our serious study of them is new—we are increasingly able to see that other elements of static theory also apply. Static [Pg 565] theory applies increasingly as understanding increases! The vaguely discerned, the novel, the imperfectly analyzed, can be stated only in qualitative terms. As things are better understood, the mind seeks system, taxonomy, quantitative measurement. ”

“ In front of every one of them hangs a multiplication table, and close by his hand is an erasing-board; [33] whenever an artisan does anything, he looks at the multiplication table, and verifies it with surety, and then writes down one thing on the board and erases another, and brings it into accuracy: what is written in figures turns out just so in fact. And when a holiday comes, they assemble in pairs, each takes a slender rod in his hand, and they go off to enjoy themselves in honorably dignified fashion, as is fitting. ”

“ They have been the servants of Nature, not the slaves of method. But the grammarian of the laboratory is often the victim of his trade. He staggers forth from his workshop, where prolonged concentration on a mechanical task, directed to a provisional and doubtful goal, has dimmed his faculties; the glaring motley of the world, bathed in sunlight, dazzles him ”