Matrix

“ Know therefore, that a pure Matrix will afford a pure conception in its own likenesse: As for example, in Animalls there are Matrixes of Women, Cowes, Mares, Bitches, and the like. So in the Minerall, and Vegetable Kingdomes, there are Metalls, Stones, Salts; for the Salts in these two Kingdomes are to bee considered of, as also their places, according to more or lesse. ”

“ For the casting of seed into the furrows of the matrix is the beginning of the generation of men, so that bone taken from bone, and flesh from flesh, by an invisible power, are fashioned into another man. ”

“ If represents the manner in which vectors are affected by a strain, will represent the manner in which surfaces are affected, and the manner in which volumes are affected. Considerations of this kind do not attach themselves so naturally to the notation nor does the subject admit so free a development with this notation, principally because the symbol refers to a special use of the matrix, and is very much in the way when we want to apply the matrix to other uses, or to subject it to various operations. ”

“ The quantities are called the elements of the determinant; the term is called a member of the determinant, and there are evidently members corresponding to the permutations of the numbers The determinant is usually written
the square array being termed the matrix of the determinant. A matrix has in many parts of mathematics a signification apart from its evaluation as a determinant. A theory of matrices has been constructed by Cayley in connexion particularly with the theory of linear transformation. The matrix consists of rows and columns. ”

“ The matrix was in most instances simple, the design giving a direct impression once and for all. But there are examples of elaborate matrices composed of several pieces, from the impressions of which the seal was built up in an ingenious fashion, both obverse and reverse being carved in hollow work, through which figures and subjects impressed on an inner layer of wax are to be seen. Such examples are the seal matrix of the Benedictine priory of St Mary and St Blaise of Boxgrave in Sussex, of the 13th century, now in the British Museum (fig. 3) ”

“ Type-Mould, without Matrix and with a Type in the Mould.would lead to fatal faults in inaccuracy of body. Exactness of body can be had only by casting all the characters in one mould, but this mould must be made to suit all the matrices. The matrices must be frequently changed, but with such nicety that the types of every letter shall be uniform in height, in line, and truly square. Any mechanic will see that the construction of an adjustable mould is work of difficulty, and that the fitting-up of a set of matrices for one mould is a very nice operation. ”

“ I may not know much math, but I did check it."
"Even after the Columbia boys said it was nonsense? Well, does it come out right?"
"No, it doesn't! You multiply the inverted matrix by the original and you not only don't get zeros for all elements outside the diagonal, you get a haphazard assortment of ones and twos. Worse still, every element in the diagonal comes out equal to zero. The product of the two matrices is about as different from the identity matrix as anything could be. ”

“ Of that freedom one may say that it is the matrix, the indispensable condition, of nearly every other form of freedom. ”

“ There is a series of finger keys representing the various characters and spaces, and the machine is so organized that on manipulating the keys it selects the matrices in the order in which their characters are to appear in print, and assembles them in a line, with wedge-shaped spaces or justifiers between the words. The series of matrices thus assembled in line forms a line matrix, or, in other words, a line of female dies adapted to mold or form a line of raised type on a slug cast against the matrices. ”

“ This defines the internal product of matrices, as This product evidently gives a matrix, the operation of which is equivalent to the successive operations of and i.e., We may express this a little more generally by saying that internal multiplication is associative when performed on a series of matrices, or on such a series terminated by a quantity of the first degree. Another kind of multiplication of binary indeterminate products is that in which the preceding factors are multiplied combinatorially, and also the following. ”

“ You have no right to look upon a single one of these infants, for they have ceased to be attached to the primitive matrix out of which they have emerged, and we will teach them all that they need to know. ”

“ Although the rules of the game are relatively simple, highly complex forms of artificial life arise on the matrix: a cell going from empty to full describes birth, from full to empty, death. Satisfaction in playing is derived from reaching complex forms of life. ”

“ The early illustrations of his theories were to be regarded as tentative rather than dogmatic, and in later life he recognized that many modifications in matters of detail were rendered necessary by new facts which had since come to light. The crystallization of solid facts out of a matrix which is necessarily partially volatile is a process requiring time. ”

“ It also furnishes us with certain Instruments to help and strengthen weak Parts, such as Pessaries, which retain the Matrix in its proper place when it is fallen, Crutches to assist feeble Persons in going, &c. ”

“ We may go further, and assert that not only must the matrices have been harmoniously justified, but the mould employed, whatever its form, must have had its adjustable parts finished with a near approach to mathematical accuracy, which left little to be accomplished in the way of further improvement. ”

“ The reason why I would not stir out of his house till I had taken a survey of his matrices was, because I was fearful that he might pick and cull (as we call it) a great {214} many things which are useful in printing besides just the alphabets; and indeed lest he might change some whole sets; though indeed the man declares he would not do a thing so ill for his life. However I having all the matrices brought into one room locked ’em up and took the key away with me, and went to dinner. ”

“ I. Your Lordship would treat me as if I were one of the vulgar who, being ignorant of Mathematics, suppose that a Woman is really a Straight Line, and only of One Dimension. No, no, my Lord; we Squares are better advised, and are as well aware of your Lordship that a Woman, though popularly called a Straight Line, is, really and scientifically, a very thin Parallelogram, possessing Two Dimensions, like the rest of us, viz., length and breadth (or thickness) . ”

“ 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. ”

“ In the study of nature, for example, we distinguish matters, forces, genera and the like, and stereotype each in its isolation. Thought is here acting in its analytic capacity, where its canon is identity, a simple reference of each attribute to itself. It is under the guidance of the same identity that the process in knowledge is effected from one scientific truth to another. Thus, for example, in mathematics magnitude is the feature which, to the neglect of any other, determines our advance. Hence in geometry we compare one figure with another, so as to bring out their identity. ”

“ Why should that absent and imagined slant-legged image displace the present and felt square image from our mind? Why, when the objective cross gives us so many shapes, as it varies its position, should we think we feel the true shape only when the cross is directly in front? And when that question is answered, how can the absent and represented feeling of a slant-legged figure so successfully intrude itself into the place of a presented square one? ”

“ 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? ”

“ I trust that some one with the right intellectual equipment will endeavor to follow this guide; and I am sanguine enough to think that a comprehensive study of the geometrical phenomena of differentiation will suggest to a penetrative mind that critical experiment which may one day reveal the meaning of spontaneous division, the mystery through which lies the road, perhaps the most hopeful, to a knowledge of the nature of life. ”

“ As a tapestry picture, however various and full of meaning, is ultimately reducible to little squares; so the world of common sense is ultimately reducible to a series of static elements conditioned by the machinery of the brain. Subtle curves, swift movement, delicate gradation, that machinery cannot represent. It leaves them out. From the countless suggestions, the tangle of many-coloured wools which the real world presents to you, you snatch one here and there. ”

“ 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. ”

“ Life spread before him like a geometrical figure, constructed of perfect circles and absolutely conformable to the rules and the principles of mathematics. That these perfect circles should ever run wild and become a square was clearly unthinkable. ”

“ James was possessed of the matrices, and consequently of the secret of adapting them to his purpose. To supply this deficiency in a specimen of the matrices (for as such the specimen is now to be considered) has been attended with trouble incredible to anyone but one who upon a like occasion shall attempt the same. And such an occasion we believe there will never be. ”

“ As we already mentioned, reality is the one generated by the model associated to a given external reality. Each time we state a truth, we have to specify first the model.
Example: There are an A and a B person. A is taller than B, as it results from measurement. The term 'length' is generated of the model 'space', as Euclid's Geometry and Newton's Mechanics understand it. These fundamental models characterize this truth as objective. If we say that "A is more attractive than B", this is a subjective truth. ”

“ 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 ”

“ Of those subjects which have no relations to space, the elementary theory of eliminations and substitutions, including the theory of matrices and determinants, seems to afford the most simple application of multiple algebra. I have already indicated what seems to me the appropriate foundation for the theory of matrices. The method is essentially that which Grassmann has sketched in his first Ausdehnungslehre under the name of the open product and has developed at length in the second. ”

“ These linear phenomena direct his attention to the linear properties of surrounding objects. Now the head becomes a circle, and now the circular line represents the head, the elliptical curve connected with it represents the body; arms and legs appear as straight or broken lines, and these again represent arms and legs; the fingers he sees as straight lines meeting in a common point, and lines so connected are, for the busy child, again hands and fingers; the eyes he sees as dots, and these again represent eyes; and thus a new world opens within and without. ”

“ This small specimen of the geometry of visibles, is intended to lead the reader to a clear and distinct conception, of the figure and extension which is presented to the mind by vision; and to demonstrate the truth of what we have affirmed above, namely, that those figures and that extension which are the immediate objects of sight, are not the figures and the extension about which common geometry is employed; that the geometrician, while he looks at his diagram, and demonstrates a proposition, hath a figure presented to his eye, which is only a sign and representative of a tangible figure ”

“ But even that accessory he became able to dispense with; for he could so clearly present the relations of space to his mind without the aid of diagrams, that he was able to draw mentally the lines illustrating the solution of complex problems and to preserve this mental image so distinctly that he could reason upon it as correctly as on the diagrams drawn upon paper required by ordinary students. When he came to solid geometry he was able by means of this power of mental representation to dispense with models, which to other students were indispensable. ”

“ Mathematical.—The development of so positive a science enabled me to establish certain mathematical or Geometric laws of cerebral action, concerning the direction and mode in which all faculties act upon the mind and body, which laws constitute the Basic Philosophy of Anthropology, the highest generalization of science. These laws constitute a compact system of science, lying at the basis of all psychology, as the bony skeleton is the basis of the human form. ”