visualization

"The authors on may be divided into... theoretical and practical... [N]one... have combined the theory with the practice... to render the subject plain and intelligible... [T]he most valuable and scientifical are... abstruse, and the practical scarcely furnish... the rationale... The object of the ensuing treatise is to simplify the theory, yet to retain a methodical and accurate... investigation, and to exemplify this theory by... important... useful examples. ...[D]emonstrations are frequently founded on principles strictly Geometrical ...and sometimes ...by algebraical signs, particularly where the Geometrical ...would require a complicated figure, or a ...tedious process. ...[T]he algebraical mode of deduction tends greatly to simplify... yet... definitions and... elementary parts... must be acquired from Geometrical principles illustrated by diagrams; otherwise a student will never obtain a clear and satisfactory knowledge... Should any person attempt to teach the elementary principles of the science by... algebraic characters, and algebraic formulae alone, without the aid of Geometry, he would... deceive both himself and his pupils."

"Geometry has two great treasures: one is the Theorem of Phythagoras, the other the division of a line in extreme and mean ratio. The first we can compare to a mass of gold; the other we may call a precious jewel."

"Geometry enlightens the intellect and sets one's mind right."

"Historically, it was Euclidean geometry that, developed to a large extent as a votive offering to the God of Reason, opened men's eyes to the possibility of design and to the possibility of uncovering it by the pursuit of mathematics."

"The use of canon raised numerous questions concerning the paths of projectiles. ...One might determine... what type of curve a projectile follows and.... prove some geometrical facts about this curve, but geometry could never answer such questions as how high the projectile would go or how far from the starting point it would land. The seventeenth century sought the quantitative or numerical information needed for practical applications, and such information is provided by algebra."

"Descartes... complained that Greek geometry was so much tied to figures "that is can exercise the understanding only on condition of greatly fatiguing the imagination." Descartes also deplored that the methods of Euclidean geometry were exceedingly diverse and specialized and did not allow for general applicability. Each theorem required a new kind of proof... What impressed Descartes especially was that algebra enables man to reason efficiently. It mechanizes thought, and hence produces almost automatically results that may otherwise be difficult to establish. ...historically it was Descartes who clearly perceived and called attention to this feature. Whereas geometry contained the truth about the universe, algebra offered the science of method. It is... paradoxical that great thinkers should be enamored with ideas that mechanize thought. Of course, their goal is to get at more difficult problems, as indeed they do."

"Over and above the specific theorems created by men such as Desargues, Pascal and La Hire, several new ideas and outlooks were beginning to appear. The first is the idea of continuous change of a mathematical entity from one state to another... [i.e., of a] a geometrical figure. It was Kepler, in his Astronomiae Optica of 1604, who first seemed to grasp the fact that parabola, ellipse, hyperbola, circle, and the degenerate conic consisting of a pair of lines are continuously derivable from each other. ...The notion of a continuous change in a figure was also employed by Pascal. He allowed two consecutive vertices of his hexagon to approach each other so that the figure became a pentagon. In the same manner he passed from pentagons to quadrilaterals. The second idea to emerge from the work of the projective geometers is that of transformation and invariance. To project a figure from some point and then take a section of that projection is to transform the figure to a new one. The properties... of interest are those that remain invariant under transformation. Other geometers of the seventeenth century, for example, Gregory of St. Vincent... and Newton, introduced transformations other than projection and section."

"A geometrician has learned to perform the most difficult demonstrations and calculations, as a monkey has learned to take his little hat off and on... All has been accomplished through signs, every species has learned what it could understand, and in this way men have acquired symbolic knowledge..."

"He said that the geometry of the dream-place he saw was abnormal, non-Euclidean, and loathsomely redolent of spheres and dimensions apart from ours."

"I claim that many patterns of Nature are so irregular and fragmented, that, compared with Euclid — a term used in this work to denote all of standard geometry — Nature exhibits not simply a higher degree but an altogether different level of complexity … The existence of these patterns challenges us to study these forms that Euclid leaves aside as being "formless," to investigate the morphology of the "amorphous.""

"I conceived, developed and applied in many areas a new geometry of nature, which finds order in chaotic shapes and processes. It grew without a name until 1975, when I coined a new word to denote it, fractal geometry, from the Latin word for irregular and broken up, fractus. Today you might say that, until fractal geometry became organized, my life had followed a fractal orbit."

"If the Greeks had had a mind to reduce mathematics to one field... their only choice would have been to reduce arithmetic to geometry... it is hardly surprising that for nearly two millennia geometry took pride of place in mathematics. And it would have been obvious to any mathematician that a geometrical problem could not be stated or solved in the language of numbers, since the geometrical universe had more structure than the numerical universe. If one desired to translate geometrical problems into the language of numbers, one would have to invent (or discover) more numbers."

"Let us calculate the motion of bodies, but also consult the plans of the Intelligence that makes them move. It seems that the ancient philosophers made the first attempts at this sort of science, in looking for metaphysical relationships between numbers and material bodies. When they said that God occupies himself with geometry, they surely meant that He unites in that science the works of His power with the perspectives of His wisdom. From the all too few ancient geometers who undertook such studies, we have little that is intelligible or well-founded. The perfection which geometry has acquired since their time puts us in a better position to succeed, and may more than compensate for the advantages that those great minds had over us."

"O king, through the country there are royal roads and roads for common citizens, but in geometry there is one road for all."

"The doctrine of Proportion, in the Fifth Book of Euclid's Elements, is obscure, and unintelligible to most readers. It is not taught either in foreign or American colleges, and is now become obsolete. It has therefore been omitted in this edition of Euclid's Elements, and a different method of treating Proportion has been substituted for it. This is the common algebraical method, which is concise, simple, and perspicuous; and is sufficient for all useful purposes in practical mathematics. The method is clear and intelligible to all persons who know the first principles of algebra. The rudiments of algebra ought to be taught before geometry, because algebra may be applied to geometry in certain cases, and facilitates the study of it."

"Geometry is almost the only subject as to which we find truths wherein all men agree; and one cause of this is, that geometers alone regard the true laws of demonstration."

"At a very early period the study of Geometry was regarded as a very important mental discipline, as may be shewn from the seventh book of the Republic of Plato. To his testimony may be added that of the celebrated Pascal (Œuvres, Tom. I. p. 66,) which Mr. Hallam has quoted in his History of the Literature of the Middle Ages. "Geometry," Pascal observes, "is almost the only subject as to which we find truths wherein all men agree; and one cause of this is, that geometers alone regard the true laws of demonstration." These are enumerated by him as eight in number. 1. To define nothing which cannot be expressed in clearer terms than those in which it is already expressed. 2. To leave no obscure or equivocal terms undefined. 3. To employ in the definition no terms not already known. 4. To omit nothing in the principles from which we argue, unless we are sure it is granted 5. To lay down no axiom which is not perfectly evident. 6. To demonstrate nothing which is as clear already as we can make it. 7. To prove every thing in the least doubtful, by means of self-evident axioms, or of propositions already demonstrated. 8. To substitute mentally the definition instead of the thing defined. Of these rules he says, "the first, fourth, and sixth are not absolutely necessary to avoid error, but the other five are indispensable; and though they may be found in books of logic, none but the geometers have paid any regard to them."

"Various relations being established in geometry between lines constituted under given conditions, as parts of geometrical figures, if we choose to adopt the idea of expressing these lines by numerical measures, we are then brought to the distinction of such lines being in some cases commensurable in their numerical values, in others not so. Their geometrical relations however are absolutely general, and do not refer to any such distinction."

"All those who have written histories [of geometry] bring to this point their account of the development of this science. Not long after these men [pupils of Plato] came Euclid… Not much younger than these [pupils of Plato] is Euclid, who put together the Elements ,…bringing to irrefragable demonstration the things which had been only loosely proved by his predecessors. This man [must have] lived in the time of the first Ptolemy; for Archimedes, who followed closely the first [Ptolemy? book?] makes mention of Euclid, and further they say that Ptolemy once asked him if there were a shorter way to study geometry…to which he replied that there was no royal road to geometry. He is therefore younger than Plato’s circle, but older than Eratosthenes and Archimedes; for these were contemporaries, as Eratosthenes somewhere says."

"It is remarkable that this generalization of plane geometry to surface geometry is identical with that generalization of geometry which originated from the analysis of the axiom of parallels. ...the construction of non-Euclidean geometries could have been equally well based upon the elimination of other axioms. It was perhaps due to an intuitive feeling for theoretical fruitfulness that the criticism always centered around the axiom of parallels. For in this way the axiomatic basis was created for that extension of geometry in which the metric appears as an independent variable. Once the significance of the metric as the characteristic feature of the plane has been recognized from the viewpoint of Gauss' plane theory, it is easy to point out, conversely, its connection with the axiom of parallels. The property of the straight line as being the shortest connection between two points can be transferred to curved surfaces, and leads to the concept of straightest line; on the surface of the sphere the great circles play the role of the shortest line of connection... analogous to that of the straight line on the plane. Yet while the great circles as "straight lines" share the most important property with those of the plane, they are distinct from the latter with respect to the axiom of the parallels: all great circles of the sphere intersect and therefore there are no parallels among these "straight lines". ...If this idea is carried through, and all axioms are formulated on the understanding that by "straight lines" are meant the great circles of the sphere and by "plane" is meant the surface of the sphere, it turns out that this system of elements satisfies the system of axioms within two dimensions which is nearly identical in all of it statements with the axiomatic system of Euclidean geometry; the only exception is the formulation of the axiom of the parallels. The geometry of the spherical surface can be viewed as the realization of a two-dimensional non-Euclidean geometry: the denial of the axiom of the parallels singles out that generalization of geometry which occurs in the transition from the plane to the curve surface."

"Visual forms are not perceived differently from colors or brightness. They are sense qualities, and the visual character of geometry consists in these sense qualities."

"The influence of geometry upon philosophy and scientific method has been profound. Geometry, as established by the Greeks, starts with axioms which are (or are deemed to be) self-evident, and proceeds, by deductive reasoning, to arrive at theorems which are very far from self-evident. The axioms and theorems are held to be true of actual space, which is something given in experience. It thus appeared to be possible to discover things about the actual world by first noticing what is self-evident and then using deduction. This view influenced Plato and Kant, and most of the intermediate philosophers... The eighteenth century doctrine of natural rights is a search for Euclidean axioms in politics. The form of Newton's Principia, in spite of its admittedly empirical material, is entirely dominated by Euclid. Theology, in its exact scholastic forms, takes its style from the same source."

"The Greeks... discovered mathematics and the art of deductive reasoning. Geometry, in particular, is a Greek invention, without which modern science would have been impossible."

"[…] nor did he [Thibaut] formulate the obvious conclusion, namely, that the Greeks were not the inventors of plane geometry, rather it was the Indians. At least this was the message that the Greek scholars saw in Thibaut’s paper. And they didn’t like it... If the Indians invented plane geometry, what was to become of Greek ‘genius’ or of the Greek ‘miracle’?"

".. general relativity is, of course, based on , which is one of the richest frameworks for our understanding of ordinary geometry. ... Many ambitious physicists and mathematicians have gone off into the in search of some fundamental generalization of geometry that would reconcile gravity with quantum mechanics but, generally speaking, they have come back empty-handed — if at all."

"One must permit his people the freedom to seek added work and greater responsibility. In my organization, there are no formal job descriptions or organization charts. Responsibilities are defined in a general way, so that people are not circumscribed. All are permitted to do as they think best and to go to anyone and anywhere for help. Each person is then limited only by his own ability."

"An organization chart is a visual display of an organization's structural skeleton. Such charts show how departments are tied together along the principal lines of authority. They show reporting relationships, not lines of communication. Organization charts are tools of management to deploy human resources and are common in both profit and nonprofit organizations."

"Organization charts are subject to important limitations. A chart shows only formal authority relationships and omits the many significant informal and informational relationships."

"In practice, the organization chart is a poor way to describe the happenings in an organization and almost worthless as a way in which to prescribe the actions of managers at the various hierarchical levels. One of the weaknesses is that the organization chart is purely hierarchical; it may defer to conventional management techniques such as matrix management or 'group working' but its only proper point of reference is that of organizational hierarchy. For a complex and changing organization form... the main purpose of the traditional organization chart seems to be to decide who to blame when something goes wrong."

"This organization chart is also the point of reference for all Job Descriptions."

"Organization charts are diagrams that show how people, operations, functions, equipment, activities, etc., are organized, arranged, structured, and/or interrelated. They are applicable with any size of organization. A typical organization chart consists of text enclosed in geometric shapes (sometimes referred to as boxes, enclosures, box enclosures, or symbols) that are connected with lines (sometimes referred to as links) or arrows. Charts of this type generally progress from top to bottom or left to right. Organization charts are sometime considered a variation of flow chart or flow diagram."

"The organization chart is a distortion of how people actually relate to each other. Each line does not represent the same process."

"McCallum quickly moved to install a management system to replace the overloaded manager. He broke his railroad into geographical divisions of manageable size. Each was headed by a superintendent responsible for the operations within his division, Each divisional superintendent was required to submit detailed reports to central headquarters, from where McCallum and his aides coordinated and gave general direction to the operations of the separate divisions. Lines of authority between each superintendent and his subordinates and between each superintendent and headquarters were clearly laid out. In sketching these lines of authority on paper, McCallum created what might have been the first organizational chart for an American business."

"The organization chart is important for several reasons. It shows each individual's position within the organization. It shows the line responsibilities within the organization and who reports to whom. It shows how the organization is structured and how the various administrative functions within the organization are grouped."

"An organization chart is, in fact, a type of . Some flowcharts that contain significantly more narrative than others are referred to as narrative flowcharts."

"One reason the informal organization chart is never drawn and printed is that it's doubtful every one knows all parts of it."

"The organization chart is a graphic representation of the departmentalization process. Most organization charts are positional; that is, they are organized by title and rank"

"An organization chart is a convenient place to begin building planning models. A Note that as an organization is an object, so are the organization units. An organization chart depicts an object aggregation hierarchy"

"It may come as a surprise to the T-oriented analyst that the typical organization chart is a poor guide regarding the locus of power in organizations: Real power does not lie in documents and memos outlining your terms of reference and area of jurisdiction: it lies in what you can achieve in practice."

"An organization chart is an example of a graph: the nodes are interpreted as positions in the organization, and the links, the reporting and authority lines."

"The organization chart is a sort of map, an if you like, in which all linkages between the individuals listed are complete and no disagreements or omissions occur."

"An organization chart is not to take the place of the printed rules and regulations. Its chief use lies in that it is likely to lead to more careful planning of the organization and the placing of responsibility as well as making the task of defining duties easier. Each superintendent of schools would find that an organization chart would be of great assistance to him in perfecting his organization."

"In preparing organization charts of an established concern, care should be exercised to see that the charts portray conditions exactly as they are, and not merely as the author thinks they should be. One of the greatest values of organization charts and write-ups is the knowledge gained through the study of conditions made necessary in compiling the data. The thorough analysis of organization conditions, the impartial study of personnel, and the actual putting down in black and white bring out forcefully loose ends and weaknesses in the organization structure that otherwise might never be recognized and would continue an ever fruitful source of waste and an unsuspected obstacle in the path of the growth and development of the company. Only too frequently some of the following conditions are found in the course of the thorough, unbiased study, which is a necessary part of the charting process."

"An organization chart is merely an administrative device which enables an executive to see the men who are responsible for performing the activities of the company."

"To the student of business structure the organization chart is what the anatomical chart is to the student of the human body. It is a device by means of which relations of the different parts of the organization can be brought out more clearly than by a verbal description. The student of business derives from them the same sort of aid that the student of medicine does from the anatomical chart, which enables him to visualize the organs of the human body. While the analogy is helpful, it is like every other analogy, in being only partial."

"In 1918 an inquiry was made by Dr. L. P. Ayres of the Division of Statistics for organization charts from as many as 105 different business enterprises. Of the 58 replies received, 30 showed that no organization charts were available tho in a number of instances it was stated that organization charts made sometime before had not been kept up to date and did not, therefore, represent the conditions then prevailing in the business. From 28 concerns the actual charts were received."

"Use of Organization Charts - Much has to be done to promote the popularity of these charts by industrial engineers, altho at the present time they are not common among ordinary business concerns. Tho they are beginning to find their way into administrative and business enterprises, considering their demonstrated value the use of these charts is comparatively slight."

"The organization chart is a diagram showing graphically the relation of one official to another, or others, of a company. It is also used to show the relation of one department to another, or others, or of one function of an organization to another, or others. This chart is valuable in that it enables one to visualize a complete organization, by means of the picture it presents. There is no accepted form for making organization charts other than putting the principal official, department or function first, or at the head of the sheet, and the others below, in the order of their rank. The titles of officials and sometimes their names are enclosed in "boxes" or circles. Lines are generally drawn from one "box" or circle to another to show the relation of one official or department to the others."

"An Organization Chart is a cross section picture covering every relationship in the bank. It is a schematic survey showing department functions and interrelations, lines of authority, responsibility, communication and counsel. Its purpose is “to bring the various human parts of the organization into effective correlation and co-operation.""

"The Finished Plan. — The particular part of the organization to be re-organized having been selected, the plan when completed should consist of:"