On Hendrik Lorentz

"We may assume the existence of an aether; only we must give up ascribing a definite state of motion to it, i. e. we must by abstraction take from it the last mechanical characteristic which Lorentz had still left it. … But this ether may not be thought of as endowed with the quality characteristic of ponderable inedia, as consisting of parts which may be tracked through time. The idea of motion may not be applied to it. (Albert Einstein, "On the irrelevance of the luminiferous aether hypothesis to physical measurements", [address] 1920)

"For the Lorentz transformation spatial measurements are also changed, because they are obtained relative to a moving system. In our example only time was transformed, while the distances between points at rest remained the same; the spatial coordinates, therefore, retain their identity." (Hans Reichenbach," The Philosophy of Space and Time", 1928)

"Why is Einstein's theory better than Lorentz's theory? It would be a mistake to argue that Einstein's theory gives an explanation of Michelson's experiment, since it does not do so. Michelson's experiment is simply taken over as an axiom." (Hans Reichenbach," The Philosophy of Space and Time", 1928)

“In a theory in which parameters are added to quantum mechanics to determine the results of individual measurements, without changing the statistical predictions, there must be a mechanism whereby the setting of one measuring device can influence the reading of another instrument, however remote. Moreover, the signal involved must propagate instantaneously, so that such a theory could not be Lorentz invariant. Of course, the situation is different if the quantum mechanical predictions are of limited validity. Conceivably they might apply only to experiments in which the settings of the instruments are made sufficiently in advance to allow them to reach some mutual rapport by exchange of signals with velocity less than or equal to that of light. In that connection, experiments of the type proposed by Bohm and Aharonov, in which the settings are changed during the flight of the particles, are crucial."(John S Bell,"On the Einstein-Podolsky-Rosen paradox", 1964)

"The search for fundamental symmetries boils down to the study of transformations that do not change fundamental physical action - such transformations as reflection, rotation, the Lorentz transformation, and the like." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"Unlike an architect, Nature does not go around expounding on the wondrous symmetries of Her design. Instead, theoretical physicists must deduce them. Some symmetries, such as parity and rotational invariances, are intuitively obvious. We expect Nature to possess these symmetries, and we are shocked if She does not. Other symmetries, such as Lorentz invariance and general covariance, are more subtle and not grounded in our everyday perceptions. But, in any case, in order to find out if Nature employs a certain symmetry, we must compare the implications of the symmetry with observation." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"In a Newtonian view, space and time are separate and different. Symmetries of the laws of physics are combinations of rigid motions of space and an independent shift in time. But... these transformations do not leave Maxwell's equations invariant. Pondering this, the mathematicians Henri Poincaré and Hermann Minkowski were led to a new view of the symmetries of space and time, on a purely mathematical level. If they had described these symmetries in physical terms, they would have beaten Einstein to relativity, but they avoided physical speculations. They did understand that symmetries in the laws of electromagnetism do not affect space and time independently but mix them up. The mathematical scheme describing these intertwined changes is known as the Lorentz group, after the physicist, Hendrik Lorentz." (Ian Stewart, "Why Beauty Is Truth: The History of Symmetry", 2008)

"The group of transformations of space-time that fixes the origin and leaves the interval invariant is called the Lorentz group after the physicist Hendrik Lorentz. The Lorentz group specifies how relative motion works in relativity, and is responsible for the theory’s counterintuitive features in which objects shrink, time slows down, and mass increases, as a body nears the speed of light."(Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

"Symmetry is not enough by itself. In electromagnetism, for example, if you write down all the symmetries we know, such as Lorentz invariance and gauge invariance, you don’t get a unique theory that predicts the magnetic moment of the electron. The only way to do that is to add the principle of renormalisability – which dictates a high degree of simplicity in the theory and excludes these additional terms that would have changed the magnetic moment of the electron from the value Schwinger calculated in 1948." (Steven Weinberg) 

On Hermann Minkowski

"The discovery of Minkowski […] is to be found […] in the fact of his recognition that the four-dimensional space-time continuum of the theory of relativity, in its most essential formal properties, shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical space. In order to give due prominence to this relationship, however, we must replace the usual time co-ordinate t by an imaginary magnitude, √-1*ct, proportional to it. Under these conditions, the natural laws satisfying the demands of the (special) theory of relativity assume mathematical forms, in which the time co-ordinate plays exactly the same role as the three space-coordinates. Formally, these four co-ordinates correspond exactly to the three space co-ordinates in Euclidean geometry." (Albert Einstein,"Relativity: The Special and General Theory", 1920)

"Mathematicians call this combination [space and time] a quadratic form of the differentials of four variables, but we may call it more briefly, with Minkowski, ‘the Universe’." (Émile Borel, "Space and Time", 1926)

"According to the special theory there is a finite limit to the speed of causal chains, whereas classical causality allowed arbitrarily fast signals. Foundational studies […] soon revealed that this departure from classical causality in the special theory is intimately related to its most dramatic consequences: the relativity of simultaneity, time dilation, and length contraction. By now it had become clear that these kinematical effects are best seen as consequences of Minkowski space-time, which in turn incorporates a nonclassical theory of causal structure. However, it has not widely been recognized that the converse of this proposition is also true: the causal structure of Minkowski space-time contains within itself the entire geometry (topological and metrical structure) of Minkowski space-time." (John A. Winnie," The Causal Theory of Space-Time", 1977)

"The rigid electron is in my view a monster in relation to Maxwell's equations, whose innermost harmony is the principle of relativity [...] the rigid electron is no working hypothesis, but a working hindrance. Approaching Maxwell's equations with the concept of the rigid electron seems to me the same thing as going to a concert with your ears stopped up with cotton wool. We must admire the courage and the power of the school of the rigid electron which leaps across the widest mathematical hurdles with fabulous hypotheses, with the hope to land safely over there on experimental-physical ground." (Hermann Minkowski [in Arthur I Miller, "Albert Einstein's Special Theory of Relativity", 1981)

"Minkowski, building on Einstein's work, had now discovered that the Universe is made of a four-dimensional ‘spacetime’ fabric that is absolute, not relative." (Kip S Thorne, "Black Holes and Time Warps: Einstein's Outrageous Legacy" , 1994)

"Minkowski calls a spatial point existing at a temporal point a world point. These coordinates are now called 'space-time coordinates'. The collection of all imaginable value systems or the set of space-time coordinates Minkowski called the world. This is now called the manifold. The manifold is four-dimensional and each of its space-time points represents an event." (Friedel Weinert," The Scientist as Philosopher: Philosophical Consequences of Great Scientific Discoveries", 2005)

"In a Newtonian view, space and time are separate and different. Symmetries of the laws of physics are combinations of rigid motions of space and an independent shift in time. But... these transformations do not leave Maxwell's equations invariant. Pondering this, the mathematicians Henri Poincaré and Hermann Minkowski were led to a new view of the symmetries of space and time, on a purely mathematical level. If they had described these symmetries in physical terms, they would have beaten Einstein to relativity, but they avoided physical speculations. They did understand that symmetries in the laws of electromagnetism do not affect space and time independently but mix them up. The mathematical scheme describing these intertwined changes is known as the Lorentz group, after the physicist, Hendrik Lorentz." (Ian Stewart, "Why Beauty Is Truth: The History of Symmetry", 2008)

On Geometry (2010-2019)

"Diagrams are information graphics that are made up primarily of geometric shapes, such as rectangles, circles, diamonds, or triangles, that are typically" (but not always) interconnected by lines or arrows. One of the major purposes of a diagram is to show how things, people, ideas, activities, etc. interrelate and interconnect. Unlike quantitative charts and graphs, diagrams are used to show interrelationships in a qualitative way." (Robbie T Nakatsu, "Diagrammatic Reasoning in AI", 2010)

"First, what are the 'graphs' studied in graph theory? They are not graphs of functions as studied in calculus and analytic geometry. They are (usually finite) structures consisting of vertices and edges. As in geometry, we can think of vertices as points (but they are denoted by thick dots in diagrams) and of edges as arcs connecting pairs of distinct vertices. The positions of the vertices and the shapes of the edges are irrelevant: the graph is completely specified by saying which vertices are connected by edges. A common convention is that at most one edge connects a given pair of vertices, so a graph is essentially just a pair of sets: a set of objects." (John Stillwell, "Mathematics and Its History", 2010)

"Strange attractors, unlike regular ones, are geometrically very complicated, as revealed by the evolution of a small phase-space volume. For instance, if the attractor is a limit cycle, a small two-dimensional volume does not change too much its shape: in a direction it maintains its size, while in the other it shrinks till becoming a 'very thin strand' with an almost constant length. In chaotic systems, instead, the dynamics continuously stretches and folds an initial small volume transforming it into a thinner and thinner 'ribbon' with an exponentially increasing length." (Massimo Cencini et al, "Chaos: From Simple Models to Complex Systems", 2010)

"A crucial difference between topology and geometry lies in the set of allowable transformations. In topology, the set of allowable transformations is much larger and conceptually much richer than is the set of Euclidean transformations. All Euclidean transformations are topological transformations, but most topological transformations are not Euclidean. Similarly, the sets of transformations that define other geometries are also topological transformations, but many topological transformations have no counterpart in these geometries. It is in this sense that topology is a generalization of geometry." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"What distinguishes topological transformations from geometric ones is that topological transformations are more 'primitive'. They retain only the most basic properties of the sets of points on which they act." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"As geometers study shape, the student of calculus examines change: the mathematics of how an object transforms from one state into another, as when describing the motion of a ball or bullet through space, is rendered pictorial in its graphs’ curves." (Daniel Tammet, "Thinking in Numbers" , 2012)

"To mathematicians who study them, moduli schemes are just as real as the regular objects in the world. […] The key idea is that an ordinary object can be studied using the set of functions on the object. […] Secondly, you can do algebra with these functions - that is, you can add or multiply two such functions and get a third function. This step makes the set of these functions into a ring. […] Then the big leap comes: If you start with any ring - that is, any set of entities that can be added and multiplied subject to the usual rules, you simply and brashly declare that this creates a new kind of geometric object. The points of the object can be given by maps from the ring to the real numbers, as in the example of the pot. But they may also be given by maps to other fields. A field is a special sort of ring in which division is possible." (David Mumford, ["The Best Writing of Mathematics: 2012"] 2012

"Visualization is what happens when you make the jump from raw data to bar graphs, line charts, and dot plots. […] In its most basic form, visualization is simply mapping data to geometry and color. It works because your brain is wired to find patterns, and you can switch back and forth between the visual and the numbers it represents. This is the important bit. You must make sure that the essence of the data isn’t lost in that back and forth between visual and the value it represents because if you can’t map back to the data, the visualization is just a bunch of shapes." (Nathan Yau, "Data Points: Visualization That Means Something", 2013)

"Geometry had its origins in the interest of working with lines, figures, and solids that could be imagined in the mind. Algebra had its origins in problems involving number - number hinged by geometric conceptions of iconic figures." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)

"Topology is a geometry in which all lengths, angles, and areas can be distorted at will. Thus a triangle can be continuously transformed into a rectangle, the rectangle into a square, the square into a circle, and so on. Similarly, a cube can be transformed into a cylinder, the cylinder into a cone, the cone into a sphere. Because of these continuous transformations, topology is known popularly as 'rubber sheet geometry'. All figures that can be transformed into each other by continuous bending, stretching, and twisting are called 'topologically equivalent'." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)

"[…] topology is concerned precisely with those properties of geometric figures that do not change when the figures are transformed. Intersections of lines, for example, remain intersections, and the hole in a torus (doughnut) cannot be transformed away. Thus a doughnut may be transformed topologically into a coffee cup (the hole turning into a handle) but never into a pancake. Topology, then, is really a mathematics of relationships, of unchangeable, or 'invariant', patterns." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)

"The association of multiplication with vector rotation was one of the geometric interpretation's most important elements because it decisively connected the imaginaries with rotary motion. As we'll see, that was a big deal." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"The primary aspects of the theory of complex manifolds are the geometric structure itself, its topological structure, coordinate systems, etc., and holomorphic functions and mappings and their properties. Algebraic geometry over the complex number field uses polynomial and rational functions of complex variables as the primary tools, but the underlying topological structures are similar to those that appear in complex manifold theory, and the nature of singularities in both the analytic and algebraic settings is also structurally very similar." (Raymond O Wells Jr, "Differential and Complex Geometry: Origins, Abstractions and Embeddings", 2017)

"We cannot draw a complete map, a complete geometry, of everything that happens in the world, because such happenings - including among them the passage of time - are always triggered only by an interaction with, and with respect to, a physical system involved in the interaction. The world is like a collection of interrelated points of view. To speak of the world 'seen from outside' makes no sense, because there is no “outside” to the world." (Carlo Rovelli, "The Order of Time", 2018)

"Because of the geometry of a circle, there’s always a quarter-cycle off set between any sine wave and the wave derived from it as its derivative, its rate of change. In this analogy, the point’s direction of travel is like its rate of change. It determines where the point will go next and hence how it changes its location. Moreover, this compass heading of the arrow itself rotates in a circular fashion at a constant speed as the point goes around the circle, so the compass heading of the arrow follows a sine-wave pattern in time. And since the compass heading is like the rate of change, voilà! The rate of change follows a sine-wave pattern too." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

On Geometry (2000-2009)

"Arithmetic and number theory study patterns of number and counting. Geometry studies patterns of shape. Calculus allows us to handle patterns of motion. Logic studies patterns of reasoning. Probability theory deals with patterns of chance. Topology studies patterns of closeness and position." (Keith Devlin, "The Math Gene: How Mathematical Thinking Evolved And Why Numbers Are Like Gossip", 2000

"The equation e^πi =-1 says that the function w= e^z, when applied to the complex number πi as input, yields the real number -1 as the output, the value of w. In the complex plane, πi is the point [0,π) - π on the i-axis. The function w=e^z maps that point, which is in the z-plane, onto the point (-1, 0) - that is, -1 on the x-axis-in the w-plane. […] But its meaning is not given by the values computed for the function w=e^z. Its meaning is conceptual, not numerical. The importance of  e^πi =-1 lies in what it tells us about how various branches of mathematics are related to one another - how algebra is related to geometry, geometry to trigonometry, calculus to trigonometry, and how the arithmetic of complex numbers relates to all of them." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)

"Complex numbers are really not as complex as you might expect from their name, particularly if we think of them in terms of the underlying two dimensional geometry which they describe. Perhaps it would have been better to call them 'nature's numbers'. Behind complex numbers is a wonderful synthesis between two dimensional geometry and an elegant arithmetic in which every polynomial equation has a solution." (David Mumford, Caroline Series & David Wright, "Indra’s Pearls: The Vision of Felix Klein", 2002)

"In the nonmathematical sense, symmetry is associated with regularity in form, pleasing proportions, periodicity, or a harmonious arrangement; thus it is frequently associated with a sense of beauty. In the geometric sense, symmetry may be more precisely analyzed. We may have, for example, an axis of symmetry, a center of symmetry, or a plane of symmetry, which define respectively the line, point, or plane about which a figure or body is symmetrical. The presence of these symmetry elements, usually in combinations, is responsible for giving form to many compositions; the reproduction of a motif by application of symmetry operations can produce a pattern that is pleasing to the senses." (Hans H Jaffé & Milton Orchin, "Symmetry in Chemistry", 2002)

"In string theory one studies strings moving in a fixed classical spacetime. […] what we call a background-dependent approach. […] One of the fundamental discoveries of Einstein is that there is no fixed background. The very geometry of space and time is a dynamical system that evolves in time. The experimental observations that energy leaks from binary pulsars in the form of gravitational waves - at the rate predicted by general relativity to the […] accuracy of eleven decimal place - tell us that there is no more a fixed background of spacetime geometry than there are fixed crystal spheres holding the planets up." (Lee Smolin, "Loop Quantum Gravity", The New Humanists: Science at the Edge, 2003)

"Quantum physics, in particular particle and string theory, has proven to be a remarkable fruitful source of inspiration for new topological invariants of knots and manifolds. With hindsight this should perhaps not come as a complete surprise. Roughly one can say that quantum theory takes a geometric object (a manifold, a knot, a map) and associates to it a (complex) number, that represents the probability amplitude for a certain physical process represented by the object." (Robbert Dijkgraaf, "Mathematical Structures", 2005)

"Lie groups turn up when we study a geometric object with a lot of symmetry, such as a sphere, a circle, or flat spacetime. Because there is so much symmetry, there are many functions from the object to itself that preserve the geometry, and these functions become the elements of the group." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"Topology is the study of geometric objects as they are transformed by continuous deformations. To a topologist the general shape of the objects is of more importance than distance, size, or angle." (Robert Messer & Philip Straffin, "Topology Now!", 2006)

"Unfortunately, if we were to use geometry to explore the concept of the square root of a negative number, we would be setting a boundary to our imagination that would be difficult to cross. To represent -1 using geometry would require us to draw a square with each side length being less than zero. To be asked to draw a square with side length less than zero sounds similar to the Zen Buddhists asking ‘What is the sound of one hand clapping?’" (Les Evans, "Complex Numbers and Vectors", 2006)

"But in mathematics there is a kind of threshold effect, an intellectual tipping point. If a student can just get over the first few humps, negotiate the notational peculiarities of the subject, and grasp that the best way to make progress is to understand the ideas, not just learn them by rote, he or she can sail off merrily down the highway, heading for ever more abstruse and challenging ideas, while an only slightly duller student gets stuck at the geometry of isosceles triangles." (Ian Stewart, "Why Beauty is Truth: A history of symmetry", 2007)

"[...] if two conics have five points in common, then they have infinitely many points in common. This geometric theorem is somewhat subtle but translates into a property of solutions of polynomial equations that makes more natural sense to a modern mathematician." (David Ruelle, "The Mathematician's Brain", 2007)

"Linear algebra is a very useful subject, and its basic concepts arose and were used in different areas of mathematics and its applications. It is therefore not surprising that the subject had its roots in such diverse fields as number theory (both elementary and algebraic), geometry, abstract algebra (groups, rings, fields, Galois theory), anal ysis (differential equations, integral equations, and functional analysis), and physics. Among the elementary concepts of linear algebra are linear equations, matrices, determinants, linear transformations, linear independence, dimension, bilinear forms, quadratic forms, and vector spaces. Since these concepts are closely interconnected, several usually appear in a given context (e.g., linear equations and matrices) and it is often impossible to disengage them." (Israel Kleiner, "A History of Abstract Algebra", 2007)

"Matrices are 'natural' mathematical objects: they appear in connection with linear equations, linear transformations, and also in conjunction with bilinear and quadratic forms, which were important in geometry, analysis, number theory, and physics. Matrices as rectangular arrays of numbers appeared around 200 BC in Chinese mathematics, but there they were merely abbreviations for systems of linear equations. Matrices become important only when they are operated on - added, subtracted, and especially multiplied; more important, when it is shown what use they are to be put to." (Israel Kleiner, "A History of Abstract Algebra", 2007)

"Geometrical truth is (as we now speak) synthetic: it states facts about the world. Such truths are not ordinary truths but essential truths, giving the reality of the empirical world in which they are imperfect embodied." (Fred Wilson, "The External World and Our Knowledge of It", 2008)

On Geometry (1975-1999)

"According to the special theory there is a finite limit to the speed of causal chains, whereas classical causality allowed arbitrarily fast signals. Foundational studies […] soon revealed that this departure from classical causality in the special theory is intimately related to its most dramatic consequences: the relativity of simultaneity, time dilation, and length contraction. By now it had become clear that these kinematical effects are best seen as consequences of Minkowski space-time, which in turn incorporates a nonclassical theory of causal structure. However, it has not widely been recognized that the converse of this proposition is also true: the causal structure of Minkowski space-time contains within itself the entire geometry (topological and metrical structure) of Minkowski space-time." (John A. Winnie," The Causal Theory of Space-Time", 1977)

"A model […] is a story with a specified structure: to explain this catch phrase is to explain what a model is. The structure is given by the logical and mathematical form of a set of postulates, the assumptions of the model. The structure forms an uninterpreted system, in much the way the postulates of a pure geometry are now commonly regarded as doing. The theorems that follow from the postulates tell us things about the structure that may not be apparent from an examination of the postulates alone." (Allan Gibbard & Hal R. Varian, "Economic Models", The Journal of Philosophy, Vol. 75, No. 11, 1978)

“Every branch of geometry can be defined as the study of properties that are unaltered when a specified figure is given specified symmetry transformations. Euclidian plane geometry, for instance, concerns the study of properties that are 'invariant' when a figure is moved about on the plane, rotated, mirror reflected, or uniformly expanded and contracted. Affine geometry studies properties that are invariant when a figure is 'stretched' in a certain way. Projective geometry studies properties invariant under projection. Topology deals with properties that remain unchanged even when a figure is radically distorted in a manner similar to the deformation of a figure made of rubber.” (Martin Gardner, "Aha! Insight", 1978)

“Geometry is the study of shapes. Although true, this definition is so broad that it is almost meaningless. The judge of a beauty contest is, in a sense, a geometrician because he is judging […] shapes, but this is not quite what we want the word to mean. It has been said that a curved line is the most beautiful distance between two points. Even though this statement is about curves, a proper element of geometry, the assertion seems more to be in the domain of aesthetics rather than mathematics.” (Martin Gardner, "Aha! Insight", 1978)

"To enter a temple constructed wholly of invariable geometric proportions is to enter an abode of eternal truth." (Robert Lawlor, "Sacred Geometry", 1982)

"The beauty that Nature has revealed to physicists in Her laws is a beauty of design, a beauty that recalls, to some extent, the beauty of classical architecture, with its emphasis on geometry and symmetry. The system of aesthetics used by physicists in judging Nature also draws its inspiration from the austere finality of geometry." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"The precise mathematical definition of symmetry involves the notion of invariance. A geometrical figure is said to be symmetric under certain operations if those operations leave it unchanged." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"Elementary functions, such as trigonometric functions and rational functions, have their roots in Euclidean geometry. They share the feature that when their graphs are 'magnified' sufficiently, locally they 'look like' straight lines. That is, the tangent line approximation can be used effectively in the vicinity of most points. Moreover, the fractal dimension of the graphs of these functions is always one. These elementary 'Euclidean' functions are useful not only because of their geometrical content, but because they can be expressed by simple formulas. We can use them to pass information easily from one person to another. They provide a common language for our scientific work. Moreover, elementary functions are used extensively in scientific computation, computer-aided design, and data analysis because they can be stored in small files and computed by fast algorithms." (Michael Barnsley, "Fractals Everwhere", 1988)

"Fractal geometry is concerned with the description, classification, analysis, and observation of subsets of metric spaces (X, d). The metric spaces are usually, but not always, of an inherently 'simple' geometrical character; the subsets are typically geometrically 'complicated'. There are a number of general properties of subsets of metric spaces, which occur over and over again, which are very basic, and which form part of the vocabulary for describing fractal sets and other subsets of metric spaces. Some of these properties, such as openness and closedness, which we are going to introduce, are of a topological character. That is to say, they are invariant under homeomorphism." (Michael Barnsley, "Fractals Everwhere", 1988)

"In deterministic geometry, structures are defined, communicated, and analysed, with the aid of elementary transformations such as affine transfor- transformations, scalings, rotations, and congruences. A fractal set generally contains infinitely many points whose organization is so complicated that it is not possible to describe the set by specifying directly where each point in it lies. Instead, the set may be defined by "the relations between the pieces." It is rather like describing the solar system by quoting the law of gravitation and stating the initial conditions. Everything follows from that. It appears always to be better to describe in terms of relationships." (Michael Barnsley, "Fractals Everwhere", 1988)

"Instead of a state of nature evolving according to a mathematical fomula, the evolution is given geometrically. The full advantage of the geometrical point of view is beginning to appear. The more traditional way of dealing with dynamics was with the use of algebraic expressions. But a description given by formulae would be cumbersome. That form of description wouldn't have led me to insights or to perceptive analysis. My background as a topologist, trained to bend objects like squares, helped to make it possible to see the horseshoe." (Steven Smale, "What is chaos?", 1990)

"Percolation is a widespread paradigm. Percolation theory can therefore illuminate a great many seemingly diverse situations. Because of its basically geometric character, it facilitates the analysis of intricate patterns and textures without needless physical complications. And the self-similarity that prevails at critical points permits profitably mining the connection with scaling and fractals." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"There are at least three (overlapping) ways that mathematics may contribute to science. The first, and perhaps the most important, is this: Since the mathematical universe of the mathematician is much larger than that of the physicist, mathematicians are able to go beyond existing frameworks and see geometrical or analytic structures unavailable to tie physicist. Instead of using the particular equations used previously to describe reality, a mathematician has at his disposal an unused world of differential equations, to be studied with no a priori constraints. New scientific phenomena, new discoveries, may thus generated. Understanding of the present knowledge may be deepened via the corresponding deductions. [...] The second way [...] has to do with the consolidation of new physical ideas. One may express this as the proof of consistency of physical theories. [...] mathematical foundations of quantum mechanics with Hilbert space, its operator theory, and corresponding differential equations. [...] The third way [...] is by describing reality in mathematical terms, or by simply constructing a mathematical model." (Steven Smale, "What is chaos?", 1990)

"Mathematics is more than doing calculations, more than solving equations, more than proving theorems, more than doing algebra, geometry or calculus, more than a way of thinking. Mathematics is the design of a snowflake, the curve of a palm frond, the shape of a building, the joy of a game, the frustration of a puzzle, the crest of a wave, the spiral of a spider's web. It is ancient and yet new. Mathematics is linked to so many ideas and aspects of the universe." (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)

"Geometry and topology most often deal with geometrical figures, objects realized as a set of points in a Euclidean space (maybe of many dimensions). It is useful to view these objects not as rigid (solid) bodies, but as figures that admit continuous deformation preserving some qualitative properties of the object. Recall that the mapping of one object onto another is called continuous if it can be determined by means of continuous functions in a Cartesian coordinate system in space. The mapping of one figure onto another is called homeomorphism if it is continuous and one-to-one, i.e. establishes a one-to-one correspondence between points of both figures." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)

"A good map tells a multitude of little white lies; it suppresses truth to help the user see what needs to be seen. Reality is three-dimensional, rich in detail, and far too factual to allow a complete yet uncluttered two-dimensional graphic scale model. Indeed, a map that did not generalize would be useless. But the value of a map depends on how well its generalized geometry and generalized content reflect a chosen aspect of reality." (Mark S Monmonier, "How to Lie with Maps" 2nd Ed., 1996)

"Geometry is the science of figures. We study various properties of figures, and classify given figures according to the results. We have the notion of invariants, which can serve as the most effective method of classification. We may briefly say that invariants describe geometric structures in terms of numbers." (Shigeyuki Morita, "Geometry of Differential Forms", 1997)

On Geometry (1950-1974)

"Probability is a mathematical discipline with aims akin to those, for example, of geometry or analytical mechanics. In each field we must carefully distinguish three aspects of the theory: (a) the formal logical content, (b) the intuitive background, (c) the applications. The character, and the charm, of the whole structure cannot be appreciated without considering all three aspects in their proper relation." (William Feller, "An Introduction to Probability Theory and Its Applications", 1950)

"Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved." (Omar Khayyam [quoted by J.J. Winter and W. Arafat, "The Algebra of ‘Umar Khayyam’", Journal of the Royal Asiatic Society of Bengal, Volume 41, 1950)

"Geometrical truth is a product of reason; that makes it superior to empirical truth, which is found through generalization of a great number of instances." (Hans Reichenbach, "The Rise of Scientific Philosophy", 1954)

“[…] no branch of mathematics competes with projective geometry in originality of ideas, coordination of intuition in discovery and rigor in proof, purity of thought, logical finish, elegance of proofs and comprehensiveness of concepts. The science born of art proved to be an art.” (Morris Kline, “Projective Geometry”, Scientific America Vol. 192 (1), 1955)

 "Conventionalism as geometrical and mathematical truths are created by our choices, not dictated by or imposed on us by scientific theory. The idea that geometrical truth is truth we create by the understanding of certain conventions in the discovery of non-Euclidean geometries." (Clifford Singer, "Engineering a Visual Field", 1955)

"Geometry, whatever others may think, is the study of different shapes, many of them very beautiful, having harmony, grace and symmetry. […] Most of us, if we can play chess at all, are content to play it on a board with wooden chess pieces; but there are some who play the game blindfolded and without touching the board. It might be a fair analogy to say that abstract geometry is like blindfold chess – it is a game played without concrete objects." (Edward Kasner & James R Newman, "New Names for Old", 1956)

"Geometry exists in its own right, and by its own strength. It can now treat accurately and coherently a range of forms and spaces that far exceeds anything that terrestrial space can provide. Today it is geometry that contains the terrestrial forms, and not vice versa, for the terrestrial forms are merely special cases in an all-embracing geometry. [...] Geometry now acts as a framework on which all terrestrial forms can find their natural place, with the relations between the various forms readily appreciable." (W Ross Ashby, "An Introduction to Cybernetics", 1956)

"A logic machine is a device, electrical or mechanical, designed specifically for solving problems in formal logic. A logic diagram is a geometrical method for doing the same thing. […] A logic diagram is a two-dimensional geometric figure with spatial relations that are isomorphic with the structure of a logical statement. These spatial relations are usually of a topological character, which is not surprising in view of the fact that logic relations are the primitive relations underlying all deductive reasoning and topological properties are, in a sense, the most fundamental properties of spatial structures. Logic diagrams stand in the same relation to logical algebras as the graphs of curves stand in relation to their algebraic formulas; they are simply other ways of symbolizing the same basic structure." (Martin Gardner, "Logic Machines and Diagrams", 1958)

"[...] it is clear that differential geometry, analysis and physics prompted the early development of differential topology (it is this that explains our admitted bias toward differential topology, that it lies close to the main stream of  mathematics). On the other hand, the combinatorial approach to manifolds was started because it was believed that these means would afford a useful attack on the differentiable case." (Steven Smale, "A survey of some recent developments in differential topology", 1961)

"Nature does not seem full of circles and triangles to the ungeometrical; rather, mastery of the theory of triangles and circles, and later of conic sections, has taught the theorist, the experimenter, the carpenter, and even the artist to find them everywhere, from the heavenly motions to the pose of a Venus." (Clifford Truesdell, "Six Lectures on Modern Natural Philosophy", 1966)

"After all, Greek thought is expressed not only mythically, in fiction, but also directly, in theorems. The gate through which the Greek world may be discussed - and without the knowledge of which, in my opinion, one’s culture can not be deemed complete - is not necessarily Homer. Greek geometry is a wider gate, through which the eye might grasp an austere, yet essential landscape." (Dan Barbilian, 1967)

"An instance of such a 'non-Euclidean' geometry is projective geometry, concerned with those properties of figures which do not change under projective transformations. Projective geometry is not merely not Euclidean geometry; it is 'very much non-Euclidean'." (Isaak Yaglom, "Geometric Transformations", 1973)

"If indeed one tries to clarify the notion of equality, which is introduced right at the beginning of Geometry, one is led to say that two figures are equal when one can go from one to the other by a specific geometric operation, called a motion. This is only a change of words; but the axiom according to which two figures equal to a third are equal to one another, subjects those operations called motions to a certain law; that is, that an operation which is the result of two successive motions is itself a motion. It is this law that mathematicians express by saying that motions form a group. Elementary Geometry can then be defined by the study of properties of figures which do not change under the operations of the group of motions." (Élie Cartan, "Notice sur les travaux scientifiques", 1974)

"[…] it is the whole logical structure of elementary Geometry which is contained in the group of motions and even, in a more precise manner, in the law according to which operations of that group compose with each other, independently of the nature of the objects on which these operations act. This law constitutes what we call the group structure." (Élie Cartan, "Notice sur les travaux scientifiques", 1974)

"The effect of magnitude or absolute size as a determinant of form shows again how space shapes the things around us. In studying polyhedrons we are unconcerned with magnitude. We assume that a cube is a cube no matter what its size. We find, however, that the geometric relations that arise from a difference in size affect structural behavior, and that a large cube is relatively weaker than a small cube. We also find, as a corollary, that in order to maintain the same structural characteristics a difference in size must be accompanied by a difference in shape." (Peter B Stevens, "Patterns in Nature", 1974)

On Geometry (1925-1949)

"As the objects of abstract geometry cannot be totally grasped by space intuition, a rigorous proof in abstract geometry can never be based only on intuition, but it must be founded on logical deduction from valid and precise axioms." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" 3rd Ed. Vol. 3, 1928)

"What had already been done for music by the end of the eighteenth century has at last been begun for the pictorial arts. Mathematics and physics furnished the means in the form of rules to be followed and to be broken. In the beginning it is wholesome to be concerned with the functions and to disregard the finished form. Studies in algebra, in geometry, in mechanics characterize teaching directed towards the essential and the functional, in contrast to apparent. One learns to look behind the façade, to grasp the root of things. One learns to recognize the undercurrents, the antecedents of the visible. One learns to dig down, to uncover, to find the cause, to analyze." (Paul Klee, "Bauhaus prospectus", 1929)

"Any mathematical science is a body of theorems deduced from a set of axioms. A geometry is a mathematical science. The question then arises why the name geometry is given to some mathematical sciences and not to others. It is likely that there is no definite answer to this question, but that a branch of mathematics is called a geometry because the name seems good, on emotional and  people." (John H C Whitehead, „The Foundation of Differential Geometry", 1932)

"Given any group of transformations in space which includes the principal group as a sub-group, then the invariant theory of this group gives a definite kind of geometry, and every possible geometry can be obtained in this way. Thus each geometry is characterized by its group, which, therefore, assumes the leading place in our considerations." (Felix Klein, "Elementary Mathematics from an Elementary Standpoint: Geometry", 1939)

"Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities, which we may call intuition and ingenuity. The activity of the intuition consists in making spontaneous judgements which are not the result of conscious trains of reasoning. [...] The exercise of ingenuity in mathematics consists in aiding the intuition through suitable arrangements of propositions, and perhaps geometrical figures or drawings." (Alan M Turing, "Systems of Logic Based on Ordinals", Proceedings of the London Mathematical Society Vol 45 (2), 1939)

"Geometry, whatever others may think, is the study of different shapes, many of them very beautiful, having harmony, grace and symmetry. […] Most of us, if we can play chess at all, are content to play it on a board with wooden chess pieces; but there are some who play the game blindfolded and without touching the board. It might be a fair analogy to say that abstract geometry is like blindfold chess – it is a game played without concrete objects." (Edward Kasner & James R Newman, "New Names for Old", 1940)

"It is difficult, however, to learn all these things from situations such as occur in everyday life. What we need is a series of abstract and quite impersonal situations to argue about in which one side is surely right and the other surely wrong. The best source of such situations for our purposes is geometry. Consequently we shall study geometric situations in order to get practice in straight thinking and logical argument, and in order to see how it is possible to arrange all the ideas associated with a given subject in a coherent, logical system that is free from contradictions. That is, we shall regard the proof of each proposition of geometry as an example of correct method in argumentation, and shall come to regard geometry as our ideal of an abstract logical system. Later, when we have acquired some skill in abstract reasoning, we shall try to see how much of this skill we can apply to problems from real life." (George D Birkhoff & Ralph Beately, "Basic Geometry", 1940)

"The main thing geometry gives us is the ideal of a logical system and of precise thinking, and some acquaintance with the language in which logical arguments are usually expressed. The answer to a problem in actual life can often be obtained by further investigation of the actual facts, while in geometry it can always be obtained by reasoning alone." (George D Birkhoff & Ralph Beately, "Basic Geometry", 1940)

"The majority of geometric relations cluster around the ideas ‘equal’ and ‘similar’. That is why the simplest notions about equal and similar triangles are taken as the basis of this geometry. From these simplest notions we derive the more complicated relations of geometry." (George D Birkhoff & Ralph Beately, "Basic Geometry", 1940)

 "Geometry, then, is the application of strict logic to those properties of space and figure which are self-evident, and which therefore cannot be disputed. But the rigor of this science is carried one step further; for no property, however evident it may be, is allowed to pass without demonstration, if that can be given. The question is therefore to demonstrate all geometrical truths with the smallest possible number of assumptions." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1943)

"With a literature much vaster than those of algebra and arithmetic combined, and as least as extensive as that of analysis, geometry is a richer treasure house of more interesting and half-forgotten things, which a hurried generation has no leisure to enjoy, than any other division of mathematics." (Eric T Bell, "The Development of Mathematics", 1945)

"The field equation may [...] be given a geometrical foundation, at least to a first approximation, by replacing it with the requirement that the mean curvature of the space vanish at any point at which no heat is being applied to the medium - in complete analogy with […] the general theory of relativity by which classical field equations are replaced by the requirement that the Ricci contracted curvature tensor vanish." (Howard P Robertson, "Geometry as a Branch of Physics", 1949)

On Geometry (1900-1924)

"Arithmetical symbols are written diagrams and geometrical figures are graphic formulas." (David Hilbert, Bulletin of the American Mathematical Society Mathematical Problems Vol. 8, 1902)

"The object of geometry is the study of a particular 'group'; but the general concept of group preexists in our minds, at least potentially. It is imposed on us not as a form of our sensitiveness, but as a form of our understanding; only, from among all possible groups, we must choose one that will be the standard, so to speak, to which we shall refer natural phenomena." (Henri Poincaré, "Science and Hypothesis", 1902)

"The truth is that other systems of geometry are possible, yet after all, these other systems are not spaces but other methods of space measurements. There is one space only, though we may conceive of many different manifolds, which are contrivances or ideal constructions invented for the purpose of determining space." (Paul Carus, Science Vol. 18, 1903)

"Surely, among the most important goals of every geometrical instruction is the strengthening of the faculty for spatial imaging and the power for spatial modelling." (Arturo M Schoenflies, 1908)

"We believe that in our reasonings we no longer appeal to intuition; the philosophers will tell us this is an illusion. Pure logic could never lead us to anything but tautologies; it could create nothing new; not from it alone can any science issue. In one sense these philosophers are right; to make arithmetic, as to make geometry, or to make any science, something else than pure logic is necessary. To designate this something else we have no word other than intuition. But how many different ideas are hidden under this same word?" (Henri Poincaré , "Intuition and Logic in Mathematics", 1905)

"But in the mathematical or pure sciences, - geometry, arithmetic, algebra, trigonometry, the calculus of variations or of curves, - we know at least that there is not, and cannot be, error in our first principles, and we may therefore fasten our whole attention upon the processes. As mere exercises in logic, therefore, these sciences, based as they all are on primary truths relating to space and number, have always been supposed to furnish the most exact discipline." (Joshua Fitch,"Lectures on Teaching", 1906)

"Geometry, accordingly, consists of the application of mathematics to experiences concerning space. Like mathematical physics, it can become an exact deductive science only on the condition of its representing the objects of experience by means of schematizing and idealizing concepts." (Ernst Mach, "Space and Geometry: In the Light of physiological, phycological and physical inquiry", 1906)

"Architecture is geometry made visible in the same sense that music is number made audible." (Claude F Bragdon, "The Beautiful Necessity: Seven Essays on Theosophy and Architecture", 1910)

"Geometry formerly was the chief borrower from arithmetic and algebra, but it has since repaid its obligation with abundant usury; and if I were asked to name, in one word, the pole-star round which the mathematical firmament revolves, the central idea which pervades as a hidden spirit the whole corpus of mathematical doctrine, I should point to Continuity as contained in our notions of space, and say, it is this, it is this!" (James J. Sylvester, Presidential Address to the British Association, [The Collected Mathematical Papers of James Joseph Sylvester Vol. 2, cca. 1904–1912])

"The ends to be attained [in mathematical teaching] are the knowledge of a body of geometrical truths to be used In the discovery of new truths, the power to draw correct inferences from given premises, the power to use algebraic processes as a means of finding results in practical problems, and the awakening of interest In the science of mathematics." (J Craig, "A Course of Study for the Preparation of Rural School Teachers", 1912)

"Time was when all the parts of the subject were dissevered, when algebra, geometry, and arithmetic either lived apart or kept up cold relations of acquaintance confined to occasional calls upon one another; but that is now at an end; they are drawn together and are constantly becoming more and more intimately related and connected by a thousand fresh ties, and we may confidently look forward to a time when they shall form but one body with one soul." (James J. Sylvester, Presidential Address to the British Association, [The Collected Mathematical Papers of James Joseph Sylvester Vol. 2, cca. 1904–1912])

"We may be thinking out a chain of reasoning in abstract Geometry, but if we draw a figure, as we usually must do in order to fix our ideas and prevent our attention from wandering owing to the difficulty of keeping a long chain of syllogisms in our minds, it is excusable if we are apt to forget that we are not in reality reasoning about the objects in the figure, but about objects which ore their idealizations, and of which the objects in the figure are only an imperfect representation. Even if we only visualize, we see the images of more or less gross physical objects, in which various qualities irrelevant for our specific purpose are not entirely absent, and which are at best only approximate images of those objects about which we are reasoning." (Ernest W Hobson, "Squaring the Circle", 1913)

"Geometers usually distinguish two kinds of geometry, the first of which they qualify as metric and the second as projective. Metric geometry is based on the notion of distance; two figures are there regarded as equivalent when they are 'congruent' in the sense that mathematicians give to this word. Projective geometry is based on the notion of straight line; in order for two figures considered there to be equivalent, it is not necessary that they be congruent; it suffices that one can pass from one to the other by a projective transformation, that is, that one be the perspective of the other. This second body of study has often been called qualitative geometry, and in fact it is if one opposes it to the first; it is clear that measure and quantity play a less important role. This is not entirely so, however. The fact that a line is straight is not purely qualitative; one cannot assure himself that a line is straight without making measurements, or without sliding on this line an instrument called a straightedge, which is a kind of instrument of measure.” (Henri Poincaré, “Dernières pensées”, 1913)

"On the other side of the subject, Geometry is an abstract rational Science which deals with the relations of objects that are no longer physical objects, although these ideal objects, points, straight lines, circles, &c., are called by the same names by which we denote their physical counterparts. At the base of this rational Science there lies a set of definitions and postulations which specify the nature of the relations between the ideal objects with which the Science deals. These postulations and definitions were suggested by our actual spatial perceptions, but they contain an element of absolute exactness which is wanting in the rough data provided by our senses. The objects of abstract Geometry possess in absolute precision properties which are only approximately realized in the corresponding objects of physical Geometry." (Ernest W Hobson, "Squaring the circle", 1913)

"A much more natural and adequate comparison would, it seems to me, liken Coordinate Geometry to a steam-hammer, which an expert may employ on any destructive or constructive work of one general kind, say the cracking of an eggshell, or the welding of an anchor. But you must have your expert to manage it, for without him it is useless. He has to toil amid the heat, smoke, grime, grease, and perpetual din of the suffocating engine-room. The work has to be brought to the hammer, for it cannot usually be taken to its work. And it is not in general, transferable; for each expert, as a rule, knows, fully and confidently, the working details of his own weapon only. Quaternions, on the other hand, are like the elephant’s trunk, ready at any moment for anything, be it to pick up a crumb or a field-gun, to strangle a tiger, or uproot a tree; portable in the extreme, applicable anywhere - alike in the trackless jungle and in the barrack square - directed by a little native who requires no special skill or training, and who can be transferred from one elephant to another without much hesitation. Surely this, which adapts itself to its work, is the grander instrument. But then, it is the natural, the other, the artificial one." (Peter G Tait [in Alexander MacFarlane's "Lectures on Ten British Mathematicians", 1916])

"Projective Geometry: a boundless domain of countless fields where reals and imaginaries, finites and infinites, enter on equal terms, where the spirit delights in the artistic balance and symmetric interplay of a kind of conceptual and logical counterpoint - an enchanted realm where thought is double and flows throughout in parallel streams." (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)

"The concept of an independent system is a pure creation of the imagination. For no material system is or can ever be perfectly isolated from the rest of the world. Nevertheless it completes the mathematician’s ‘blank form of a universe’ without which his investigations are impossible. It enables him to introduce into his geometrical space, not only masses and configurations, but also physical structure and chemical composition." (Lawrence J Henderson, "The Order of Nature: An Essay", 1917)

"Imagine any sort of model and a copy of it done by an awkward artist: the proportions are altered, lines drawn by a trembling hand are subject to excessive deviation and go off in unexpected directions. From the point of view of metric or even projective geometry these figures are not equivalent, but they appear as such from the point of view of geometry of position [that is, topology]." (Henri Poincaré, "Dernières pensées", 1920)

"The discovery of Minkowski […] is to be found […] in the fact of his recognition that the four-dimensional space-time continuum of the theory of relativity, in its most essential formal properties, shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical space. In order to give due prominence to this relationship, however, we must replace the usual time co-ordinate t by an imaginary magnitude, √-1*ct, proportional to it. Under these conditions, the natural laws satisfying the demands of the (special) theory of relativity assume mathematical forms, in which the time co-ordinate plays exactly the same role as the three space-coordinates. Formally, these four co-ordinates correspond exactly to the three space co-ordinates in Euclidean geometry." (Albert Einstein,"Relativity: The Special and General Theory", 1920)

On Geometry (1875-1899)

"The culture of the geometric imagination, tending to produce precision in remembrance and invention of visible forms will, therefore, tend directly to increase the appreciation of works of belles-letters." (Thomas Hill, "Uses of Mathesis", Bibliotheca Sacra Vol. 32, 1875)

“Algebra is but written geometry and geometry is but figured algebra.” (Sophie Germain, Mémoire sur les Surfaces Élastiques”, 1880)

“In geometry, as in most sciences, it is very rare that an isolated proposition is of immediate utility. But the theories most powerful in practice are formed of propositions which curiosity alone brought to light, and which long remained useless without its being able to divine in what way they should one day cease to be so. In this sense it may be said, that in real science, no theory, no research, is in effect useless.” (Voltaire, “A Philosophical Dictionary”, 1881)

"This symbol [v-1] is restricted to a precise signification as the representative of perpendicularity in quaternions, and this wonderful algebra of space is intimately dependent upon the special use of the symbol for its symmetry, elegance, and power. The immortal author of quaternions has shown that there are other significations which may attach to the symbol in other cases. But the strongest use of the symbol is to be found in its magical power of doubling the actual universe, and placing by its side an ideal universe, its exact counterpart, with which it can be compared and contrasted, and, by means of curiously connecting fibres, form with it an organic whole, from which modern analysis has developed her surpassing geometry." (Benjamin Peirce, "On the Uses and Transformations of Linear Algebras", American Journal of Mathematics Vol. 4, 1881)

“We admit, in geometry, not only infinite magnitudes, that is to say, magnitudes greater than any assignable magnitude, but infinite magnitudes infinitely greater, the one than the other. This astonishes our dimension of brains, which is only about six inches long, five broad, and six in depth, in the largest heads.” (Voltaire, “A Philosophical Dictionary”, 1881)

"It is this combination of observation at the foundation and geometry at the summit that I wished to express by naming this method Geometric Statistics. It cannot be subject to the usual criticisms directed at the use of pure mathematics in economic matters, which are said to be too complex to be confined within a formula." (Emile Cheysson, "La Statistique géométrique", 1888)

"This method is what I call Geometric Statistics. But despite its somewhat forbidding name-which I’ll explain in a moment - it is not a mathematical abstraction or a mere intellectual curiosity accessible only to a select few. It is intended, if not for all merchants and industrialists, then at least for that elite who lead the masses behind them. Practice is both its starting point and its destination. It was inspired in me more than fifteen years ago by the demands of the profession, and if I’ve decided to present it today, it’s because I’ve since verified its advantages through various applications, both in private industry and in public service." (Emile Cheysson, "La Statistique géométrique", 1888)

"A Logarithmic Table is a small table by the use of which we can obtain a knowledge of all geometrical dimensions and motions in space, by a very easy calculation. It is deservedly called very small, because it does not exceed in size a table of sines; very easy, because by it all multiplications, divisions, and the more difficult extractions of roots are avoided; for by only a very few most easy additions, subtractions, and divisions by two, it measures quite generally all figures and motions."  (John Napier, "The Construction of the Wonderful Canon of Logarithms", 1889)

“Geometry exhibits the most perfect example of logical stratagem.” (Henry T Buckle, “History of Civilization in England” Vol. 2, 1891)

"I do think [...] that you would find it would lose nothing by omitting the word 'vector' throughout. It adds nothing to the clearness or simplicity of the geometry, whether of two dimensions or three dimensions. Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell." (William T Kelvin, [Letter to Robert B Hayward] 1892)

"Instead of the points of a line, plane, space, or any manifold under investigation, we may use instead any figure contained within the manifold: a group of points, curve, surface, etc. As there is nothing at all determined at the outset about the number of arbitrary parameters upon which these figures should depend, the number of dimensions of the line, plane, space, etc. is likewise arbitrary and depends only on the choice of space element. But so long as we base our geometrical investigation on the same group of transformations, the geometrical content remains unchanged. That is, every theorem resulting from one choice of space element will also be a theorem under any other choice; only the arrangement and correlation of the theorems will be changed. The essential thing is thus the group of transformations; the number of dimensions to be assigned to a manifold is only of secondary importance." (Felix Klein, "A comparative review of recent researches in geometry", Bulletin of the American Mathematoical Society 2(10), 1893)

"Instead of the points of a line, plane, space, or any manifold under investigation, we may use instead any figure contained within the manifold: a group of points, curve, surface, etc. As there is nothing at all determined at the outset about the number of arbitrary parameters upon which these figures should depend, the number of dimensions of the line, plane, space, etc. is likewise arbitrary and depends only on the choice of space element. But so long as we base our geometrical investigation on the same group of transformations, the geometrical content remains unchanged. That is, every theorem resulting from one choice of space element will also be a theorem under any other choice; only the arrangement and correlation of the theorems will be changed. The essential thing is thus the group of transformations; the number of dimensions to be assigned to a manifold is only of secondary importance." (Felix Klein, "A comparative review of recent researches in geometry", Bulletin of the American Mathematoical Society 2(10), 1893)
"[…] geometry is the art of reasoning well from badly drawn figures; however, these figures, if they are not to deceive us, must satisfy certain conditions; the proportions may be grossly altered, but the relative positions of the different parts must not be upset." (Henri Poincaré, 1895)

"Geometry has been, throughout, of supreme importance in the history of knowledge." (Bertrand Russell, :Foundations of Geometry", 1897)

"[…] there are terms which cannot be defined, such as number and quantity. Any attempt at a definition would only throw difficulty in the student’s way, which is already done in geometry by the attempts at an explanation of the terms point, straight line, and others, which are to be found in treatise on that subject." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1898 

"[…] geometry is the art of reasoning well from badly drawn figures; however, these figures, if they are not to deceive us, must satisfy certain conditions; the proportions may be grossly altered, but the relative positions of the different parts must not be upset." (Henri Poincaré, 1895)

"Geometry, then, is the application of strict logic to those properties of space and figure which are self-evident, and which therefore cannot be disputed. But the rigor of this science is carried one step further; for no property, however evident it may be, is allowed to pass without demonstration, if that can be given. The question is therefore to demonstrate all geometrical truths with the smallest possible number of assumptions." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1898)

"Mathematics in its pure form, as arithmetic, algebra, geometry, and the applications of the analytic method, as well as mathematics applied to matter and force, or statics and dynamics, furnishes the peculiar study that gives to us, whether as children or as men, the command of nature in this its quantitative aspect; mathematics furnishes the instrument, the tool of thought, which we wield in this realm." (William T  Harris, "Psychologic Foundations of Education", 1898)

"[…] there are terms which cannot be defined, such as number and quantity. Any attempt at a definition would only throw difficulty in the student’s way, which is already done in geometry by the attempts at an explanation of the terms point, straight line, and others, which are to be found in treatise on that subject." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1898) 

On Geometry (1850-1874)

"It is well known that geometry presupposes not only the concept of space but also the first fundamental notions for constructions in space as given in advance. It only gives nominal definitions for them, while the essential means of determining them appear in the form of axioms. The relationship of these presumptions is left in the dark; one sees neither whether and in how far their connection is necessary, nor a priori whether it is possible." (Bernhard Riemann, "On the hypotheses which lie at the foundation of geometry", 1854)

"Mathematics is peculiarly and preeminently the science of relations, and whether quantity or direction may severally form its object, these are never contemplated in characters purely absolute, but invariably in comparison with other objects like themselves; and it is hence that relations once established by the unerring theorems of the science, we are enabled, disregarding magnitude in itself, to pass indifferently from the finite to the infinite, from the limited regions of sense to those of conception, and with all the assurance and all the certainty that even the geometry of the ancients could confer." (John H W Waugh, Mathematical Essays", 1854)

“Geometry in every proposition speaks a language which experience never dares to utter; and indeed of which she but half comprehends the meaning. Experience sees that the assertions are true, but she sees not how profound and absolute is their truth. She unhesitatingly assents to the laws which geometry delivers, but she does not pretend to see the origin of their obligation. She is always ready to acknowledge the sway of pure scientific principles as a matter of fact, but she does not dream of offering her opinion on their authority as a matter of right; still less can she justly claim to herself the source of that authority.” (William Whewell, “The Philosophy of the Inductive Sciences”, 1858)

"The ideas which these sciences, Geometry, Theoretical Arithmetic and Algebra involve extend to all objects and changes which we observe in the external world; and hence the consideration of mathematical relations forms a large portion of many of the sciences which treat of the phenomena and laws of external nature, as Astronomy, Optics, and Mechanics. Such sciences are hence often termed Mixed Mathematics, the relations of space and number being, in these branches of knowledge, combined with principles collected from special observation; while Geometry, Algebra, and the like subjects, which involve no result of experience, are called Pure Mathematics." (William Whewell, "The Philosophy of the Inductive Sciences" , 1858)

"These sciences, Geometry, Theoretical Arithmetic and Algebra, have no principles besides definitions and axioms, and no process of proof but deduction; this process, however, assuming a most remarkable character; and exhibiting a combination of simplicity and complexity, of rigour and generality, quite unparalleled in other subjects." (William Whewell, "The Philosophy of the Inductive Sciences", 1858)

"This science, Geometry, is one of indispensable use and constant reference, for every student of the laws of nature; for the relations of space and number are the alphabet in which those laws are written. But besides the interest and importance of this kind which geometry possesses, it has a great and peculiar value for all who wish to understand the foundations of human knowledge, and the methods by which it is acquired. For the student of geometry acquires, with a degree of insight and clearness which the unmathematical reader can but feebly imagine, a conviction that there are necessary truths, many of them of a very complex and striking character; and that a few of the most simple and self-evident truths which it is possible for the mind of man to apprehend, may, by systematic deduction, lead to the most remote and unexpected results." (William Whewell, "The Philosophy of the Inductive Sciences", 1858)

"I hold: 1) that small portions of space are, in fact, of a nature analogous to little hills on a surface that is on the average fiat; namely, that the ordinary laws of geometry are not valid in them; 2) that this property of being curved or distorted is constantly being passed on from one portion of space to another after the manner of a wave; 3) that this variation of the curvature of space is what really happens in the phenomenon that we call the motion of matter, whether ponderable or ethereal; 4) that in the physical world nothing else takes place but this variation, subject (possibly) to the law of continuity." (William K Clifford, "On the Space Theory of Matter", [paper delivered before the Cambridge Philosophical Society, 1870)

"Riemann has shewn that as there are different kinds of lines and surfaces, so there are different kinds of space of three dimensions; and that we can only find out by experience to which of these kinds the space in which we live belongs. In particular, the axioms of plane geometry are true within the limits of experiment on the surface of a sheet of paper, and yet we know that the sheet is really covered with a number of small ridges and furrows, upon which (the total curvature not being zero) these axioms are not true. Similarly, he says although the axioms of solid geometry are true within the limits of experiment for finite portions of our space, yet we have no reason to conclude that they are true for very small portions; and if any help can be got thereby for the explanation of physical phenomena, we may have reason to conclude that they are not true for very small portions of space." (William K Clifford, "On the Space-Theory of Matter", [in Cambridge Philosophical Society's Proceedings II, 1876] 1870)

"Although geometrical representations of propositions in the thermodynamics of fluids are in general use and have done good service in disseminating clear notions in this science, yet they have by no means received the extension in respect to variety and generality of which they are capable." (J Willard Gibbs, Graphical Methods in the Thermodynamics of Fluids, 1873)

"It [geometry] escapes the tedious and troublesome task of collecting experimental facts, which is the province of the natural sciences in the strict sense of the word; the sole form of its scientific method is deduction." (Hermann von Helmholtz, "Popular Lectures on Scientific Subjects", 1873)

“When we consider that the whole of geometry rests ultimately on axioms which derive their validity from the nature of our intuitive faculty, we seem well justified in questioning the sense of imaginary forms, since we attribute to them properties which not infrequently contradict all our intuitions.” (Gottlob Frege, “On a Geometrical Representation of Imaginary forms in the Plane”, 1873) 

“[…] with few exceptions all the operations and concepts that occur in the case of real numbers can indeed be carried over unchanged to complex ones. However, the concept of being greater cannot very well be applied to complex numbers. In the case of integration, too, there appear differences which rest on the multplicity of possible paths of integration when we are dealing with complex variables. Nevertheless, the large extent to which imaginary forms conform to the same laws as real ones justifies the introduction of imaginary forms into geometry.” (Gottlob Frege, “On a Geometrical Representation of Imaginary forms in the Plane”, 1873)

“When we consider complex numbers and their geometrical representation, we leave the field of the original concept of quantity, as contained especially in the quantities of Euclidean geometry: its lines, surfaces and volumes. According to the old conception, length appears as something material which fills the straight line between its end points and at the same time prevents another thing from penetrating into its space by its rigidity. In adding quantities, we are therefore forced to place one quantity against another. Something similar holds for surfaces and solid contents. The introduction of negative quantities made a dent in this conception, and imaginary quantities made it completely impossible. Now all that matters is the point of origin and the end point; whether there is a continuous line between them, and if so which, appears to make no difference whatsoever; the idea of filling space has been completely lost. All that has remained is certain general properties of addition, which now emerge as the essential characteristic marks of quantity. The concept has thus gradually freed itself from intuition and made itself independent. This is quite unobjectionable, especially since its earlier intuitive character was at bottom mere appearance. Bounded straight lines and planes enclosed by curves can certainly be intuited, but what is quantitative about them, what is common to lengths and surfaces, escapes our intuition.” (Gottlob Frege, “Methods of Calculation based on an Extension of the Concept of Quantity”, 1874)