🧊On Geometry: On Curves (2020-2029)

"Skewed data means data that is shifted in one direction or the other. Skewness can cause machine learning models to underperform. Many machine learning models assume normally distributed data or data structures to follow the Gaussian structure. Any deviation from the assumed Gaussian structure, which is the popular bell curve, can affect model performance. A very effective area where we can apply feature engineering is by looking at the skewness of data and then correcting the skewness through normalization of the data." (Anthony So et al, "The Data Science Workshop" 2nd Ed., 2020)

"Correlation does not imply causation: often some other missing third variable is influencing both of the variables you are correlating. […] The need for a scatterplot arose when scientists had to examine bivariate relations between distinct variables directly. As opposed to other graphic forms - pie charts, line graphs, and bar charts - the scatterplot offered a unique advantage: the possibility to discover regularity in empirical data (shown as points) by adding smoothed lines or curves designed to pass 'not through, but among them', so as to pass from raw data to a theory-based description, analysis, and understanding." (Michael Friendly & Howard Wainer, "A History of Data Visualization and Graphic Communication", 2021)

"Standard graphs, like bar and line charts, are so common because they are perceptually more accurate, familiar to people, and easy to create. Nonstandard graphs - those that use circles or curves, for instance - may not allow the reader to most accurately perceive the exact data values. But perceptual accuracy is not always the goal. And sometimes it's not a goal at all. Spurring readers to engage with a graph is sometimes just as important. Sometimes, it's more important. And nonstandard chart types may do just that. In some cases, nonstandard graphs may help show underlying patterns and trends in better ways that standard graphs. In other cases, the fact that these nonstandard graphs are different may make them more engaging, which we may sometimes need to first attract attention to the visualization."  (Jonathan Schwabish, "Better Data Visualizations: A guide for scholars, researchers, and wonks", 2021)

"Ignoring sampling weights can give a misleading presentation of a distribution. Whether for a histogram, bar plot, box plot, two-dimensional contour, or smooth curve, we need to use the weights to get a representative plot." (Sam Lau et al, "Learning Data Science: Data Wrangling, Exploration, Visualization, and Modeling with Python", 2023)

"Side-by-side box plots offer a similar comparison of distributions across groups. The box plot offers a simpler approach that can give a crude understanding of a distribution. Likewise, violin plots sketch density curves along an axis for each group. The curve is flipped to create a symmetric 'violin' shape. The violin plot aims to bridge the gap between the density curve and box plot." (Sam Lau et al, "Learning Data Science: Data Wrangling, Exploration, Visualization, and Modeling with Python", 2023)

"Smoothing and aggregating can help us see important features and relationships, but when we have only a handful of observations, smoothing techniques can be misleading. With just a few observations, we prefer rug plots over histograms, box plots, and density curves, and we use scatterplots rather than smooth curves and density contours. This may seem obvious, but when we have a large amount of data, the amount of data in a subgroup can quickly dwindle. This phenomenon is an example of the curse of dimensionality." (Sam Lau et al, "Learning Data Science: Data Wrangling, Exploration, Visualization, and Modeling with Python", 2023)

"Stacked line plots are even more difficult to read because we have to judge the gap between curves as they jiggle up and down." (Sam Lau et al, "Learning Data Science: Data Wrangling, Exploration, Visualization, and Modeling with Python", 2023)

🧊On Geometry: On Curves (2000-2009)

"A great virtue of nomograms is that they are usually multivariate, showing relationships among variables in quite complex systems. It is surely helpful to have both an analysis of the underlying equation along with nomogram visualization of the curves generated by the equation. Nomograms show how equations perform. Nomograms remain useful for understanding; their computational use has passed. Computational power is so cheap now, we don’t need look-up tables or nomograms; we can just plug the numbers into the equations and solve." (Edward Tufte, 2002)

"Every network has its own fitness distribution, which tells us how similar or different the nodes in the network are. In networks where most of the nodes have comparable fitness, the distribution follows a narrowly peaked bell curve. In other networks, the range of fitnesses is very wide such that a few nodes are much more fit than most others. […] the mathematical tools developed decades earlier to describe quantum gases enabled us to see that, independent of the nature of links and nodes, a network's behavior and topology are determined by the shape of its fitness distribution. But even though each system, from the Web to Holywood, has a unique fitness distribution, Bianconi's calculation indicated that in terms of topology all networks fall into one of only two possible categories. In most networks the competition does not have an easily noticeable impact on the network's topology. In some networks, however, the winner takes all the links, a clear signature of Bose-Einstein condensation." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"Nature normally hates power laws. In ordinary systems all quantities follow bell curves, and correlations decay rapidly, obeying exponential laws. But all that changes if the system is forced to undergo a phase transition. Then power laws emerge-nature's unmistakable sign that chaos is departing in favor of order. The theory of phase transitions told us loud and clear that the road from disorder to order is maintained by the powerful forces of self-organization and is paved by power laws. It told us that power laws are not just another way of characterizing a system's behavior. They are the patent signatures of self-organization in complex systems." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"A closed (periodic) plane curve is said to be simple if it is a one-to-one map up to the period (this for a parameterized curve). For a geometric curve, simple means a curve which has the global topology of a circle (in the jargon, it is a differentiable embedding in the plane of the circle seen as an abstract one dimensional manifold). We also will assume the speed never vanishes, indeed that it has unit speed." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003) 

"Consider geodesics of a surface not as curves on the surface, but as flow lines of a vector field on the unit tangent bundle. To appreciate that such a viewpoint is possible, we can simply write out the geodesic equation, and think of it this way. Now we may ask what portion of the unit tangent bundle is taken up by periodic geodesics. For example, for geodesics of the two dimensional torus, the unit tangent bundle is a three dimensional torus, and the geodesic flow consists of straight lines. Those which are periodic are in fact dense. On the sphere, all geodesics are periodic, so again the periodic geodesics are dense in the unit tangent bundle." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)

"If the intensity of the material world is plotted along the horizontal axis, and the response of the human mind is on the vertical, the relation between the two is represented by the logarithmic curve. Could this rule provide a clue to the relationship between the objective measure of information, and our subjective perception of it?" (Hans Christian von Baeyer, "Information, The New Language of Science", 2003)

"Riemann forged two simultaneous innovations: first, he defined (not too rigorously) a differentiable manifold to be a set of any dimension n, where one can perform differential calculus, change coordinates, etc. In particular, one has differentiable curves, tangent vectors (velocities) of those curves, and a tangent space at each point (i.e. all possible velocities of any curves through that point). Then he asked that a geometry on a manifold be simply an arbitrary positive definite quadratic form on each of those tangent spaces, thought of as the analogue of Gauß’s first fundamental form. One could use the same expression to define length of curves, look for shortest curves, etc."  (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)

"The inner geometry of curves does not differ from that of straight lines, but the geometry is radically different if we look at the way a curve sits in the plane." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)

"General relativity explains gravitation as a curvature, or bending, or warping, of the geometry of space-time, produced by the presence of matter. Free fall in a space shuttle around Earth, where space is warped, produces weightlessness, and is equivalent from the observer's point of view to freely moving in empty space where there is no large massive body producing curvature. In free fall we move along a 'geodesic' in the curved space-time, which is essentially a straight-line motion over small distances. But it becomes a curved trajectory when viewed at large distances. This is what produces the closed elliptical orbits of planets, with tiny corrections that have been correctly predicted and measured. Planets in orbits are actually in free fall in a curved space-time!" (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"The incommensurability of the diagonal of a square was initially a problem of measuring length but soon moved to the very theoretical level of introducing irrational numbers. Attempts to compute the length of the circumference of the circle led to the discovery of the mysterious number. Measuring the area enclosed between curves has, to a great extent, inspired the development of calculus." (Heinz-Otto Peitgen et al, "Chaos and Fractals: New Frontiers of Science" 2nd Ed., 2004)

"The central limit theorem […] states that regardless of the shape of the curve of the original population, if you repeatedly randomly sample a large segment of your group of interest and take the average result, the set of averages will follow a normal curve." (Charles Livingston & Paul Voakes, "Working with Numbers and Statistics: A handbook for journalists", 2005)

"The statements on asymptotes are qualitative views of what happens to the curves when x or y is large (quite vague, even arguable for those in quotes). But we have not so far said what asymptotic directions or points at infinity actually are, which is a disadvantage in discussing asymptotes formally or in calculating with them. Making sense of asymptotes (of algebraic plane curves), and providing a simple framework for calculating with them is one thing that projective geometry does very well." (Miles Reid & Balazs Szendröi, "Geometry and Topology", 2005)

"There is weirdness in non-Euclidean geometry, but not because of anything that geometers might say about the ordinary fond familiar world in which space is flat, angles sharp, and only curves are curved. Non-Euclidean geometry is an instrument in the enlargement of the mathematician’s self-consciousness, and so comprises an episode in a long, difficult, and extended exercise in which the human mind attempts to catch sight of itself catching sight of itself, and so without end." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

"Apparent Impossibilities that Are New Truths […] irrational numbers, imaginary numbers, points at infinity, curved space, ideals, and various types of infinity. These ideas seem impossible at first because our intuition cannot grasp them, but they can be captured with the help of mathematical symbolism, which is a kind of technological extension of our senses." (John Stillwell,"Yearning for the impossible: the surpnsing truths of mathematics", 2006)

"Bell curves don't differ that much in their bells. They differ in their tails. The tails describe how frequently rare events occur. They describe whether rare events really are so rare. This leads to the saying that the devil is in the tails." (Bart Kosko, "Noise", 2006)

"Mathematical language is littered with pejorative and mystical terms - such as irrational, imaginary, surd, transcendental - that were once used to ridicule supposedly impossible objects. And these are just terms applied to numbers. Geometry also has many concepts that seem impossible to most people, such as the fourth dimension, finite universes, and curved space - yet geometers" (and physicists) cannot do without them. Thus there is no doubt that mathematics flirts with the impossible, and seems to make progress by doing so." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"The central limit theorem differs from laws of large numbers because random variables vary and so they differ from constants such as population means. The central limit theorem says that certain independent random effects converge not to a constant population value such as the mean rate of unemployment but rather they converge to a random variable that has its own Gaussian bell-curve description." (Bart Kosko, "Noise", 2006)

"The flaw in the classical thinking is the assumption that variance equals dispersion. Variance tends to exaggerate outlying data because it squares the distance between the data and their mean. This mathematical artifact gives too much weight to rotten apples. It can also result in an infinite value in the face of impulsive data or noise. [...] Yet dispersion remains an elusive concept. It refers to the width of a probability bell curve in the special but important case of a bell curve. But most probability curves don't have a bell shape. And its relation to a bell curve's width is not exact in general. We know in general only that the dispersion increases as the bell gets wider. A single number controls the dispersion for stable bell curves and indeed for all stable probability curves - but not all bell curves are stable curves." (Bart Kosko, "Noise", 2006)

"It is important to pay heed to the following detail: a disadvantage of logarithmic diagrams is that a graphical integration is not possible, i.e., the area under the curve (the integral) is of no relevance." (Manfred Drosg, "Dealing with Uncertainties: A Guide to Error Analysis", 2007)

"Since the ellipse is a closed curve it has a total length, λ say, and therefore f(l + λ) = f(l). The elliptic function f is periodic, with 'period' λ, just as the sine function is periodic with period 2π. However, as Gauss discovered in 1797, elliptic functions are even more interesting than this: they have a second, complex period. This discovery completely changed the face of calculus, by showing that some functions should be viewed as functions on the plane of complex numbers. And just as periodic functions on the line can be regarded as functions on a periodic line - that is, on the circle - elliptic functions can be regarded as functions on a doubly periodic plane - that is, on a 2-torus." (John Stillwell, "Yearning for the impossible: the surpnsing truths of mathematics", 2006)

"The flaw in the classical thinking is the assumption that variance equals dispersion. Variance tends to exaggerate outlying data because it squares the distance between the data and their mean. This mathematical artifact gives too much weight to rotten apples. It can also result in an infinite value in the face of impulsive data or noise. [...] Yet dispersion remains an elusive concept. It refers to the width of a probability bell curve in the special but important case of a bell curve. But most probability curves don't have a bell shape. And its relation to a bell curve's width is not exact in general. We know in general only that the dispersion increases as the bell gets wider. A single number controls the dispersion for stable bell curves and indeed for all stable probability curves - but not all bell curves are stable curves."  (Bart Kosko, "Noise", 2006)

"Wherever we look in our world the complex systems of nature and time seem to preserve the look of details at finer and finer scales. Fractals show a holistic hidden order behind things, a harmony in which everything affects everything else, and, above all, an endless variety of interwoven patterns. Fractal geometry allows bounded curves of infinite length, as well as closed surfaces with infinite area. It even allows curves with positive volume and arbitrarily large groups of shapes with exactly the same boundary." (Philip Tetlow, "The Web’s Awake: An Introduction to the Field of Web Science and the Concept of Web Life", 2007)

"[...] the term statistical misconception refers to any of several widely held but incorrect notions about statistical concepts, about procedures for analyzing data and about the meaning of results produced by such analyses. To illustrate, many people think that (1) normal curves are bell shaped, (2) a correlation coeffi cient should never be used to address questios of causality, and (3) the level of signifi cance dictates the probability of a Type I error. Some people, of course, have only one or two (rather than all three) of these misconceptions, and a few individuals realize that all three of those beliefs are false."(Schuyler W Huck, "Statistical Misconceptions", 2008)

"We can describe general relativity using either of two mathematically equivalent ideas: curved space-time or metric field. Mathematicians, mystics and specialists in general relativity tend to like the geometric view because of its elegance. Physicists trained in the more empirical tradition of high-energy physics and quantum field theory tend to prefer the field view, because it corresponds better to how we" (or our computers) do concrete calculations." (Frank Wilczek, "The Lightness of Being: Mass, Ether, and the Unification of Forces", 2008)

🧊On Geometry: On Curves (1990-1999)

"Exploratory regression methods attempt to reveal unexpected patterns, so they are ideal for a first look at the data. Unlike other regression techniques, they do not require that we specify a particular model beforehand. Thus exploratory techniques warn against mistakenly fitting a linear model when the relation is curved, a waxing curve when the relation is S-shaped, and so forth." (Lawrence C Hamilton, "Regression with Graphics: A second course in applied statistics", 1991)

"First, strange attractors look strange: they are not smooth curves or surfaces but have 'non-integer dimension' - or, as Benoit Mandelbrot puts it, they are fractal objects. Next, and more importantly, the motion on a strange attractor has sensitive dependence on initial condition. Finally, while strange attractors have only finite dimension, the time-frequency analysis reveals a continuum of frequencies." (David Ruelle, "Chance and Chaos", 1991)

"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)

"String theory promises to take a further step beyond that taken by Einstein's picture of force subsumed within curved space and time geometry. Indeed, string theory contains Einstein's theory of gravitation within itself. Loops of string behave like the exchange particles of the gravitational forces, or 'gravitons' as they are called in the point-particle picture of things. But it has been argued that it must be possible to extract even the geometry of space and time from the characteristics of the strings and their topological properties. At present, it is not known how to do this and we merely content ourselves with understanding how strings behave when they sit in a background universe of space and time." (John D. Barrow, "Theories of Everything: The Quest for Ultimate Explanation", 1991)

"History too has an inertia. In the four dimensions of spacetime, particles" (or events) have directionality; mathematicians, trying to show this, draw what they call 'world lines' on graphs. In human affairs, individual world lines form a thick tangle, curling out of the darkness of prehistory and stretching through time: a cable the size of Earth itself, spiraling round the sun on a long curved course. That cable of tangled world lines is history. Seeing where it has been, it is clear where it is going - it is a matter of simple extrapolation." (Kim S Robinson, "Red Mars", 1992)

"Skewness is a measure of symmetry. For example, it's zero for the bell-shaped normal curve, which is perfectly symmetric about its mean. Kurtosis is a measure of the peakedness, or fat-tailedness, of a distribution. Thus, it measures the likelihood of extreme values." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"A bell curve shows the 'spread' or variance in our knowledge or certainty. The wider the bell the less we know. An infinitely wide bell is a flat line. Then we know nothing. The value of the quantity, position, or speed could lie anywhere on the axis. An infinitely narrow bell is a spike that is infinitely tall. Then we have complete knowledge of the value of the quantity. The uncertainty principle says that as one bell curve gets wider the other gets thinner. As one curve peaks the other spreads. So if the position bell curve becomes a spike and we have total knowledge of position, then the speed bell curve goes flat and we have total uncertainty (infinite variance) of speed." (Bart Kosko, "Fuzzy Thinking: The new science of fuzzy logic", 1993)

"An attractor that consists of an infinite number of curves, surfaces, or higher-dimensional manifolds - generalizations of surfaces to multidimensional space - often occurring in parallel sets, with a gap between any two members of the set, is called a strange attractor." (Edward N Lorenz, "The Essence of Chaos", 1993)

"Topology deals with those properties of curves, surfaces, and more general aggregates of points that are not changed by continuous stretching, squeezing, or bending. To a topologist, a circle and a square are the same, because either one can easily be bent into the shape of the other. In three dimensions, a circle and a closed curve with an overhand knot in it are topologically different, because no amount of bending, squeezing, or stretching will remove the knot." (Edward N Lorenz, "The Essence of Chaos", 1993)

"Since geometry is the mathematical idealization of space, a natural way to organize its study is by dimension. First we have points, objects of dimension O. Then come lines and curves, which are one-dimensional objects, followed by two-dimensional surfaces, and so on. A collection of such objects from a given dimension forms what mathematicians call a 'space'. And if there is some notion enabling us to say when two objects are 'nearby' in such a space, then it's called a topological space." (John L Casti, "Five Golden Rules", 1995)

"Grouped area graphs sometimes cause confusion because the viewer cannot determine whether the areas for the data series extend down to the zero axis. […] Grouped area graphs can handle negative values somewhat better than stacked area graphs but they still have the problem of all or portions of data curves being hidden by the data series towards the front." (Robert L Harris, "Information Graphics: A Comprehensive Illustrated Reference", 1996)

"[...] if we consider a topological space instead of a plane, then the question of whether the coordinates axes in that space are curved or straight becomes meaningless. The way we choose coordinate systems is related to the way we observe the property of smoothness in a topological space." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"When approximations are all that are needed, stacked area graphs are usually adequate. When accuracy is desired, this type of graph is generally not used, particularly when the values fluctuate significantly and/or the slopes of the curves are steep." (Robert L Harris, "Information Graphics: A Comprehensive Illustrated Reference", 1996)

"Spacetime tells matter how to move; matter tells spacetime how to curve." (John A Wheeler, "Geons, Black Holes and Quantum Foam: A Life in Physics" , 1998)

"The spirit of mathematics and the essence of its beauty is remarkably fragile, because mathematics is about ideas and about thought. Mathematics takes place in the mind, and no two minds are the same. After many years of study and work, a mathematician may stumble on a vast and beautiful vista that unifies and simplifies many things that once appeared disparate and complicated. Mathematicians can share a beautiful mathematical vista with one another, but there is no camera that can easily capture an image of such a vista to convey it in full to people who have not trudged along many of the same trails." (Silvio Levy,"The Eightfold Way: The Beauty of Klein’s Quartic Curve", 1999)

"[...] mathematics is about ideas and about thought." (Silvio Levy, "The Eightfold Way: The Beauty of Klein’s Quartic Curve", 1999)

"It is in the nature of exponential growth that events develop extremely slowly for extremely long periods of time, but as one glides through the knee of the curve, events erupt at an increasingly furious pace. And that is what we will experience as we enter the twenty-first century." (Ray Kurzweil, "The Age of Spiritual Machines: When Computers Exceed Human Intelligence", 1999)

"The recurrence of regression fallacies is testimony to its subtlety, deceptive simplicity, and, I speculate, to the wide use of the word regression to describe least squares fitting of curves, lines, and surfaces." (Stephen M Stigler, Statistics on the Table: the history of statistical concepts and methods, 1999) 

🧊On Geometry: On Curves (1980-1989)

"However, it turns out that a one-to-one mapping of the points in a square into the points on a line cannot be continuous. As we move smoothly along a curve through the square, the points on the line which represent the successive points on the square necessarily jump around erratically, not only for the mapping described above but for any one-to-one mapping whatever. Any one-to-one mapping of the square onto the line is discontinuous." (John R Pierce, "An Introduction to Information Theory: Symbols, Signals & Noise" 2nd Ed., 1980)

"The whole fabric of the space-time continuum is not merely curved, it is in fact totally bent." (Douglas N Adams, "The Restaurant at the End of the Universe", 1980)

"The ‘eyes of the mind’ must be able to see in the phase space of mechanics, in the space of elementary events of probability theory, in the curved four-dimensional space-time of general relativity, in the complex infinite dimensional projective space of quantum theory. To comprehend what is visible to the ‘actual eyes’, we must understand that it is only the projection of an infinite dimensional world on the retina." (Yuri I Manin, "Mathematics and Physics", 1981)

"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 Mathematics All Around You", 1986)

"A surface which can be regarded as the set of successive position of a curve moving in space is said to be generated by the curve. The utility of this notion in constructing a surface geometrically, in a picture or as a model is increased as the complexity of the generator and its motion is decreased. When the generator is a straight line, it is called a ruled surface. Since you can exchange X and Y in the above analysis, the hyperbolic paraboloid is generated by a line in two ways. It is a doubly ruled surface." (George K Francis, "A Topological Picturebook", 1987)

"Linking topology and dynamical systems is the possibility of using a shape to help visualize the whole range of behaviors of a system. For a simple system, the shape might be some kind of curved surface; for a complicated system, a manifold of many dimensions. A single point on such a surface represents the state of a system at an instant frozen in time. As a system progresses through time, the point moves, tracing an orbit across this surface. Bending the shape a little corresponds to changing the system's parameters, making a fluid more visous or driving a pendulum a little harder. Shapes that look roughly the same give roughly the same kinds of behavior. If you can visualize the shape, you can understand the system." (James Gleick, "Chaos: Making a New Science", 1987)

"[...] the image of a stable map of a surface into space looks like in the neighborhood of each point. If no neighborhood, no matter how small, of a given point looks like a mildly bent disc, then it is a singular point. A stable map can have three kinds of singular points. In a neighborhood of a double point a surface looks like two sheets of some fabric crossing along a so-called double curve. A neighborhood of a triple point looks like three surface sheets crossing transversely. Thus triple points are isolated. You can see why a quadruple point is unstable. A slight perturbation of one of the sheets would make four sheets cross each other so as to produce a little tetrahedral cell. Double curves are either closed, extend to infinity, terminate on the border or simply end at very special points, called pinch points." (George K Francis, "A Topological Picturebook", 1987)

"These then are the elements of descriptive topology: normal surfaces with their border curves and double curves, triple points and pinch points, and their pictures with cusps and contours." (George K Francis, "A Topological Picturebook", 1987)

"Thus the unit normal N/|N is independent of the parametrization and defines a mapping from the surface to the unit sphere, called the Gauss map. The Jacobian of the Gauss map is called the Gauss curvature function on the surface. Here is a less complicated way of thinking about this function. For each plane normal to a surface at a point, the curvature of the curve of intersection is a sectional curvature. The Gauss curvature is the product of the maximum and the minimum sectional curvatures at the point. (The average sectional curvature is the mean curvature and this vanishes on surfaces minimizing area locally; soap films, for example.)" (George K Francis, "A Topological Picturebook", 1987)

"Bodies like the earth are not made to move on curved orbits by a force called gravity; instead, they follow the nearest thing to a straight path in a curved space, which is called a geodesic. A geodesic is the shortest" (or longest) path between two nearby points." (Stephen Hawking, "A Brief History of Time", 1988)

"Nonlinear systems" (the graph of at least one relationship displays some curved feature) are notoriously more difficult to comprehend than linear systems, that is, they are more complex. Consequently they are also more difficult to control. This is exemplified by the volumes of elegant mathematics that have been developed in the search for optimal control of linear systems." (Robert L Flood & Ewart R Carson, "Dealing with Complexity: An introduction to the theory and application of systems", 1988)

"The metaphor never goes very far, anymore than a curve can long be confused with its tangent." (Henri Bergson,"A World of Ideas", 1989)

"It is now known to science that there are many more dimensions than the classical four. Scientists say that these don’t normally impinge on the world because the extra dimensions are very small and curve in on themselves, and that since reality is fractal most of it is tucked inside itself. This means either that the universe is more full of wonders than we can hope to understand or, more probably, that scientists make things up as they go along." (Terry Pratchett, Pyramids, 1989)

"The metaphor never goes very far, anymore than a curve can long be confused with its tangent." (Henri Bergson, "A World of Ideas", 1989)

🧊On Geometry: On Curves (1960-1969)

"A cylindrical surface is any surface generated by a moving straight line which constantly intersects a given fixed curve and which remains parallel to a fixed straight line. The fixed curve is called the directrix, and the generating line is known as the generatrix. Thus a cylinder is a ruled surface, that is, the surface is comprised of straight lines (the generators), all of which are parallel." (William L Schaaf, "Analytic Geometry; a college course guide", 1962) 

"In algebra, when we plot the graph of an equation, the curve so obtained represents the locus of the equation. In analytic geometry, the word 'curve' refers to any locus; it may be a straight line, a curved line, or a group of lines. We therefore see that: (1) The locus of an equation is a curve containing all those points, and only those points, whose coordinates satisfy the given equation. (2) The equation of a locus is an equation such that the coordinates of every point on the locus satisfy the equation, and every ordered pair of numbers which satisfy the equation are the coordinates of a point on the locus." (William L Schaaf, "Analytic Geometry; a college course guide", 1962)

"It is possible, however, to determine the general shape and salient features of a curve by a more searching method. This is accomplished by examining more closely the equation of the curve, rather than by computing a more extensive table of values. Certain characteristic features of curves can be determined by inspection of the form of the equation, and by computing only certain special values of the variables. These characteristic properties include the intercepts, extent, symmetry, and asymptotes of the curve." (William L Schaaf, "Analytic Geometry; a college course guide", 1962)

"It is to be understood that by an algebraic equation is meant an equation which involves only integral and fractional powers of the variables x and y. If the equation is of degree higher than two, the equation represents an algebraic higher plane curve. Any equation which is not algebraic is known as a transcendental equation; functions defined by such equations are said to be transcendental functions. For example, y = cos x, y = a log x, and y = k"" are transcendental functions."  (William L Schaaf, "Analytic Geometry; a college course guide", 1962)

"The empirical data are usually available in the form of a table exhibiting pairs of corresponding values of the variables. Since the given values do not lie exactly along a straight hne, a parabola, an exponential curve or a power curve, it is necessar}' to find a curve that fits the given data approximately. This process is called curve fitting, and the equation of the curve which fits the data approximately is called an empirical equation. When fitting a curve to a set of given points, we must assume a general form for the equation which we are going to use to represent the curve." (William L Schaaf, "Analytic Geometry; a college course guide", 1962) 

"When either the translation or the rotation transformations are applied to an equation, they change the coordinates of all points in the plane (except the origin in the case of a rotation). These transformations actually move the axes with relation to the curve, but they do not alter the shape of the curve represented by the original equation and by the transformed equation. Beside this constant or unvarying feature, i.e., the preservation of the geometric form of the curve, there are also certain algebraic expressions whose values remain unchanged in both equations. Mathematical forms which are preserved or do not vary when other changes take place are called invariants." (William L Schaaf, "Analytic Geometry; a college course guide", 1962)

"Mathematical statistics provides an exceptionally clear example of the relationship between mathematics and the external world. The external world provides the experimentally measured distribution curve; mathematics provides the equation (the mathematical model) that corresponds to the empirical curve. The statistician may be guided by a thought experiment in finding the corresponding equation." (Marshall J Walker, "The Nature of Scientific Thought", 1963)

"The mathematicians and physics men Have their mythology; they work alongside the truth, Never touching it; their equations are false But the things work. Or, when gross error appears, They invent new ones; they drop the theory of waves In universal ether and imagine curved space." (Robinson Jeffers," The Beginning and the End and Other Poems, The Great Wound", 1963)

"Mathematical statistics provides an exceptionally clear example of the relationship between mathematics and the external world. The external world provides the experimentally measured distribution curve; mathematics provides the equation" (the mathematical model) that corresponds to the empirical curve. The statistician may be guided by a thought experiment in finding the corresponding equation." (Marshall J Walker, "The Nature of Scientific Thought", 1963)

"The mathematicians and physics men Have their mythology; they work alongside the truth, Never touching it; their equations are false But the things work. Or, when gross error appears, They invent new ones; they drop the theory of waves In universal ether and imagine curved space." (Robinson Jeffers," The Beginning and the End and Other Poems, The Great Wound", 1963)

"When a curve approaches an axis or any other straight line in this fashion, the line is said to be an asymptote of the curve. As a working definition, we may say: An asymptote of a curve is any straight line which a curve approaches continuously as the curve moves on to infinity." (William L Schaaf, "The Calculus, a college course guide", 1963)

"A manifold, roughly, is a topological space in which some neighborhood of each point admits a coordinate system, consisting of real coordinate functions on the points of the neighborhood, which determine the position of points and the topology of that neighborhood; that is, the space is locally cartesian. Moreover, the passage from one coordinate system to another is smooth in the overlapping region, so that the meaning of 'differentiable' curve, function, or map is consistent when referred to either system." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"The mathematical models for many physical systems have manifolds as the basic objects of study, upon which further structure may be defined to obtain whatever system is in question. The concept generalizes and includes the special cases of the cartesian line, plane, space, and the surfaces which are studied in advanced calculus. The theory of these spaces which generalizes to manifolds includes the ideas of differentiable functions, smooth curves, tangent vectors, and vector fields. However, the notions of distance between points and straight lines" (or shortest paths) are not part of the idea of a manifold but arise as consequences of additional structure, which may or may not be assumed and in any case is not unique." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"To abstract the qualitative features of a differential equation on M, the concept of a phase portrait become important. Usually the phase portrait means the picture of the solution curves of the differential equation. [...] Then two differential equations on M have the same phase portrait if they are topologically equivalent. A definition of phase portrait is thus a topological  equivalence class of differential equations on M. A main goal of the qualitative study of ordinary differential equations is to obtain information on the phase portrait of differential equations." (Steven Smale, "What is global analysis?", American Mathematical Monthly Vol. 76 (1), 1969)

🧊On Geometry: On Curves (1950-1959)

"A piece of self-deception - often dear to the heart of apprentice scientists - is the drawing of a 'smooth curve' (how attractive it sounds!) through a set of points which have about as much trend as the currants in plum duff. Once this is done, the mind, looking for order amidst chaos, follows the Jack-o'-lantern line with scant attention to the protesting shouts of the actual points. Nor, let it be whispered, is it unknown for people who should know better to rub off the offending points and publish the trend line which their foolish imagination has introduced on the flimsiest of evidence. Allied to this sin is that of overconfident extrapolation, i.e. extending the graph by guesswork beyond the range of factual information. Whenever extrapolation is attempted it should be carefully distinguished from the rest of the graph, e.g. by showing the extrapolation as a dotted line in contrast to the full line of the rest of the graph. [...] Extrapolation always calls for justification, sooner or later. Until this justification is forthcoming, it remains a provisional estimate, based on guesswork." (Michael J Moroney, "Facts from Figures", 1951)

"The concept of an average, the equation to a curve, the description of a froth or cellular tissue, all come within the scope of mathematics for no other reason than that they are summations of more elementary principles or phenomena. Growth and Form are throughout of this composite nature; therefore the laws of mathematics are bound to underlie them, and her methods to be peculiarly fitted to interpret them." (D'Arcy W Thompson, "On Growth and Form", 1951)

"The waves of the sea, the little ripples on the shore, the sweeping curve of the sandy bay between the headlands, the outline of the hills, the shape of the clouds, all these are so many riddles of form, so many problems of morphology." (Sir D’Arcy W Thompson, "On Growth and Form", 1951)

"The number of grid lines should be kept to a minimum. This means that there should be just enough coordinate lines in the field so that the eye can readily interpret the values at any point on the curve. No definite rule can be specified as to the optimum number of lines in a grid. This must be left to the discretion of the chart-maker and can come only from experience. The size of the chart, the type and range of the data. the number of curves, the length and detail of the period covered, as well as other factors, will help to determine the number of grid lines." (Calvin F Schmid, "Handbook of Graphic Presentation", 1954)

"When you are told that something is an average you still don't know very much about it unless you can find out which of the common kinds of average it is-mean, median, or mode. [...] The different averages come out close together when you deal with data, such as those having to do with many human characteristics, that have the grace to fall close to what is called the normal distribution. If you draw a curve to represent it you get something shaped like a bell, and mean, median, and mode fall at the same point." (Darell Huff, "How to Lie with Statistics", 1954)

"Is evolution a theory, a system or a hypothesis? It is much more: it is a general condition to which all theories, all hypotheses, all systems must bow and which they must satisfy henceforward if they are to be thinkable and true. Evolution is a light illuminating all facts, a curve that all lines must follow." (Pierre Teilhard de Chardin, "The Phenomenon of Man", 1955)

"A regular curve on a Riemannian manifold is a curve with a continuously turning nontrivial tangent vector. A regular homotopy is a homotopy which at every stage is a regular curve, keeps end points and directions fixed and such that the tangent vector moves continuously with the homotopy. A regular curve is closed if its initial point and tangent  coincides with its end point and tangent." (Steven Smale,"Regular Curves on Riemann Manifolds", 1956)

"In the mathematical theory of the maximum and minimum problems in calculus of variations, different methods are employed. The old classical method consists in finding criteria -as to whether or not for a given curve the corresponding number assumes a maximum or minimum. In order to find such criteria a considered curve is a little varied, and it is from this method that the name 'calculus of variations' for the whole branch of mathematics is derived." (Karl Menger, "What Is Calculus of Variations and What Are Its Applications?" [James R Newman, "The World of Mathematics" Vol. II], 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)

"A primary goal of any learning model is to predict correctly the learning curve - proportions of correct responses versus trials. Almost any sensible model with two or three free parameters, however, can closely fit the curve, and so other criteria must be invoked when one is comparing several models." (Robert R Bush & Frederick Mosteller, "A Comparison of Eight Models?", Studies in Mathematical Learning Theory, 1959)

"This is the essential characteristic of a logarithmic scale. Any given increase, regardless of its absolute size, is related to a given base quantity. Thus, a perfectly straight line on such a graph denotes a constant percentage rate of increase, and not a constant absolute increase. It is the slope of the line or curve which is significant in such a graph. The steeper the slope, whether it be downwards or upwards, the more marked is the rate of change." (Alfred R Ilersic, "Statistics", 1959)

🧊On Geometry: On Curves (1925-1949)

"In order to regain in a rigorously defined function those properties that are analogous to those ascribed to an empirical curve with respect to slope and curvature" (first and higher difference quotients), we need not only to require that the function is continuous and has a finite number of maxima and minima in a finite interval, but also assume explicitly that it has the first and a series of higher derivatives" (as many as one will want to use)." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

"It is possible to pass continuously from any non-singular curve to any other such curve by interposition of other curves in such a way that during this procedure one will meet no other occurrence of singularities, apart from a finite number of times a curve with an ordinary double point, no matter whether the curve has at that point real or imaginary branches." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

"[...] the time stream is curved helically in some higher dimension. In your case, a still further distortion brought two points of the coil into contact, and a sort of short circuit threw you into the higher curve." (Robert H Wilson, "A Flight Into Time", Wonder Stories, 1931)

"The underlying notion of the integral calculus is also that of finding a limiting value, but this time it is the limiting value of a sum of terms when the number of terms increases without bound at the same time that the numerical value of each term approaches Zero. The area bounded by one or more curves is found as the limiting value of a sum of small rectangles; the length of an arc of a curve is found as the limiting value of a sum of lengths of straight lines (chords of the arc); the volume of a solid bounded by one or more curved surfaces is found as the limiting value of a sum of volumes of small solids bounded by planes; etc." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"The words 'maximum' and 'minimum' are used here in a technical sense. Maximum value of the function, for example, does not mean (as one might well suppose) the greatest value which the function attains for any value of x but, merely, the greatest value which it attains when, having been increasing, it ceases increasing and begins to decrease. In other words, the ordinate of a maximum point on a curve is greater than the ordinates of other nearby points. In a similar manner the ordinate of a minimum point is less than the ordinates of other nearby points." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"The term nomography serves to designate the general study of the graphic representation of equations in any number of variables on a plane surface. Its practical applications consist in the representation of the numerical relations between the variables by calibrated systems (straight lines or curves) constructed once for all and permitting the determination by a single reading of one or more of the variables when the others are given." (Howard G Funkhouser," Historical Development of the Graphical Representation of Statistical Data", 1937)

"The graphic language is modern. We are learning its alphabet. That it will develop a lexicon and a literature marvelous for its vividness and the variety of application is inevitable. Graphs are dynamic, dramatic. They may epitomize an epoch, each dot a fact, each slope an event, each curve a history. Wherever there are data to record, inferences to draw, or facts to tell, graphs furnish the unrivalled means whose power we are just beginning to realize and to apply."  (Henry D Hubbard [foreword to Willard C Brinton, "Graphic Presentation", 1939)])

"There is a magic in graphs. The profile of a curve reveals in a flash a whole situation - the life history of an epidemic, a panic, or an era of prosperity. The curve informs the mind, awakens the imagination, convinces." (Henry D Hubbard [in William Brinton's "Graphic Presentation", 1939])

"The curves treated by the calculus are normal and healthy; they possess no idiosyncrasies. But mathematicians would not be happy merely with simple, lusty configurations. Beyond these their curiosity extends to psychopathic patients, each of whom has an individual case history resembling no other; these are the pathological curves in mathematics." (Edward Kasner & James R Newman, "Mathematics and the Imagination", 1940)

"Geometric lines are fictions in the sense that, while we draw them, we think of them as having no width, simply length. Lines may be curved or straight." (William L Schaaf, "Mathematics For Everyday Use", 1942)

"It is clear that for any given point on a graph, its horizontal distance from the vertical scale (abscissa) represents the magnitude of the independent variable, while the vertical distance above or below the horizontal scale (ordinate) represents the corresponding magnitude of the dependent variable. Thus the position of the curve with respect to the axes depicts the actual magnitudes of the variables. But in studying changing variables and functional relationships, it is frequently desirable to inquire as to the rate at which a quantity is changing, i.e., how fast it is increasing or decreasing, rather than how large or how small it is. Rate implies a ratio; a rate of change means the amount of change in the function (or dependent variable) per unit change in the independent variable." (William L Schaaf, "Mathematics for Mechanics", 1942)

"When statistical data are of such a nature that it is permissible to assume that 'in-between values' vary continuously and uniformly (or very nearly so) from one observed or measured value to the next, a modification of the broken-line graph may be used. Instead of connecting the plotted points with straightline segments, a 'smooth' curved line is drawn between the points [...]. Such curvedline graphs may be drawn either 'free hand' or with the aid of drafting instruments known as French curves." (William L Schaaf, "Mathematics For Everyday Use", 1942)

"Any region of space-time that has no gravitating mass in its vicinity is uncurved, so that the geodesics here are straight lines, which means that particles move in straight courses at uniform speeds" (Newton's first law). But the world-lines of planets, comets and terrestrial projectiles are geodesics in a region of space-time which is curved by the proximity of the sun or earth. […] No force of gravitation is […] needed to impress curvature on world-lines; the curvature is inherent in the space […]" (James H Jeans," The Growth of Physical Science", 1947)

"An important rule in the drafting of curve charts is that the amount scale should begin at zero. In comparisons of size the omission of the zero base, unless clearly indicated, is likely to give a misleading impression of the relative values and trend." (Rufus R Lutz, "Graphic Presentation Simplified", 1949)

"Space-time is curved in the neighborhood of material masses, but it is not clear whether the presence of matter causes the curvature of space-time or whether this curvature is itself responsible for the existence of matter." (Gerald J Whitrow, "The Structure of the Universe: An Introduction to Cosmology", 1949)

🧊On Geometry: On Curves (1900-1924)

"By [diagrams] it is possible to present at a glance all the facts which could be obtained from figures as to the increase, fluctuations, and relative importance of prices, quantities, and values of different classes of goods and trade with various countries; while the sharp irregularities of the curves give emphasis to the disturbing causes which produce any striking change." (Arthur L Bowley, "A Short Account of England's Foreign Trade in the Nineteenth Century, its Economic and Social Results", 1905)

"Mathematicians, however, are well aware that it is childish to try to show by drawing curves that every continuous function has a derivative. Though differentiable functions are the simplest and the easiest to deal with, they are exceptional. Using geometrical language, curves that have no tangents are the rule, and regular curves, such as the circle, are interesting but quite special." (Jean-Baptiste Perrin, 1906)

"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions. […] A curve is the totality of points, whose coordinates are functions of a parameter which may be differentiated as often as may be required." (Felix Klein, "Elementar Mathematik vom hoheren Standpunkte aus" Vol. 2, 1909)

"But it is a third geometry from which quantity is completely excluded and which is purely qualitative; this is analysis situs. In this discipline, two figures are equivalent whenever one can pass from one to the other by a continuous deformation; whatever else the law of this deformation may be, it must be continuous. Thus, a circle is equivalent to an ellipse or even to an arbitrary closed curve, but it is not equivalent to a straight-line segment since this segment is not closed. A sphere is equivalent to any convex surface; it is not equivalent to a torus since there is a hole in a torus and in a sphere there is not. Imagine an arbitrary design and a copy of this same design executed by an unskilled draftsman; the properties are altered, the straight lines drawn by an inexperienced hand have suffered unfortunate deviations and contain awkward bends. From the point of view of metric geometry, and even of projective geometry, the two figures are not equivalent; on the contrary, from the point of view of analysis situs, they are." (Henri Poincaré, "Dernières pensées", 1913)

"The number was first studied in respect of its rationality or irrationality, and it was shown to be really irrational. When the discovery was made of the fundamental distinction between algebraic and transcendental numbers, i. e. between those numbers which can be, and those numbers which cannot be, roots of an algebraical equation with rational coefficients, the question arose to which of these categories the number π belongs. It was finally established by a method which involved the use of some of the most modern of analytical investigation that the number π was transcendental. When this result was combined with the results of a critical investigation of the possibilities of a Euclidean determination, the inferences could be made that the number π, being transcendental, does not admit of a construction either by a Euclidean determination, or even by a determination in which the use of other algebraic curves besides the straight line and the circle are permitted." (Ernest W Hobson, "Squaring the Circle", 1913)

"Human knowledge is not" (or does not follow) a straight line, but a curve, which endlessly approximates a series of circles, a spiral. Any fragment, segment, section of this curve can be transformed" (transformed one-sidedly) into an independent, complete, straight line [...]" (Vladimir I Lenin, "On the Question of Dialectics", 1915)

"The practice of drawing several curves on the same sheet is not to be commended except in cases where the curves will not intersect. A crowded chart on which the curves frequently intersect resembles a Chinese puzzle more than a graphic record, and a report submitted in figures is to be preferred to a chart of this kind. Even when the curves do not intersect, they should be made in different colors in order that they may be readily distinguished, one from the other." (Allan C Haskell, "How to Make and Use Graphic Charts", 1919)

"The principles of charting and curve plotting are not at all complex, and it is surprising that many business men dodge the simplest charts as though they involved higher mathematics or contained some sort of black magic. [...] The trouble at present is that there are no standards by which graphic presentations can be prepared in accordance with definite rules so that their interpretation by the reader may be both rapid and accurate. It is certain that there will evolve for methods of graphic presentation a few useful and definite rules which will correspond with the rules of grammar for the spoken and written language." (Willard C Brinton, "Graphic Methods for Presenting Facts", 1919)

"Under certain conditions, however, the ordinary form of graphic chart is slightly misleading. It will be conceded that its true function is to portray comparative fluctuations. This result is practically secured when the factors or quantities compared are nearly of the same value or volume, but analysis will show that this is not accomplished when the amounts compared differ greatly in value or volume. [...] The same criticism applies to charts which employ or more scales for various curve. If the different scale are in proper proportion, the result is the same as with one scale, but when two or more scales are used which are not proportional an indication may be given with respect to comparative fluctuations which is absolutely false." (Allan C Haskell, "How to Make and Use Graphic Charts", 1919)

"The chief problems in the technique of historigram [aka histogram] plotting are those of base line scales, types of lines to use for the graphs and methods of and purposes of smoothing these curves. The size of page, ability of grasp by the eye, subsequent treatment of the illustration, etc., are determining factors. The variable factor is usually plotted from a base line along the ordinate axis. Spacing and rules for scales apply as in frequency diagrams." (William C Marshall, "Graphical methods for schools, colleges, statisticians, engineers and executives", 1921)

🧊On Geometry: On Curves (2010-2019)

"Scatterplots are the preferred medium for adding smooth curves to show a causal functional relationship or an association […] However, despite the advantage of the scatterplot for seeing some types of patterns, the linked micromap design adds geographic location to the information displayed and so enables searches for geographic patterns that the scatterplot omits." (Daniel B Carr & Linda W Pickle, "Visualizing Data Patterns with Micromaps", 2010)

"[…] according to the bell-shaped curve the likelihood of a very-large-deviation event (a major outlier) located in the striped region appears to be very unlikely, essentially zero. The same event, though, is several thousand times more likely if it comes from a set of events obeying a fat-tailed distribution instead of the bell-shaped one." (John L Casti, "X-Events: The Collapse of Everything", 2012)

"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)

"Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values" (the bifurcation parameters) of a system causes a sudden"qualitative" or topological change in its behavior. Bifurcations can occur in both continuous systems" (described by ODEs, DDEs, or PDEs) and discrete systems" (described by maps)." (Tianshou Zhou, "Bifurcation", 2013)

"[...] affine curvature can be viewed as being about transporting a vector around a closed curve and returning to the point of departure with a different vector. Torsion, on the other hand, is about representing in affine space small curves of general affinely connected differentiable manifolds. That the paths have to be very small for integration of vector-valued integrals to mean anything at all is seldom if ever considered in books on differential geometry, more interested in formal aspects than on geometric interpretations. Or they have it wrong for failing to refer to very small paths." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)

"[...]  if one wants to study the velocity field of a curve which lies on a plane, then the concept of vector fields used up to now is not suitable for the following two reasons. Firstly, the velocity field is not defined on the whole surface, but only along the curve. Secondly, the curve may intersect itself, leading to the problem that the velocity vector has two different values at this point of intersection. Considered as a vector field on the surface it would not be uniquely determined at such points." (Christian Bär, "Elementary Differential Geometry", 2010)

"In Riemann’s and Cartan’s theories, a surface is a differentiable manifold of dimension two, meaning that we need two independent coordinates to label its points. But, as we shall see in an appendix, the theory of differentiable manifolds of dimension two and the theory of surfaces developed before Riemann do not coincide. Thus, whereas we may speak of the torsion of a differentiable 2-manifold, we may not speak of the torsion of a surface in the theory of curves and surfaces of that time." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)

"The integrands of the integrals on a differential manifold (on its curves, surfaces, etc.; all of them oriented) constitute a structure called an algebra. To be more specific, it is a graded algebra. [...] The subspaces constituted by all the elements of a graded algebra that have the same grade constitute modules (generalization of the concept of vector space)." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)

"To summarize, affine torsion is about representing on affine space itself (i.e. on the corresponding Klein space) tiny closed curves of an affinely connected manifold and 'quantifying' its failure to close in the form of a vector, which one obtains through integration of the torsion. Let it not be forgotten that the representation of closed curves fails to close even if the torsion is zero but the affine curvature is not. But the reason for the failure to close is not the same one when the torsion is zero as when the affine curvature is zero. In general, both torsion and curvature will contribute to the failure to close." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)

"Curvature is a central concept in differential geometry. There are conceptually different ways to define it, associated with different mathematical objects, the metric tensor, and the affine connection. In our case, however, the affine connection may be derived from the metric. The 'affine curvature' is associated with the notion of parallel transport of vectors as introduced by Levi-Civita. This is most simply illustrated in the case of a two- dimensional surface embedded in three- dimensional space. Let us take a closed curve on that surface and attach to a point on that curve a vector tangent to the surface. Let us now transport that vector along the curve, keeping it parallel to itself. When it comes back to its original position, it will coincide with the original vector if the surface is flat or deviate from it by a certain angle if the surface is curved. If one takes a small curve around a point on the surface, then the ratio of the angle between the original and the final vector and the area enclosed by the curve is the curvature at that point. The curvature at a point on a two-dimensional surface is a pure number." (Hanoch Gutfreund, "The Road to Relativity", 2015)

"To make progress, every field of science needs to have data commensurate with the complexity of the phenomena it studies. [...] With big data and machine learning, you can understand much more complex phenomena than before. In most fields, scientists have traditionally used only very limited kinds of models, like linear regression, where the curve you fit to the data is always a straight line. Unfortunately, most phenomena in the world are nonlinear. [...] Machine learning opens up a vast new world of nonlinear models." (Pedro Domingos, "The Master Algorithm", 2015)

"Underfitting is when a model doesn’t take into account enough information to accurately model real life. For example, if we observed only two points on an exponential curve, we would probably assert that there is a linear relationship there. But there may not be a pattern, because there are only two points to reference. [...] It seems that the best way to mitigate underfitting a model is to give it more information, but this actually can be a problem as well. More data can mean more noise and more problems. Using too much data and too complex of a model will yield something that works for that particular data set and nothing else." (Matthew Kirk, "Thoughtful Machine Learning", 2015)

"When memorization happens, you may have the illusion that everything is working well because your machine learning algorithm seems to have fitted the in sample data so well. Instead, problems can quickly become evident when you start having it work with out-of-sample data and you notice that it produces errors in its predictions as well as errors that actually change a lot when you relearn from the same data with a slightly different approach. Overfitting occurs when your algorithm has learned too much from your data, up to the point of mapping curve shapes and rules that do not exist [...]. Any slight change in the procedure or in the training data produces erratic predictions." (John P Mueller & Luca Massaron, "Machine Learning for Dummies", 2016)

"Bias is error from incorrect assumptions built into the model, such as restricting an interpolating function to be linear instead of a higher-order curve. [...] Errors of bias produce underfit models. They do not fit the training data as tightly as possible, were they allowed the freedom to do so. In popular discourse, I associate the word 'bias' with prejudice, and the correspondence is fairly apt: an apriori assumption that one group is inferior to another will result in less accurate predictions than an unbiased one. Models that perform lousy on both training and testing data are underfit." (Steven S Skiena, "The Data Science Design Manual", 2017)

"Sometimes mathematical advances happen by just looking at something in a slightly different way, which doesn’t mean building something new or going somewhere different, it just means changing your perspective and opening up huge new possibilities as a result. This particular insight leads to calculus and hence the understanding of anything curved, anything in motion, anything fluid or continuously changing." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)

"With time series though, there is absolutely no substitute for plotting. The pertinent pattern might end up being a sharp spike followed by a gentle taper down. Or, maybe there are weird plateaus. There could be noisy spikes that have to be filtered out. A good way to look at it is this: means and standard deviations are based on the naïve assumption that data follows pretty bell curves, but there is no corresponding 'default' assumption for time series data" (at least, not one that works well with any frequency), so you always have to look at the data to get a sense of what’s normal. [...] Along the lines of figuring out what patterns to expect, when you are exploring time series data, it is immensely useful to be able to zoom in and out." (Field Cady, "The Data Science Handbook", 2017)

🧊On Geometry: On Curves (1800-1899)

"The curves which the law of attraction may oblige the heavenly bodies to describe, are not confined to the elipse and circle; the sections of the cone are equally within its province, because they all have a focus wherein attraction may reside." (Johann H Lambert, "The System of the World", 1800)

"What should one understand by ∫ ϕx · dx for x = a + bi? Obviously, if we want to start from clear concepts, we have to assume that x passes from the value for which the integral has to be 0 to x = a + bi through infinitely small increments" (each of the form x = a + bi), and then to sum all the ϕx · dx. Thereby the meaning is completely determined. However, the passage can take placein infinitely many ways: Just like the realm of all real magnitudes can be conceived as an infinite straight line, so can the realm of all magnitudes, real and imaginary, be made meaningful by an infinite plane, in which every point, determined by abscissa = a and ordinate = b, represents the quantity a+bi. The continuous passage from one value of x to another a+bi then happens along a curve and is therefore possible in infinitely many ways. I claim now that after two different passages the integral ∫ ϕx · dx acquires the same value when ϕx never becomes equal to ∞ in the region enclosed by the two curves representing the two passages." (Carl F Gauss, [letter to Bessel] 1811)

"The analytical equations, unknown to the ancients, which Descartes first introduced into the study of curves and surfaces, are not restricted to the properties of figures, and to those properties which are the object of rational mechanics; they apply to all phenomena in general. There cannot be a language more universal and more simple, more free from errors and obscurities, that is to say, better adapted to express the invariable relations of nature." (Jean-Baptiste-Joseph Fourier, "The Analytical Theory of Heat", 1822)

"The integrals which we have obtained are not only general expressions which satisfy the differential equation, they represent in the most distinct manner the natural effect which is the object of the phenomenon [...] when this condition is fulfilled, the integral is, properly speaking, the equation of the phenomenon; it expresses clearly the character and progress of it, in the same manner as the finite equation of a line or curved surface makes known all the properties of those forms." (Jean-Baptiste-Joseph Fourier, "Théorie Analytique de la Chaleur", 1822)

"But since equal reliance can probably not be placed on all the observations, we must take care to distinguish those points which correspond to observations entitled to the greatest confidence. [...] These should be marked on the chart in some special manner [....]so as to strike the eye at a general glance; for example, by larger or darker points. [...] And when we draw our curve, we must take care to make it pass either through or very near all those points which are thus most distinguished." (John F W Herschel, "On the investigation of the orbits of revolving double stars: Being a supplement to a paper entitled 'micrometrical measures of double stars'", Memoirs of the Royal Astronomical Society  1833)

"The analytical equations, unknown to the ancient geometers, which Descartes was the first to introduce into the study of curves and surfaces, are not restricted to the properties of figures, and to those properties which are the object of rational mechanics; they extend to all general phenomena. There cannot be a language more universal and more simple, more free from errors and from obscurities, that is to say more worthy to express the invariable relations of natural things." (Joseph Fourier, "The Analytical Theory of Heat", 1822)

"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)

"Even if a curve is not drawn nor it is assumed to be tracked by the eye, but we have a 'perception' of it, it has in any case a limited precision and does not therefore correspond to the exact concept of a function of precision mathematics but rather to the idea of a function stripe." (Felix Klein, 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)

"It then becomes a method of graphical interpolation or extrapolation, which involves hypothetically extending a curve within or beyond the range of known data points, assuming the continuity of its pattern. In this way, one can fill in gaps in past observations and even probe the depths of the future." (Emile Cheysson, "La Statistique géométrique", 1888)

"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)

🧊On Geometry: On Curves (1700-1799)

"Even though these are called imaginary, they continue to be useful and even necessary in expressing real magnitudes analytically. For example, it is impossible to express the analytic value of a straight line necessary to trisect a given angle without the aid of imaginaries. Just so it is impossible to establish our calculus of transcendent curves without using differences which are on the point of vanishing, and at last taking the incomparably small in place of the quantity to which we can assign smaller values to infinity." (Gottfried W Leibniz, [letter to Varignon], 1702)

"It is held because of this law in particular, that no change may occur suddenly, but rather that every change always passes by infinitely small stages, of which the trajectory of a point in a curved line provides a first example." (Abraham G Kästner, "Anfangsgründe der Analysis des Unendlichen" ["Beginnings of the Analysis of the Infinite"], 1766)

"In order to know the curvature of a curve, the determination of the radius of the osculating circle furnishes us the best measure, where for each point of the curve we find a circle whose curvature is precisely the same. However, when one looks for the curvature of a surface, the question is very equivocal and not at all susceptible to an absolute response, as in the case above. There are only spherical surfaces where one would be able to measure the curvature, assuming the curvature of the sphere is the curvature of its great circles, and whose radius could be considered the appropriate measure. But for other surfaces one doesn’t know even how to compare a surface with a sphere, as when one can always compare the curvature of a curve with that of a circle. The reason is evident, since at each point of a surface there are an infinite number of different curvatures. One has to only consider a cylinder, where along the directions parallel to the axis, there is no curvature, whereas in the directions perpendicular to the axis, which are circles, the curvatures are all the same, and all other oblique sections to the axis give a particular curvature. It’s the same for all other surfaces, where it can happen that in one direction the curvature is convex, and in another it is concave, as in those resembling a saddle." (Leonhard Euler, "Recherches sur la courbure des surfaces", 1767)

"This method of subjecting the infinite to algebraic manipulations is called differential and integral calculus. It is the art of numbering and measuring with precision things the existence of which we cannot even conceive. Indeed, would you not think that you are being laughed at, when told that there are lines infinitely great which form infinitely small angles? Or that a line which is straight so long as it is finite would, by changing its direction infinitely little, become an infinite curve? Or that there are infinite squares, infinite cubes, and infinities of infinities, one greater than another, and that, as compared with the ultimate infinitude, those which precede it are as nought. All these things at first appear as excess of frenzy; yet, they bespeak the great scope and subtlety of the human spirit, for they have led to the discovery of truths hitherto undreamt of." (Voltaire, "Lettres philosophiques", 1778)

"The ordinary operations of algebra suffice to resolve problems in the theory of curves." (Joseph-Louis de Lagrange, "Théorie des fonctions analytiques", 1797)

🧊On Geometry: On Curves (1970-1979)

"Well it's a matter of continuity. Most people's lives have ups and downs that are gradual, a sinuous curve with first derivatives at every point. They're the ones who never get struck by lightning. No real idea of cataclysm at all. But the ones who do get hit experience a singular point. a discontinuity in the curve of life - do you know what the time rate of change is at a cusp? Infinity, that's what! A-and right across the point, it's minus infinity! How's that for sudden change, eh?" (Thomas Pynehon, "Gravity's Rainbow", 1973)

"Curves and surfaces in Euclidean space were studied since the earliest days of geometry and, after they were invented, both analytic geometry and calculus were systematically used in these studies. However, the discoveries of Gauss, announced in 1827, profoundly altered the course of differential geometry and pointed the way to the concept of abstract difierentiable manifolds-the underlying spaces of every geometry and of other important mathematical theories as well. In his celebrated 'Theorema Egregium' Gauss showed that there is a measure of curvature of a surface (now called the Gaussian curvature) which depends only on one‘s ability to measure the lengths of curves on the surface. This means that this curvature is unchanged by alterations of shape of the surface which leave arclength unchanged. [...] This discovery of an 'inner' geometry, independent of the shape of the surface in E3, led very naturally toward the invention of abstract surfaces (2-manifolds) on which a measure of arclength is (somehow) provided." (William M Boothby, "Riemannian geometry: An introduction to differentiable Manifolds and Riemannian Geometry", 1975)

"Is evolution a theory, a system or a hypothesis? It is much more: it is a general condition to which all theories, all hypotheses, all systems must bow and which they must satisfy henceforth if they are to be thinkable and true. Evolution is a light illuminating all facts, a curve that all lines must follow." (Pierre T de Chardin, "The Phenomenon of Man", 1975)

"Probably one of the most common misuses (intentional or otherwise) of a graph is the choice of the wrong scale - wrong, that is, from the standpoint of accurate representation of the facts. Even though not deliberate, selection of a scale that magnifies or reduces - even distorts - the appearance of a curve can mislead the viewer." (Peter H Selby, "Interpreting Graphs and Tables", 1976)

"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)

"At a simpler level, some elementary but important suggestions for the clarity of graphs are as follows: (i) the axes should be clearly labelled with the names of the variables and the units of measurement; (ii) scale breaks should be used for false origins; (iii) comparison of related diagrams should be made easy, for example by using identical scales of measurement and placing diagrams side by side; (iv) scales should be arranged so that systematic and approximately linear relations are plotted at roughly 45° to the x-axis; (v) legends should make diagrams as nearly self-explanatory, i.e. independent of the text, as is feasible; (vi) interpretation should not be prejudiced by the technique of presentation, for example by superimposing thick smooth curves on scatter diagrams of points faintly reproduced." (David R Cox,"Some Remarks on the Role in Statistics of Graphical Methods", Applied Statistics 27 (1), 1978)

"Because of its foundation in topology, catastrophe theory is qualitative, not quantitative. Just as geometry treated the properties of a triangle without regard to its size, so topology deals with properties that have no magnitude, for example, the property of a given point being inside or outside a closed curve or surface. This property is what topologists call 'invariant' -it does not change even when the curve is distorted. A topologist may work with seven-dimensional space, but he does not and cannot measure" (in the ordinary sense) along any of those dimensions. The ability to classify and manipulate all types of form is achieved only by giving up concepts such as size, distance, and rate. So while catastrophe theory is well suited to describe and even to predict the shape of processes, its descriptions and predictions are not quantitative like those of theories built upon calculus. Instead, they are rather like maps without a scale: they tell us that there are mountains to the left, a river to the right, and a cliff somewhere ahead, but not how far away each is, or how large." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)

🧊On Geometry: On Projective Geometry

"But it is a third geometry from which quantity is completely excluded and which is purely qualitative; this is analysis situs. In this discipline, two figures are equivalent whenever one can pass from one to the other by a continuous deformation; whatever else the law of this deformation may be, it must be continuous. Thus, a circle is equivalent to an ellipse or even to an arbitrary closed curve, but it is not equivalent to a straight-line segment since this segment is not closed. A sphere is equivalent to any convex surface; it is not equivalent to a torus since there is a hole in a torus and in a sphere there is not. Imagine an arbitrary design and a copy of this same design executed by an unskilled draftsman; the properties are altered, the straight lines drawn by an inexperienced hand have suffered unfortunate deviations and contain awkward bends. From the point of view of metric geometry, and even of projective geometry, the two figures are not equivalent; on the contrary, from the point of view of analysis situs, they are." (Henri Poincaré,"Dernières pensées", 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)

"The disadvantage of differential geometry, as compared with Euclidean or projective geometry and also topology, is that we are not in a position to found it on invariant basic concepts" (fundamental relations) and axioms therefor. The situation is no different for conformal geometry on a Riemann surface." (Hermann Weyl, "The Concept of a Riemann Surface", 1913)

"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)

"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)

"[…] 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)

"There are many useful connections between these two disciplines [geometry and algebra]. Many applications of algebra to geometry and of geometry to algebra were known in antiquity; nearer to our time there appeared the important subject of analytical geometry, which led to algebraic geometry, a vast and rapidly developing science, concerned equally with algebra and geometry. Algebraic methods are now used in projective geometry, so that it is uncertain whether projective geometry should be called a branch of geometry or algebra. In the same way the study of complex numbers, which arises primarily within the bounds of algebra, proved to be very closely connected with geometry; this can be seen if only from the fact that geometers, perhaps, made a greater contribution to the development of the theory than algebraists." (Isaak M Yaglom, "Complex Numbers in Geometry", 1968)

"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)

"Projective geometry has opened up for us with the greatest facility new territories in our science, and has rightly been called the royal road to our particular field of knowledge." (Felix Klein)

On Geometrical Figures: On Parabola

"To a given right line to apply a parallelogram equal to a given triangle in an angle which is equal to a given right lined angle. According to the Familiars of Eudemus, the inventions respecting the application, excess, and defect of spaces, is ancient and belongs to the Pythagoric muse. But junior mathematicians receiving names from these, transferred them to the lines which are called conic, because one of these they denominate a parabola, but the other an hyperbola, and the third an ellipsis; since, indeed these ancient and divine men, in the plane description of spaces on a terminated right line, regarded the things indicated by these appellations. For when a right line being proposed, you adapt a given space to the whole right line, then that space is said to be applied, but when you make the longitude of the space greater than that of the right line, then the space is said to exceed; but when less, so that some part of the right line is external to the described space, then the space is said to be deficient." (Proclus Lycaeus, cca 5th century)

"Whenever two unknown magnitudes appear in a final equation, we have a locus, the extremity of one of the unknown magnitudes describing a straight line or a curve. The straight line is simple and unique; the classes of curves are indefinitely many, - circle, parabola, hyperbola, ellipse, etc." (Pierre de Fermat, "Introduction aux Lieux Plans et Solides", 1679)

"The Ellipse is the most simple of the Conic Sections, most known, and nearest of Kin to a Circle, and easiest describ'd by the Hand in plano. Though many prefer the Parabola before it, for the Simplicity of the Æquation by which it is express'd. But by this Reason the Parabola ought to be preferr'd before the Circle it self, which it never is. Therefore the reasoning from the Simplicity of the Æquation will not hold. The modern Geometers are too fond of the Speculation of Æquations." (Isaac Newton, "Arithmetica Universalis", 1707)

"Euclidean mathematics assumes the completeness and invariability of mathematical forms; these forms it describes with appropriate accuracy and enumerates their inherent and related properties with perfect clearness, order, and completeness, that is, Euclidean mathematics operates on forms after the manner that anatomy operates on the dead body and its members. On the other hand, the mathematics of variable magnitudes - function theory or analysis - considers mathematical forms in their genesis. By writing the equation of the parabola, we express its law of generation, the law according to which the variable point moves. The path, produced before the eyes of the student by a point moving in accordance to this law, is the parabola. [...] If, then, Euclidean mathematics treats space and number forms after the manner in which anatomy treats the dead body, modern mathematics deals, as it were, with the living body, with growing and changing forms, and thus furnishes an insight, not only into nature as she is and appears, but also into nature as she generates and creates, - reveals her transition steps and in so doing creates a mind for and understanding of the laws of becoming. Thus modern mathematics bears the same relation to Euclidean mathematics that physiology or biology […] bears to anatomy." (Christian H Dillmann," Die Mathematik die Fackelträgerin einer neuen Zeit", 1889)

"A surface which can be regarded as the set of successive position of a curve moving in space is said to be generated by the curve. The utility of this notion in constructing a surface geometrically, in a picture or as a model is increased as the complexity of the generator and its motion is decreased. When the generator is a straight line, it is called a ruled surface. Since you can exchange X and Y in the above analysis, the hyperbolic paraboloid is generated by a line in two ways. It is a doubly ruled surface." (George K Francis, "A Topological Picturebook", 1987)