"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)
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