"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. [...]
"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)
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