🧊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)