„Every measurable thing except numbers is imagined in the manner of a continuous quantity. Therefore, for the mensuration of such a thing, it is necessary that points, lines, and surfaces, or their properties, be imagined. For in them… measure or ratio is initially found… Therefore, every intensity which can be acquired successively ought to be imagined by a straight line perpendicularly erected on some point of the space or subject of the intensible thing, e. g., a quality… And since the quantity or ratio of lines is better known and is more readily conceived by us—nay the line is in the first species of continua, therefore such intensity ought to be imagined by lines… Therefore, equal intensities are designated by equal lines, a double intensity by a double line, and always in the same way if one proceeds proportionally.“

Tractatus de Configurationibus et Qualitatibus et Motuum (c. 1350)

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Nikolaus von Oresme Foto
Nikolaus von Oresme
französischer Bischof, Naturwissenschaftler und Philosoph, … 1323 - 1382

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Bernhard Riemann Foto

„Let us imagine that from any given point the system of shortest lines going out from it is constructed; the position of an arbitrary point may then be determined by the initial direction of the geodesic in which it lies, and by its distance measured along that line from the origin. It can therefore be expressed in terms of the ratios dx0 of the quantities dx in this geodesic, and of the length s of this line. …the square of the line-element is \sum (dx)^2 for infinitesimal values of the x, but the term of next order in it is equal to a homogeneous function of the second order… an infinitesimal, therefore, of the fourth order; so that we obtain a finite quantity on dividing this by the square of the infinitesimal triangle, whose vertices are (0,0,0,…), (x1, x2, x3,…), (dx1, dx2, dx3,…). This quantity retains the same value so long as… the two geodesics from 0 to x and from 0 to dx remain in the same surface-element; it depends therefore only on place and direction. It is obviously zero when the manifold represented is flat, i. e., when the squared line-element is reducible to \sum (dx)^2, and may therefore be regarded as the measure of the deviation of the manifoldness from flatness at the given point in the given surface-direction. Multiplied by -¾ it becomes equal to the quantity which Privy Councillor Gauss has called the total curvature of a surface. …The measure-relations of a manifoldness in which the line-element is the square root of a quadric differential may be expressed in a manner wholly independent of the choice of independent variables. A method entirely similar may for this purpose be applied also to the manifoldness in which the line-element has a less simple expression, e. g., the fourth root of a quartic differential. In this case the line-element, generally speaking, is no longer reducible to the form of the square root of a sum of squares, and therefore the deviation from flatness in the squared line-element is an infinitesimal of the second order, while in those manifoldnesses it was of the fourth order. This property of the last-named continua may thus be called flatness of the smallest parts. The most important property of these continua for our present purpose, for whose sake alone they are here investigated, is that the relations of the twofold ones may be geometrically represented by surfaces, and of the morefold ones may be reduced to those of the surfaces included in them…“

—  Bernhard Riemann German mathematician 1826 - 1866

On the Hypotheses which lie at the Bases of Geometry (1873)

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Archimedes Foto
Wassily Kandinsky Foto
John Dee Foto
Simone Weil Foto
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Leigh Hunt Foto

„There are two worlds: the world that we can measure with line and rule, and the world we feel with our hearts and imagination.“

—  Leigh Hunt English critic, essayist, poet and writer 1784 - 1859

As quoted in The Farmer's Wife, Vol. 36 (1933), p. 72

Jonathan Safran Foer Foto
Burkard Schliessmann Foto
Immanuel Kant Foto
Piet Mondrian Foto

„[the double line in his paintings] is still one line, as in the case of your grooves [= the wide sunken lines in the relief's, the artist Gorin made then]... In my last things the double line widens to form a plane, and yet it remains a line. Be that as it may, I believe that this question is one of those which lie beyond the realm of theory, and which are of such subtlety that they are rooted in the mystery of 'art.“

—  Piet Mondrian Peintre Néerlandais 1872 - 1944

But all that is not yet clear in my mind.
Quote in Mondrian's letter to artist Gorin, [who stated that the double line broke the necessary symmetry], 31 January, 1934; as quoted in Mondrian, - The Art of Destruction, Carel Blotkamp, Reaktion Books LTD. London 2001, p. 215

Aristarchus of Samos Foto
Gerald James Whitrow Foto
Fritz Houtermans Foto

„On a double-log plot, my grandmother fits on a straight line.“

—  Fritz Houtermans German physicist 1903 - 1966

as quoted in Herbert Kroemer's Autobiography http://nobelprize.org/nobel_prizes/physics/laureates/2000/kroemer-autobio.html, The Nobel Prize in Physics 2000.

Victor Hugo Foto

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