“The very fact that a Line is visible implies that it possesses yet another Dimension.”

—  Edwin Abbott Abbott , book Flatland

Source: Flatland: A Romance of Many Dimensions (1884), PART II: OTHER WORLDS, Chapter 16. How the Stranger Vainly Endeavoured to Reveal to Me in Words the Mysteries of Spaceland

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Edwin Abbott Abbott photo
Edwin Abbott Abbott 87
British theologian and author 1838–1926

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“Dogmatism as Nietzsche means it implies that one possesses the truth, or at least the most important or the most valuable truth. Yet the truth is elusive like that woman of whom he spoke at the very beginning. Elsewhere he says we are the first generation which no longer believes that it possesses the truth. That is what he means by the end of dogmatism.”

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“But the Moderns advancing yet much farther, have receiv'd into Geometry all Lines that can be express'd by Æquations, and have distinguish'd, according to the Dimensions of the Æquations, those Lines into Kinds; and have made it a Law, that you are not to construct a Problem by a Line of a superior Kind, that may be constructed by one of an inferior one.”

Isaac Newton book Arithmetica Universalis

Arithmetica Universalis (1707)
Context: The Antients, as we learn from Pappus, in vain endeavour'd at the Trisection of an Angle, and the finding out of two mean Proportionals by a right line and a Circle. Afterwards they began to consider the Properties of several other Lines. as the Conchoid, the Cissoid, and the Conick Sections, and by some of these to solve these Problems. At length, having more throughly examin'd the Matter, and the Conick Sections being receiv'd into Geometry, they distinguish'd Problems into three Kinds: viz. (1.) Into Plane ones, which deriving their Original from Lines on a Plane, may be solv'd by a right Line and a Circle; (2.) Into Solid ones, which were solved by Lines deriving their Original from the Consideration of a Solid, that is, of a Cone; (3.) And Linear ones, to the Solution of which were requir'd Lines more compounded. And according to this Distinction, we are not to solve solid Problems by other Lines than the Conick Sections; especially if no other Lines but right ones, a Circle, and the Conick Sections, must be receiv'd into Geometry. But the Moderns advancing yet much farther, have receiv'd into Geometry all Lines that can be express'd by Æquations, and have distinguish'd, according to the Dimensions of the Æquations, those Lines into Kinds; and have made it a Law, that you are not to construct a Problem by a Line of a superior Kind, that may be constructed by one of an inferior one. In the Contemplation of Lines, and finding out their Properties, I like their Distinction of them into Kinds, according to the Dimensions thy Æquations by which they are defin'd. But it is not the Æquation, but the Description that makes the Curve to be a Geometrical one.<!--pp.227-228

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