"To a given right line to apply a parallelogram equal to a given triangle in an angle which is equal to a given right lined angle. According to the Familiars of Eudemus, the inventions respecting the application, excess, and defect of spaces, is ancient and belongs to the Pythagoric muse. But junior mathematicians receiving names from these, transferred them to the lines which are called conic, because one of these they denominate a parabola, but the other an hyperbola, and the third an ellipsis; since, indeed these ancient and divine men, in the plane description of spaces on a terminated right line, regarded the things indicated by these appellations. For when a right line being proposed, you adapt a given space to the whole right line, then that space is said to be applied, but when you make the longitude of the space greater than that of the right line, then the space is said to exceed; but when less, so that some part of the right line is external to the described space, then the space is said to be deficient." (Proclus Lycaeus, cca 5th century)
"Whenever two unknown magnitudes appear in a final equation, we have a locus, the extremity of one of the unknown magnitudes describing a straight line or a curve. The straight line is simple and unique; the classes of curves are indefinitely many, - circle, parabola, hyperbola, ellipse, etc." (Pierre de Fermat, "Introduction aux Lieux Plans et Solides", 1679)
"The operations performed with imaginary characters, though destitute of meaning themselves, are yet notes of reference to others which are significant. They, point out indirectly a method of demonstrating a certain property of the hyperbola, and then leave us to conclude from analogy, that the same property belongs also to the circle. All that we are assured of by the imaginary investigation is, that its conclusion may, with all the strictness of mathematical reasoning, be proved of the hyperbola; but if from thence we would transfer that conclusion to the circle, it must be in consequence of the principle just now mentioned. The investigation therefore resolves itself ultimately into an argument from analogy; and, after the strictest examination, will be found without any other claim to the evidence of demonstration." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)
"It is well known that an initial value problem for a nonlinear ordinary differential equation may very well fail to have a solution for all time; the solution may blow up after a finite time. The same is true for quasi-linear hyperbolic partial differential equations: solutions may break down after a finite time when their first derivatives blow up." (Peter D Lax, "Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves", 1974)
"A surface which can be regarded as the set of successive position of a curve moving in space is said to be generated by the curve. The utility of this notion in constructing a surface geometrically, in a picture or as a model is increased as the complexity of the generator and its motion is decreased. When the generator is a straight line, it is called a ruled surface. Since you can exchange X and Y in the above analysis, the hyperbolic paraboloid is generated by a line in two ways. It is a doubly ruled surface." (George K Francis, "A Topological Picturebook", 1987)
"The Smale's horseshoe is the classical example of a structurally stable chaotic system: Its dynamical properties do not change under small perturbations, such as changes in control parameters. This is due to the horseshoe map being hyperbolic (i.e., the stable and unstable manifolds are transverse at each point of the invariant set)." (Robert Gilmore & Marc Lefranc, "TheTopologyof Chaos: Alice in Stretch and Squeezeland", 2002)
"When you encounter the classical wave equation, it’s likely to be accompanied by some or all of the words 'linear, homogeneous, second-order partial differential equation'. You may also see the word 'hyperbolic' included in the list of adjectives. Each of these terms has a very specific mathematical meaning that’s an important property of the classical wave equation. But there are versions of the wave equation to which some of these words don’t apply, so it’s useful to spend some time understanding them." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)
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