Αναλυτική γεωμετρία (Greek Wikipedia)
Analysis of information sources in references of the Wikipedia article "Αναλυτική γεωμετρία" in Greek language version.
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- Boyer, Carl B. (1991). «The Age of Plato and Aristotle». A History of Mathematics (Second έκδοση). John Wiley & Sons, Inc. σελίδες 94–95. ISBN 0-471-54397-7.
Menaechmus apparently derived these properties of the conic sections and others as well. Since this material has a strong resemblance to the use of coordinates, as illustrated above, it has sometimes been maintained that Menaechmus had analytic geometry. Such a judgment is warranted only in part, for certainly Menaechmus was unaware that any equation in two unknown quantities determines a curve. In fact, the general concept of an equation in unknown quantities was alien to Greek thought. It was shortcomings in algebraic notations that, more than anything else, operated against the Greek achievement of a full-fledged coordinate geometry.
- Boyer, Carl B. (1991). «Apollonius of Perga». A History of Mathematics (Second έκδοση). John Wiley & Sons, Inc. σελίδες 142. ISBN 0-471-54397-7.
The Apollonian treatise On Determinate Section dealt with what might be called an analytic geometry of one dimension. It considered the following general problem, using the typical Greek algebraic analysis in geometric form: Given four points A, B, C, D on a straight line, determine a fifth point P on it such that the rectangle on AP and CP is in a given ratio to the rectangle on BP and DP. Here, too, the problem reduces easily to the solution of a quadratic; and, as in other cases, Apollonius treated the question exhaustively, including the limits of possibility and the number of solutions.
- Boyer, Carl B. (1991). «Apollonius of Perga». A History of Mathematics (Second έκδοση). John Wiley & Sons, Inc. σελίδες 156. ISBN 0-471-54397-7.
The method of Apollonius in the Conics in many respects are so similar to the modern approach that his work sometimes is judged to be an analytic geometry anticipating that of Descartes by 1800 years. The application of references lines in general, and of a diameter and a tangent at its extremity in particular, is, of course, not essentially different from the use of a coordinate frame, whether rectangular or, more generally, oblique. Distances measured along the diameter from the point of tangency are the abscissas, and segments parallel to the tangent and intercepted between the axis and the curve are the ordinates. The Apollonian relationship between these abscissas and the corresponding ordinates are nothing more nor less than rhetorical forms of the equations of the curves. However, Greek geometric algebra did not provide for negative magnitudes; moreover, the coordinate system was in every case superimposed a posteriori upon a given curve in order to study its properties. There appear to be no cases in ancient geometry in which a coordinate frame of reference was laid down a priori for purposes of graphical representation of an equation or relationship, whether symbolically or rhetorically expressed. Of Greek geometry we may say that equations are determined by curves, but not that curves are determined by equations. Coordinates, variables, and equations were subsidiary notions derived from a specific geometric situation; [...] That Apollonius, the greatest geometer of antiquity, failed to develop analytic geometry, was probably the result of a poverty of curves rather than of thought. General methods are not necessary when problems concern always one of a limited number of particular cases.
- Boyer (1991). «The Arabic Hegemony»
. A History of Mathematics. σελίδες 241–242. ISBN 9780471543978. Omar Khayyam (ca. 1050–1123), the "tent-maker," wrote an Algebra that went beyond that of al-Khwarizmi to include equations of third degree. Like his Arab predecessors, Omar Khayyam provided for quadratic equations both arithmetic and geometric solutions; for general cubic equations, he believed (mistakenly, as the sixteenth century later showed), arithmetic solutions were impossible; hence he gave only geometric solutions. The scheme of using intersecting conics to solve cubics had been used earlier by Menaechmus, Archimedes, and Alhazan, but Omar Khayyam took the praiseworthy step of generalizing the method to cover all third-degree equations (having positive roots). For equations of higher degree than three, Omar Khayyam evidently did not envision similar geometric methods, for space does not contain more than three dimensions, ... One of the most fruitful contributions of Arabic eclecticism was the tendency to close the gap between numerical and geometric algebra. The decisive step in this direction came much later with Descartes, but Omar Khayyam was moving in this direction when he wrote, "Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved."
- Cooke, Roger (1997). «The Calculus». The History of Mathematics: A Brief Course. Wiley-Interscience. σελίδες 326. ISBN 0-471-18082-3.
The person who is popularly credited with being the discoverer of analytic geometry was the philosopher René Descartes (1596–1650), one of the most influential thinkers of the modern era.
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- Cooper, Glen M. (2003). «Review: Omar Khayyam, the Mathmetician by R. Rashed, B. Vahabzadeh». The Journal of the American Oriental Society 123 (1): 248–249. doi:.
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- Cooper, Glen M. (2003). «Review: Omar Khayyam, the Mathmetician by R. Rashed, B. Vahabzadeh». The Journal of the American Oriental Society 123 (1): 248–249. doi:.
- Stillwell, John (2004). «Analytic Geometry». Mathematics and its History (Second έκδοση). Springer Science + Business Media Inc. σελ. 105. ISBN 0-387-95336-1.
the two founders of analytic geometry, Fermat and Descartes, were both strongly influenced by these developments.
- Cooke, Roger (1997). «The Calculus». The History of Mathematics: A Brief Course. Wiley-Interscience. σελίδες 326. ISBN 0-471-18082-3.
The person who is popularly credited with being the discoverer of analytic geometry was the philosopher René Descartes (1596–1650), one of the most influential thinkers of the modern era.