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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Porism | 2/2 | https://en.wikipedia.org/wiki/Porism | reference | science, encyclopedia | 2026-05-05T07:24:49.702070+00:00 | kb-cron |
under certain conditions a problem becomes impossible; under certain other conditions, indeterminate or capable of an infinite number of solutions. These cases could be defined separately, were in a manner intermediate between theorems and problems, and were called "porisms." Playfair defined a porism as "[a] proposition affirming the possibility of finding such conditions as will render a certain problem indeterminate or capable of innumerable solutions." Although Playfair's definition of a porism appears to be most favoured in England, Simson's view has been most generally accepted abroad, and had the support of Michel Chasles. However, in Liouville's Journal de mathematiques pures et appliquées (vol. xx., July, 1855), P. Breton published Recherches nouvelles sur les porismes d'Euclide, in which he gave a new translation of the text of Pappus, and sought to base a view of the nature of a porism that conforms more closely to Pappus's definition. This was followed in the same journal and in La Science by a controversy between Breton and A. J. H. Vincent, who disputed the interpretation given by the former of Pappus's text, and declared himself in favour of Frans van Schooten's idea, put forward in his Mathematicae exercitationes (1657). According to Schooten, if the various relations between straight lines in a figure are written down in the form of equations or proportions, then the combination of these equations in all possible ways, and of new equations thus derived from them leads to the discovery of innumerable new properties of the figure. The discussions between Breton and Vincent, which C. Housel joined, did not carry forward the work of restoring Euclid's Porisms, which was left for Chasles. His work (Les Trois livres de porismes d'Euclide, Paris, 1860) makes full use of all the material found in Pappus. An interesting hypothesis about porisms was put forward by H. G. Zeuthen (Die Lehre von den Kegelschnitten im Altertum, 1886, ch. viii.). Zeuthen observed, for example the intercept-porism is still true if the two fixed points are points on a conic, and the straight lines drawn through them intersect on the conic instead of on a fixed straight line. He conjectured that the porisms were a by-product of a fully developed projective geometry of conics.
== See also ==
Poncelet's porism Steiner's porism
== Notes ==
== References == Alexander Jones (1986) Book 7 of the Collection, part 1: introduction, text, translation ISBN 0-387-96257-3, part 2: commentary, index, figures ISBN 3-540-96257-3, Springer-Verlag . J. L. Heiberg's Litterargeschichtliche Studien über Euklid (Leipzig, 1882) A valuable chapter on porisms (from a philological standpoint) is included. August Richter. Porismen nach Simson bearbeitet (Elbing, 1837) M. Cantor, "Über die Porismen des Euklid and deren Divinatoren," in Schlomilch's Zeitsch. f. Math. u. Phy. (1857), and Literaturzeitung (1861), p. 3 seq. Th. Leidenfrost, Die Porismen des Euklid (Programm der Realschule zu Weimar, 1863) John J. Milne (1911) An Elementary Treatise on Cross-Ratio Geometry with Historical Notes, page 115, Cambridge University Press. Fr. Buch-binder, Euclids Porismen und Data (Programm der kgl. Landesschule Pforta, 1866). Attribution:
This article incorporates text from a publication now in the public domain: Heath, Thomas Little (1911). "Porism". In Chisholm, Hugh (ed.). Encyclopædia Britannica. Vol. 22 (11th ed.). Cambridge University Press. pp. 102–103.