Porism
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A porism is a mathematical
proposition In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
or
corollary In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...
. It has been used to refer to a direct consequence of a
proof Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a con ...
, analogous to how a corollary refers to a direct consequence of a
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
. In modern usage, it is a relationship that holds for an infinite range of values but only if a certain condition is assumed, such as
Steiner's porism In geometry, a Steiner chain is a set of circles, all of which are tangent to two given non-intersecting circles (blue and red in Figure 1), where is finite and each circle in the chain is tangent to the previous and next circles in the chain. ...
. The term originates from three books of
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
that have been lost. A proposition may not have been proven, so a porism may not be a
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
or true.


Origins

The book that talks about porisms first is
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
's ''Porisms''. What is known of it is in
Pappus of Alexandria Pappus of Alexandria (; grc-gre, Πάππος ὁ Ἀλεξανδρεύς; AD) was one of the last great Greek mathematicians of antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem i ...
's ''Collection'', who mentions it along with other geometrical treatises, and gives several
lemma Lemma may refer to: Language and linguistics * Lemma (morphology), the canonical, dictionary or citation form of a word * Lemma (psycholinguistics), a mental abstraction of a word about to be uttered Science and mathematics * Lemma (botany), a ...
s necessary for understanding it. Pappus states: :The porisms of all classes are neither theorems nor problems, but occupy a position intermediate between the two, so that their enunciations can be stated either as theorems or problems, and consequently some geometers think that they are theorems, and others that they are problems, being guided solely by the form of the enunciation. But it is clear from the definitions that the old geometers understood better the difference between the three classes. The older geometers regarded a theorem as directed to proving what is proposed, a problem as directed to constructing what is proposed, and finally a porism as directed to finding what is proposed ('). Pappus said that the last definition was changed by certain later geometers, who defined a porism as an accidental characteristic as (''to leîpon hypothései topikoû theōrḗmatos''), that which falls short of a locus-theorem by a (or in its) hypothesis. Proclus pointed out that the word ''porism'' was used in two senses: one sense is that of "corollary", as a result unsought but seen to follow from a theorem. In the other sense, he added nothing to the definition of "the older geometers", except to say that the finding of the center of a circle and the finding of the greatest common measure are porisms.
Proclus Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor ( grc-gre, Πρόκλος ὁ Διάδοχος, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers ...
, ed. Friedlein, p. 301


Pappus on Euclid's porism

Pappus rejected Euclid's definition of ''porism''. A porism, expressed in modern language, asserts that given four straight lines, of which three turn about the points in which they meet the fourth if two of the points of intersection of these lines lie each on a fixed straight line, the remaining point of intersection will also lie on another straight line. The general definition applies to any number, ''n'', of straight lines, of which ''n'' can turn about as many points fixed on the (''n'' + 1)th. These ''n'' straight lines cut two and two into ''n''(''n'' − 1) points, ''n''(''n'' − 1) being a triangular number whose side is ''n'' − 1. If they are made to turn about the ''n'' fixed points so that any ''n'' − 1 of their ''n''(''n'' − 1) points of intersection, chosen subject to a certain limitation, lie on ''n'' − 1 given fixed straight lines, then each of the remaining points of intersection, ''n''(''n'' − 1)(''n'' − 2) in number, describes a straight line. The above can be expressed as: If about two fixed points, P and Q, one makes the turn two straight lines meeting on a given straight line, L, and if one of them cuts off a segment, AM, from a fixed straight line, AX, given in position, another fixed straight line BY, and a point B fixed on it can be determined, such that the segment BM' made by the second moving line on this second fixed-line measured from B has a given ratio X to AM. The lemmas which Pappus gives in connection with the porisms are: #the fundamental theorem that the cross or anharmonic ratio of a pencil of four straight lines meeting in a point is constant for all transversals; #the proof of the harmonic properties of a complete quadrilateral; #the theorem that, if the six vertices of a hexagon lie three and three on two straight lines, the three points of the concourse of opposite sides lie on a straight line.


Later analysis

Robert Simson explained the only three propositions which Pappus indicates with any completeness, which was published in the ''Philosophical Transactions'' in 1723. Later he investigated the subject of porisms generally in a work entitled ''De porismatibus traclatus; quo doctrinam porisrnatum satis explicatam, et in posterum ab oblivion tutam fore sperat auctor'', and published after his death in a volume, ''Roberti Simson opera quaedam reliqua'' (Glasgow, 1776). Simson's treatise, ''De porismatibus'', begins with the definitions for theorem, problem, datum, porism, and locus. Simon wrote that Pappus's definition is too general, and that he substituted it as:
Porisma est propositio in qua proponitur demonstrare rem aliquam, vel plures datas esse, cui, vel quibus, ut et cuilibet ex rebus innumeris, non quidem datis, sed quae ad ea quae data sunt eandem habent rationem, convenire ostendendum est affectionem quandam communem in propositione descriptam. Porisma etiam in forma problematis enuntiari potest, si nimirum ex quibus data demonstranda sunt, invenienda proponantur.
Simson said that a locus is a species of porism. Then follows a Latin translation of Pappus's note on the porisms, and the propositions which form the bulk of the treatise. John Playfair's memoir (''Trans. Roy. Soc. Edin.'', 1794, vol. iii.), a sort of sequel to Simson's treatise, explored the probable origin of porisms, or the steps that led ancient geometers to discover them. Playfair remarked that the careful investigation of all possible particular cases of a proposition would show that # 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 " 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 Hieronymus Georg Zeuthen (15 February 1839 – 6 January 1920) was a Denmark, Danish mathematician. He is known for work on the enumerative geometry of conic sections, algebraic surfaces, and history of mathematics. Biography Zeuthen was born i ...
(''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 In geometry, a Steiner chain is a set of circles, all of which are tangent to two given non-intersecting circles (blue and red in Figure 1), where is finite and each circle in the chain is tangent to the previous and next circles in the chain. ...


Notes


References

*Alexander Jones (1986) ''Book 7 of the Collection'', part 1: introduction, text, translation , part 2: commentary, index, figures ,
Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...
. * 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 Moritz Benedikt Cantor (23 August 1829 – 10 April 1920) was a German historian of mathematics. Biography Cantor was born at Mannheim. He came from a Sephardi Jewish family that had emigrated to the Netherlands from Portugal, another branch of ...
, "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 Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing hou ...
. * Fr. Buch-binder, ''Euclids Porismen und Data'' (''Programm der kgl. Landesschule Pforta'', 1866). Attribution: *{{EB1911 , last=Heath , first=Thomas Little , author-link=Thomas Heath (classicist) , wstitle=Porism , volume=24 , pages=102-103 Mathematical terminology