Porism
Encyclopedia
A porism is a mathematical proposition
or corollary
. In particular, the term porism has been used to refer to a direct result of a proof, analogous to how a corollary refers to a direct result of a theorem
.
, the author of the Elements
. For as much as we know of this lost treatise we are indebted to the Collection of Pappus of Alexandria
, who mentions it along with other geometrical treatises, and gives a number of lemma
s necessary for understanding it. Pappus states
Pappus goes on to say that this last definition was changed by certain later geometers, who defined a porism on the ground of an accidental characteristic as
, that which falls short of a locus-theorem by a (or in its) hypothesis. Proclus points out that the word was used in two senses. One sense is that of "corollary," as a result unsought, as it were, but seen to follow from a theorem. On the "porism" in the other sense he adds 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
, ed. Friedlein, p. 301).
This may be expressed thus: If about two fixed points P, Q we make turn two straight lines meeting on a given straight line L, and if one of them cut off a segment AM from a fixed straight line AX, given in position, we can determine another fixed straight line BY, and a point B fixed on it, 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 the first segment AM. The rest of the enunciations given by Pappus are incomplete, and he merely says that he gives thirty-eight lemmas for the three books of porisms; and these include 171 theorems. The lemmas which Pappus gives in connexion with the porisms are interesting historically, because he gives:
During the last three centuries this subject seems to have had great fascination for mathematicians, and many geometers have attempted to restore the lost porisms. Thus Albert Girard
says in his Traité de trigonometrie (1626) that he hopes to publish a restoration. About the same time Pierre de Fermat
wrote a short work under the title Porismatum euclidaeorum renovata doctrina et sub forma isagoges recentioribus geometeis exhibita (see Œuvres de Fermat, i., Paris, 1891); but two at least of the five examples of porisms which he gives do not fall within the classes indicated by Pappus.
was the first to throw real light upon the subject. He first succeeded in explaining the only three propositions which Pappus indicates with any completeness. This explanation 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 definitions of theorem, problem, datum, porism and locus. Respecting the porism Simson says that Pappus's definition is too general, and therefore he will substitute for it the following:
A locus (says Simson) 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. These are Pappus's thirty-eight lemmas relating to the porisms, ten cases of the proposition concerning four straight lines, twenty-nine porisms, two problems in illustration and some preliminary lemmas.
John Playfair
's memoir (Trans. Roy. Soc. Edin., 1794, vol. iii.), a sort of sequel to Simson's treatise, had for its special object the inquiry into the probable origin of porisms, that is, into the steps which led the ancient geometers to the discovery of them. Playfair remarked that the careful investigation of all possible particular cases of a proposition would show that (1) under certain conditions a problem becomes impossible; (2) under certain other conditions, indeterminate or capable of an infinite number of solutions. These cases could be enunciated separately, were in a manner intermediate between theorems and problems, and were called "porisms." Playfair accordingly defined a porism thus: "A proposition affirming the possibility of finding such conditions as will render a certain problem indeterminate or capable of innumerable solutions."
Though this 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 thereon a view of the nature of a porism more closely conforming to the definitions in Pappus. 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 the text of Pappus, and declared himself in favour of the idea of Schooten, put forward in his Mathematicae exercitationes (1657), in which he gives the name of "porism" to one section. According to Frans van 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, and here we have "porisms."
The discussions, however, between Breton and Vincent, in which C. Housel also 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. But we may doubt its being a successful reproduction of Euclid's actual work. Thus, in view of the ancillary relation in which Pappus's lemmas generally stand to the works to which they refer, it seems incredible that the first seven out of thirty-eight lemmas should be really equivalent (as Chasles makes them) to Euclid's first seven Porisms. Again, Chasles seems to have been wrong in making the ten cases of the four-line Porism begin the book, instead of the intercept-Porism fully enunciated by Pappus, to which the "lemma to the first Porism" relates intelligibly, being a particular case of it.
An interesting hypothesis as to the Porisms was put forward by H. G. Zeuthen (Die Lehre von den Kegelschnitten im Altertum, 1886, ch. viii.). Observing, e.g., that 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, Zeuthen conjectures that the Porisms were a by-product of a fully developed projective geometry of conics. It is a fact that Lemma 31 (though it makes no mention of a conic) corresponds exactly to Apollonius
's method of determining the foci of a central conic (Conics, iii. 4547 with 42). The three porisms stated by Diophantus
in his Arithmetica are propositions in the theory of numbers which can all be enunciated in the form "we can find numbers satisfying such and such conditions"; they are sufficiently analogous therefore to the geometrical porism as defined in Pappus and Proclus
.
Proposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...
or corollary
Corollary
A corollary is a statement that follows readily from a previous statement.In mathematics a corollary typically follows a theorem. The use of the term corollary, rather than proposition or theorem, is intrinsically subjective...
. In particular, the term porism has been used to refer to a direct result of a proof, analogous to how a corollary refers to a direct result of a theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
.
Beginnings
The treatise which has given rise to this subject is the Porisms of EuclidEuclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
, the author of the Elements
Euclid's Elements
Euclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
. For as much as we know of this lost treatise we are indebted to the Collection of Pappus of Alexandria
Pappus of Alexandria
Pappus of Alexandria was one of the last great Greek mathematicians of Antiquity, known for his Synagoge or Collection , and for Pappus's Theorem in projective geometry...
, who mentions it along with other geometrical treatises, and gives a number of lemma
Lemma (mathematics)
In mathematics, a lemma is a proven proposition which is used as a stepping stone to a larger result rather than as a statement in-and-of itself...
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 really 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 goes on to say that this last definition was changed by certain later geometers, who defined a porism on the ground of an accidental characteristic as
, that which falls short of a locus-theorem by a (or in its) hypothesis. Proclus points out that the word was used in two senses. One sense is that of "corollary," as a result unsought, as it were, but seen to follow from a theorem. On the "porism" in the other sense he adds 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
Proclus Lycaeus , called "The Successor" or "Diadochos" , was a Greek Neoplatonist philosopher, one of the last major Classical philosophers . He set forth one of the most elaborate and fully developed systems of Neoplatonism...
, ed. Friedlein, p. 301).
Pappus on Euclid's porism
Pappus gives a complete enunciation of a porism derived from Euclid, and an extension of it to a more general case. This porism, expressed in modern language, asserts the following: 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 enunciation applies to any number of straight lines, say n + 1, of which n can turn about as many points fixed on the (n + 1)th. These n straight lines cut, two and two, in 1/2n(n − 1) points, 1/2n(n − 1) being a triangular number whose side is n − 1. If, then, they are made to turn about the n fixed points so that any n − 1 of their 1/2n(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, 1/2n(n − 1)(n − 2) in number, describes a straight line. Pappus gives also a complete enunciation of one porism of the first book of Euclid's treatise.This may be expressed thus: If about two fixed points P, Q we make turn two straight lines meeting on a given straight line L, and if one of them cut off a segment AM from a fixed straight line AX, given in position, we can determine another fixed straight line BY, and a point B fixed on it, 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 the first segment AM. The rest of the enunciations given by Pappus are incomplete, and he merely says that he gives thirty-eight lemmas for the three books of porisms; and these include 171 theorems. The lemmas which Pappus gives in connexion with the porisms are interesting historically, because he gives:
- the fundamental theorem that the cross or an harmonic 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 concourse of opposite sides lie on a straight line.
During the last three centuries this subject seems to have had great fascination for mathematicians, and many geometers have attempted to restore the lost porisms. Thus Albert Girard
Albert Girard
Albert Girard was a French-born mathematician. He studied at the University of Leiden. He "had early thoughts on the fundamental theorem of algebra" and gave the inductive definition for the Fibonacci numbers....
says in his Traité de trigonometrie (1626) that he hopes to publish a restoration. About the same time Pierre de Fermat
Pierre de Fermat
Pierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus, including his adequality...
wrote a short work under the title Porismatum euclidaeorum renovata doctrina et sub forma isagoges recentioribus geometeis exhibita (see Œuvres de Fermat, i., Paris, 1891); but two at least of the five examples of porisms which he gives do not fall within the classes indicated by Pappus.
Later analysis
Robert SimsonRobert Simson
Robert Simson was a Scottish mathematician and professor of mathematics at the University of Glasgow. The pedal line of a triangle is sometimes called the "Simson line" after him.-Life:...
was the first to throw real light upon the subject. He first succeeded in explaining the only three propositions which Pappus indicates with any completeness. This explanation 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 definitions of theorem, problem, datum, porism and locus. Respecting the porism Simson says that Pappus's definition is too general, and therefore he will substitute for it the following:
"Porisma est propositio in qua proponitur demonstrate rem aliquam vel plures Batas ease, cui vel quibus, ut et cuilibet ex rebus innumeris non quidem datis, sed quae ad ea quae data sunt eandem habent relationem, convenire ostendendum est affectionem quandam communem in propositione descriptam. Porisma etiam in forma problematis enuntiari potest, si nimirum ex quibus data demonstranda aunt, invenienda proponantur."
A locus (says Simson) 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. These are Pappus's thirty-eight lemmas relating to the porisms, ten cases of the proposition concerning four straight lines, twenty-nine porisms, two problems in illustration and some preliminary lemmas.
John Playfair
John Playfair
John Playfair FRSE, FRS was a Scottish scientist and mathematician, and a professor of natural philosophy at the University of Edinburgh. He is perhaps best known for his book Illustrations of the Huttonian Theory of the Earth , which summarized the work of James Hutton...
's memoir (Trans. Roy. Soc. Edin., 1794, vol. iii.), a sort of sequel to Simson's treatise, had for its special object the inquiry into the probable origin of porisms, that is, into the steps which led the ancient geometers to the discovery of them. Playfair remarked that the careful investigation of all possible particular cases of a proposition would show that (1) under certain conditions a problem becomes impossible; (2) under certain other conditions, indeterminate or capable of an infinite number of solutions. These cases could be enunciated separately, were in a manner intermediate between theorems and problems, and were called "porisms." Playfair accordingly defined a porism thus: "A proposition affirming the possibility of finding such conditions as will render a certain problem indeterminate or capable of innumerable solutions."
Though this 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
Michel Chasles
Michel Floréal Chasles was a French mathematician.He was born at Épernon in France and studied at the École Polytechnique in Paris under Siméon Denis Poisson. In the War of the Sixth Coalition he was drafted to fight in the defence of Paris in 1814...
. However, in Liouville
Joseph Liouville
- Life and work :Liouville graduated from the École Polytechnique in 1827. After some years as an assistant at various institutions including the Ecole Centrale Paris, he was appointed as professor at the École Polytechnique in 1838...
'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 thereon a view of the nature of a porism more closely conforming to the definitions in Pappus. 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 the text of Pappus, and declared himself in favour of the idea of Schooten, put forward in his Mathematicae exercitationes (1657), in which he gives the name of "porism" to one section. According to Frans van Schooten
Frans van Schooten
Franciscus van Schooten was a Dutch mathematician who is most known for popularizing the analytic geometry of René Descartes.-Life:...
, 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, and here we have "porisms."
The discussions, however, between Breton and Vincent, in which C. Housel also 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. But we may doubt its being a successful reproduction of Euclid's actual work. Thus, in view of the ancillary relation in which Pappus's lemmas generally stand to the works to which they refer, it seems incredible that the first seven out of thirty-eight lemmas should be really equivalent (as Chasles makes them) to Euclid's first seven Porisms. Again, Chasles seems to have been wrong in making the ten cases of the four-line Porism begin the book, instead of the intercept-Porism fully enunciated by Pappus, to which the "lemma to the first Porism" relates intelligibly, being a particular case of it.
An interesting hypothesis as to the Porisms was put forward by H. G. Zeuthen (Die Lehre von den Kegelschnitten im Altertum, 1886, ch. viii.). Observing, e.g., that 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, Zeuthen conjectures that the Porisms were a by-product of a fully developed projective geometry of conics. It is a fact that Lemma 31 (though it makes no mention of a conic) corresponds exactly to Apollonius
Apollonius of Perga
Apollonius of Perga [Pergaeus] was a Greek geometer and astronomer noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and René Descartes...
's method of determining the foci of a central conic (Conics, iii. 4547 with 42). The three porisms stated by Diophantus
Diophantus
Diophantus of Alexandria , sometimes called "the father of algebra", was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica. These texts deal with solving algebraic equations, many of which are now lost...
in his Arithmetica are propositions in the theory of numbers which can all be enunciated in the form "we can find numbers satisfying such and such conditions"; they are sufficiently analogous therefore to the geometrical porism as defined in Pappus and Proclus
Proclus
Proclus Lycaeus , called "The Successor" or "Diadochos" , was a Greek Neoplatonist philosopher, one of the last major Classical philosophers . He set forth one of the most elaborate and fully developed systems of Neoplatonism...
.