Arbitrary section name
This article needs an intro and headings. Ben Finn 12:58, 2 May 2007 (UTC)
Sure, here's my stab at the Latin. Corrections welcome.
"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."
A porism is a proposition in which one claims to prove that some thing (or many things) have been given. One must show that a certain shared "feeling" described in the proposition matches up with this thing, or with those things (or with any of a countless number of things) which have not actually been given, but have the same relationship to the things that have been given. A porism may also be expressed in the form of a problem, if, for example, one is asked to find those things from which what has been given is to be proved. 184.108.40.206 (talk) 01:08, 28 September 2013 (UTC)
Definition and current use
Would someone who's more of a specialist in the history of mathematics than I am please re-write that very first sentence, and write a lead section to put it in? The entire lead section as I write this is: "A porism is a mathematical proposition." This is, imo, almost worse than useless. Possibly some of the following may be of some assistance.
My sense of this word is that its nearest synonym is simply "corollary", i.e. a deduction from a previous demonstration. Or, more formally, if you like, Corollary: "A theorem that follows so obviously from the proof of some other theorem that no, or almost no proof is necessary; a by-product of another theorem." ( Mathematics Dictionary, James & James, 4th ed. )
This is what the Oxford English Dictionary ( Draft Rev. March 2010 ) gives as the definition of "porism":
"In Euclidean geometry: a proposition arising during the investigation of some other proposition by immediate deduction from it (= COROLLARY n. 1); (in later use) a special case of a problem in which the particular values of its parameters result in the solution being indeterminate. Other widely different definitions have also been given." (emphasis mine)
In post-classical Latin the word porisma meant a deduction, a corollary. Its predecessor in Hellenistic Greek meant a deduction from a previous demonstration, a corollary (Euclid), or - somewhat less helpfully, for our purposes - a kind of proposition intermediate between a theorem and a problem. The Hellenistic Greek word derives from an ancient Greek one which meant to carry, or to provide.
The term isn't found in Mathematics Dictionary, James & James, 4th ed. I know it's is used in connection with Steiner and Poncelet, of course, but it's my impression that the word isn't a current one at all. It appears to me that it's rarely or never used by working mathematicians today outside of its historical context. – OhioStandard (talk) 09:15, 6 October 2010 (UTC)
The article does explain historical usages
I wonder whether the public-domain article from EB (1911), which this article reuses, does not sufficiently (if unclearly) describe the various uses of the term "porism" a century ago? In particular, the author was pretty clear in telling us that "Pappus states":
the difference between the three classes... a theorem [is] directed to proving what is proposed, a problem [is] directed to constructing what is proposed, and finally a porism [is] directed to finding what is proposed
Further, he also gives the meanings assigned to the term by two later authors (Simson and Playfair), and states that:
[Playfair's] definition of a porism appears to be most favoured in England, Simson's view has been most generally accepted abroad
with the proviso that:
Breton ... sought to base thereon a view of the nature of a porism more closely conforming to the definitions in Pappus
further qualified by
a controversy between Breton and A. J. H. Vincent, who ... declared himself in favour of the idea of ... Frans van Schooten,
according to whom
...the discovery of innumerable new properties of the figure ... here we have "porisms".
Since, as you indicate, only the Poncelet-Steiner porism now regularly receives that epithet, it might be worth stating that to be the case - provided we can find a suitable reference for it. It might also be worth summarising the various uses as at 1911 (much along the lines above) and indicating that those were the usages then current.
Notwithstanding the status quo, there might be some value for today's mathematicians to return to Pappus' clear threefold division between things sought to be proven (theorems), constructed (problems) or found (porisms). But to say so would be non-encyclopaedic, or OR. yoyo (talk) 14:44, 12 September 2011 (UTC)
The article on Steiner's porism contains the following explanation:
A porism is a type of theorem relating to the number of solutions and the conditions on it. Porisms often describe a geometrical figure that cannot exist unless a condition is met, but otherwise may exist in infinite number; another example is Poncelet's porism.
Though this statement is unreferenced, it does show that at least one current writer on mathematics (possibly a mathematician!) does ascribe a clear meaning to the term "porism", and one specifically tied to the potential multiplicity or absence of solutions. yoyo (talk) 14:49, 12 September 2011 (UTC)
Pappas on Euclid's porism
The explanation includes the statement, "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."
First, I think what is meant is, "if two of the points of intersection of these lines lie each on one of the four lines". Then, "the remaining point of intersection" implies there are only 3 points of intersection, when there are up to 6. What must be meant there is "if two of the points of intersection of the first three lines lie each on one of those three lines." But then, with that revision, the claim is uninteresting: of course a point of intersection between three straight lines lies on at least one of them. The claim can't be interesting unless all 4 lines are being considered, in which case there are up to 6 points of intersection.
The word "another" in "another line" seems to have no possible meaning. If it means "another line besides the 4" it is trivial. If it means "another besides the 3" then it means the fourth, and should say so (and would be a false claim). If it means "another besides the 3 lines involved in determining these 2 points of intersection"... well, there aren't any other lines. Philgoetz (talk) 15:31, 15 June 2017 (UTC)