porism

The term originates from three books of Euclid with porism, that have been lost.

A locus (says Simson) is a species of porism.

A porism is a type of theorem relating to the number of solutions and the conditions on it.

A porism is a mathematical proposition or corollary.

A fundamental result is Steiner's porism, which states:

Pappus gives a complete enunciation of a porism derived from Euclid, and an extension of it to a more general case.

Pappus gives also a complete enunciation of one porism of the first book of Euclid's treatise.

Simson's treatise, De porismatibus, begins with definitions of theorem, problem, datum, porism and locus.

The required chord AC (in this example corresponding to sin(30+6)) is then calculated by application of the "Porism".

In modern textbooks the term porism [Eve, pg137] is considered an informal notion, applying to a fairly but not exactly defined class of propositions.

Respecting the porism Simson says that Pappus's definition is too general, and therefore he will substitute for it the following:

With the mention of the Porisms of Euclid we have an account of the relation of porism to theorem and problem.

The pentagram map is similar in spirit to the constructions underlying Desargues' Theorem and Poncelet's porism.

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.

Copernicus who used Ptolemy's theorem extensively in his trigonometrical work refers to this result as a 'Porism' or self-evident corollary:

Note that a proposition may not have been proven, so a porism may not be a theorem, or for that matter, it may not be true.

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."

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.

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.

A porism is a proposition in which there is a condition for a certain relation to subsists, but if the condition subsists the the relation subsists infinitely often.

Poncelet's porism can be proved via elliptic curves; geometrically this depends on the representation of an elliptic curve as the double cover of C with four ramification points.

Thus in cyclic quadrilateral BEDC, sides BE, BC and ED are known along with diagonals CE and BD by application of the "Porism" (Pythagoras Thm).

In geometry, Poncelet's porism (sometimes referred to as Poncelet's closure theorem), named after French engineer and mathematician Jean-Victor Poncelet, states the following: Let C and D be two plane conics.

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.

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.