The word has changed its meaning more than once, and one trouble is that
the famous "Lost book of Porisms" by His-name's-on-the-tip-of-my-tongue
was indeed lost quite long ago, so that we're not quite sure what it
contained. The common thread seems to be "subtle proposition".
The word has been applied to geometric fallacies such as the well-known
(and apparently old) "theorem" that all triangles are equilateral; to
clever identities such as the above (especially when expressed in geometrical
language); and (what I think is the surviving use) to results having
properties like "Poncelet's Porism". Let me explain the latter before
I try to recall the formal definition I recall being taught at Cambridge.
Poncelet's porism says that if two conics are such that there's one
N-gon that's inscribed in one and circumscibed to the other, then there
are infinitely many such - indeed there's one starting at any point of
the circumscribing conic. This is an instance of a generic phenomenon -
that a construction requires some identity to hold for it to be possible,
but that when the identity DOES hold, the construction is possible in
an infinity of ways, and in traditional algebraic geometry such constructions
are called poristic.
I think this specialized use might also be ancient. Take the old
favorite of Martin Gardner: "I drill a cylindrical hole through a sphere,
the height of the cylinder being 6 inches. What was the volume removed?".
Here there doesn't seem to be enough information to answer the problem,
but in fact there is, since the 6-inch-high "holes" in spheres of
different sizes all happen to have the same volume. I think an ancient
Greek might have called this result a porism.
John Conway