I am currently revising a paper in which I need to use both the terms

topology and

geometry in adjective form on many occasions (and as a contrast to each other).

This poses an interest expository conundrum: both "geometric" and geometrical" are correct adjectives for geometry, although the former is a good deal more common in mathematics than the latter. Nevertheless, both are correct, and you can find both used in the Wikipedia entry on geometry to which I linked above. However, I have

*never* seen the word "topologic" used (as "topological" seems to be used all of the time), though

at least one online dictionary claims that it is technically correct.

The expository issue is that it is rather awkward to use "topological" yet "geometric", so the best solution appears to be to use the term "geometrical", even though it is less standard and seems a bit less nice when considered on its own. (Part of the issue is that certain things, like "geometric series", appear to almost always use the other form of the word.)

I have decided that I would like to get to the bottom of this --- because I find this interesting --- so I decided to e-mail my colleague

Peter Neumann, who has a keen interest in the history of mathematics and who I judged most likely among all of my Mathematical Institute colleagues to know the history behind how the adjective forms of geometry and topology developed in different ways. (One possibility that has crossed my mind was because of the "ology" ending in "topology" but not the other, but I have not tried to check whether that difference is worth pursuing to try to figure out what's going on.)

Peter responded as follows:

*What a very interesting question. French has just the one form topologique, g\'eom\'etrique, arithm\'etique, alg\'ebrique. Similarly, German has just one form topologisch, geometrisch, arithmetisch, algebraisch.*

Yes, in English, most of these words used adjectivally have two forms. I think that topological is the only one to have just one. All the others can be geometric or geometrical, arithmetic or arithmetical, algebraic or algebraical (though this last is very uncommon now). I suspect that the -al forms may have been created in the 19th Century by people like Cayley, Sylvester, Hamilton, but that is a pure guess. I'll copy this to Alan Hughes of the Oxford English Dictionary. He may well know.

We do use geometrical, I think, in phrases like `geometrical drawing', geometrical argument', `geometrical proof'. But the geometric mean could not possibly be the geometrical mean, could it?
So that is where we now stand. Peter has cc'ed a person from the Oxford English Dictionary, who I hope will be able to shed some further light on the subject. (If Alan doesn't know or can't point out someone else to ask, I will try to figure out which of my humanities colleagues might be a good person to ask. Surely somebody at Oxford can help me get to the bottom of this?)

**Update (5/07/13):** I have now heard from Alan Hughes from the Oxford English Dictionary. His response was very illuminating:

*Topologic does occur, but much less frequently than topological: in Google
Books, 31k against 2.6m; in the Oxford English Corpus, 8 vs 1400; say three
orders of magnitude difference.*

Geometric and geometrical seem to be the oldest English words to have the
endings -metric and -metrical (geometrical is 14th c., geometric is 16th c.).
The revised (OED3) entry for -metrical says "Where matching formations in
-metric and -metrical exist, there is a tendency for the formation in -metrical
to be earlier." Geometric mean occurs from 1701, but we have no OED entry for
geometrical mean, supporting Peter's comment below [map: above].

Words in -ology generally > adjectives in -logical; but American English often
uses -logic for scientific words, e.g. hematologic instead of haematological,
geologic instead of geological. This may account for modern occurrences of
topologic (in OED1 the word is attested with quotations of 1872 and 1903, but
neither is to do with maths). The ending is later in English formations. OED3
notes at -ology "From the late 18th cent. onwards the element is freely used
with first elements of classical origin to form the names of branches of study."
Perhaps the lateness of -ology means there is less variability (between -ic and
-ical) than is the case with words in -metric(al).

There is sometimes a semantic difference between -ic and -ical: a historical
novel (about subjects in history) but a historic feat (it will go down in
history). The OED3 entry for algebraical (1571-, earlier than algebraic, 1653-)
says "Originally: = algebraic adj. 1. In later use chiefly: characteristic or
reminiscent of algebra."

My view is that you need have no qualms at all about using geometric and
topological in the same context, If one departs from what is usual usage, a
reader is apt to be distracted by the language away from the meaning intended.
So there you have it. A dude from the OED has spoken, and it turns out that my speculation that the difference might pertain to the ending 'ology' has some basis in truth.

I hope that you have enjoyed this grammatical excursion into the history of mathematics. Or maybe this has actually been a grammatic excursion. :)