3

I want to say:

This technique is based on an observation we made about/on the theorem of ..."

Question: In the above sentence which one should I use, "about" or "on" or neither ?

To clarify the theorem has already exist but we made a new observation about/on the theorem.

2

Either is correct, and you can find current examples of both.

This American tourist's list of observations about small-town Britain has gone surprisingly viral

A 5 year old's observations on the Republican debate

In my opinion, "observations on" is slightly more old-fashioned and formal. Here is a Google ngrams search showing the declining use of "observation on" and the increasing use of "observation about": https://books.google.com/ngrams/graph?content=observation+on%2Cobservation+about&year_start=1800&year_end=2000&corpus=15&smoothing=3&share=&direct_url=t1%3B%2Cobservation%20on%3B%2Cc0%3B.t1%3B%2Cobservation%20about%3B%2Cc0

  • I'd say it's almost certain both of your cited usages carry the sense a statement based on something one has seen, heard, or noticed, whereas in OP's context he means the action or process of closely observing or monitoring something or someone. For the former sense, make [an] observation[s] is such a well-established usage that native speakers usually use a different verb such as conduct, carry out, perform when they mean the latter. – FumbleFingers Oct 29 '15 at 18:21
  • (The "decline" shown by your chart is entirely due to the fact that the "statement" sense is increasingly seen as a dated/formal/flowery usage.) – FumbleFingers Oct 29 '15 at 18:24

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