thy is an obsolete form of your.
To come under your observation = to become observed by you.
A rough paraphrase might be "all that crosses your path" or simply "everything that you notice" (and which you are able to continue observing).
Nothing has as much power to broaden the mind as the ability to investigate systematically and truly everything in life which crosses your path and can be studied further.
The legal matter came under the federal court's jurisdiction.
The phrase "thy observation in life" might be loosely paraphrased as "your (professional) bailiwick".
... the ability to study systematically all that falls under your professional scrutiny (or purview).