Why do the titles of scholarly works sometimes begin with the word "on"?

Solution 1:

This naming convention extends back to ancient times.

• Many of Aristotle's (384–322 BC) works are titled On ---
• Lucretius (99 - 55 BC) wrote De rerum natura or On the Nature of Things
• In 44 BC Cicero wrote De officiis, or On Obligation

I am not certain whether these writers titled their own works. I'm vaguely remembering that these titles are simply how subsequent scholars referred to the works, but I haven't been able to verify this.

Scholars carried this convention forward. Montaigne, the Renaissance writer and father of the essay, titled many of his essays in this way.

When an author (or whoever is naming the article) uses On in their titles today they are joining this academic lineage, whether consciously or unconsciously. It may be an attempt to seem scholarly.

As for a difference in meaning,

I personally read a difference if the preposition is present or missing, though not enough to cause any real confusion. The On indicates that the writer is commenting on a known concept; the lack indicates they will be announcing a new one.

• "On the fibration method for zero-cycles and rational points" would mean that we already know about these fibration methods; the writer is adding to the discourse.
• "The fibration method for zero-cycles and rational points" would mean that the writer is announcing the discovery of these methods.

Solution 2:

The phenomenon appears to be a result of translation that dates back to early Greek writings. Early works by Greek writers would introduce a topic with Περὶ, meaning about, which Latin scholars translated to De, and were then translated to English as On. The trend of starting essays with On then stuck with English writers, who adopted it in their own titling.

Observation suggests that Περὶ was used to introduce an examination of a subject, such as medicine, grammar, sleep, dreams, etc. Titles by Aristotle that lack this introductory word tend to be those focused on more original concepts, such as Metaphysics (Greek: τὰ μετὰ τὰ φυσικά).

On who titled the works

It's not clear to me whether Aristotle provided titles for his own works, or if they were cataloged by other Greek scholars, particularly the scholars at the Library of Alexandria, where treatises were collected. Several sources, including Encyclopedia Britannica, seem to indicate that the titles were added by Alexandrian scholars, and as such the original titles would have been in Greek, even if they weren't named by Aristotle himself.*

This article from Standford indicates that much of the editorial work related to Aristotle's writings might have been done by Andronicus of Rhodes. Credit to Matt from History.SE for that finding.

Notably, translators used their discretion in applying the original naming to their well-known English names. For example, Aristotle's Poetics is seldom translated in English as On Poetics, though the original Greek is still Περὶ ποιητικῆς, and the Latin is De Poetica.

Another classic example from Aristotle:

• Greek: Περὶ οὐρανοῦ, 350 BC // Latin: De Caelo // English: On the Heavens (Aristotle)

Aristotle was not the first Greek writer to use this style. Writings by Hippocrates have similar titles in Greek.

• Greek: Περὶ Ἀρχαίας Ἰατρικῆς ?450-400BC // English: On Ancient Medicine (Hippocrates)

An example of an English writer adopting the style:

• On Liberty, 1859, John Stewart Mill [credit to WS2 in the comments for pointing to this example]

Additional examples from the Aristotle canon:

• Greek: Περὶ Ψυχῆς // Latin: De Anima // English: On the soul

• Greek: Περὶ αἰσθήσεως καὶ αἰσθητῶν // Latin: De sensu et sensibilibus // English: On Sense and the Sensible

• Greek: Περὶ μνήμης καὶ ἀναμνήσεως // Latin: De memoria et reminiscentia // English: On Memory

*Seeking some added clarity on the origin of the titles of the works, I've posted a question on the History Stack Exchange: Who titled the works of Aristotle?

Solution 3:

While all the other answers are very well thought out, let me point out a case where "On" keeps a title from being misleading.

The paper

On large subsets of 𝔽_qⁿ with no three-term arithmetic progression,

gives an upper bound on how large such subsets can be. In other words, it shows that no extremely large subsets exist. If this paper had been titled Large subsets ..., it would be misleading, as it might give the impression that it actually constructs large subsets with no three-term arithmetic progression.