How are Euclid’s postulates different from his common notions?

In his foundational work, the Elements, Euclid categorized his initial unproved assumptions into two distinct groups: postulates and common notions.12

While the exact principle distinguishing them remains a subject of historical debate, the primary distinction is often viewed through the scope of their application:

• Postulates: These were considered specific to the study of geometry. They were introduced with the phrase "let there be demanded," suggesting they were required assumptions specific to geometric constructions and properties, such as the ability to draw a straight line between two points or the equality of all right angles.23
• Common Notions: These were viewed as general principles or axioms that apply across all sciences concerned with quantity, or even to all sciences universally. They represent self-evident logical truths, such as "things equal to the same thing are equal" or "the whole is greater than a part".123

Although Proclus noted that some philosophers considered these terms synonymous, Euclid’s classification reflects a structural separation between domain-specific geometric requirements and universal logical rules.2