Euclid — "If equals be added to equals, the wholes are equal."
If equals be added to equals, the wholes are equal.
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More from Euclid
"To describe a circle with any centre and radius."
"And the greater is a multiple of the less when it is measured by the less."
"A prime number is that which is measured by a unit alone."
"The extremities of a line are points."
"A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another."
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Understanding this quote
What it means
When you add the same amount to two already-equal quantities, the results remain equal. Equality is preserved through identical operations. This is a foundational logical principle: fairness and balance maintained through consistent treatment. It underpins arithmetic, algebra, and formal reasoning — if two things start the same and receive the same, they end the same.
Relevance to Euclid
Euclid codified this as Common Notion 2 in his Elements, the axiomatic system he constructed around 300 BCE. Rather than accepting mathematical truths informally, Euclid insisted every conclusion follow from explicit axioms. This quote embodies his core mission: build all of geometry on irrefutable, self-evident starting points, leaving nothing assumed.
The era
In ancient Alexandria under Ptolemaic rule, Greek intellectual culture demanded rigorous argumentation over intuition. Euclid worked at the Library of Alexandria during a period of unprecedented knowledge synthesis. Mathematics was philosophy — proving things beyond doubt mattered deeply. His axiomatic method directly responded to Platonic ideals of absolute truth and influenced every formal system that followed.
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