Explanation: 1.3: Subtracting Equations Preserves Equality

(related to Subsection: Common Notions (all Books))

(Common Notion 3 from Book 1 of Euclid's “Elements”)

And if equal things are subtracted from equal things then the remainders are equal.

Modern Formulation

Since $x-a=x-b$ is equivalent to $x+(-a)=x+(-b)$, this is only a special case of the previous lemma. In Ancient mathematics, subtraction was considered as a separate operation rather than a special case of addition.

Proofs: 1 2 3 4

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Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"