Explanation: 1.2: Adding Equations Preserves Equality

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

(Common Notion 2 from Book 1 of Euclid's “Elements”)

And if equal things are added to equal things then the wholes are equal.

Modern Formulation

see1 addition of real numbers is cancellative.

Explanations: 1
Proofs: 2

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Adapted from (subject to copyright, with kind permission)

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


  1. In modern mathematics, the Euclidean space \(\mathbb R^n\) constitutes a normed vector space. In Euclid's "Elements", the set of real numbers is represented by the measures of geometrical objects like the lengths of segments, the magnitudes of angles, the areas of plane figures, the volumes of solids, etc.