THE ELEMENTS, BOOK 1.

Common Notions.

original: Koinai ennoiai. While the Postulates on the previous page were specific to geometry, these Common Notions are general logical truths that apply to all mathematics and reasoning.

1. Things equal to the same thing are also equal to one another.
2. And if equals are added to equals, the wholes are equal.
3. And if equals are subtracted from equals, the remainders
5 are equal.
[4. And if equals are added to unequals, the wholes are unequal.]
[5. And doubles of the same thing are equal to one another.]
[6. And halves of the same thing are equal to one another.]
10 7. And things that coincide with one another are equal to one another. original: epharmodzonta. This refers to the idea of "superposition," where one shape is placed exactly over another to prove they are identical in size and shape.
8. And the whole is greater than the part.
[9. And two straight lines do not enclose a space.] Brackets in this edition indicate text that Heiberg believes was added by later editors rather than Euclid himself.

1.

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This is the first proposition of the Elements. It demonstrates how to build a triangle with three equal sides using only a compass and a straightedge.
Let AB be the given finite straight line.
It is required to construct an equilateral triangle on the straight line AB.
With center A and distance AB, let a circle...

The following citations refer to ancient and medieval authors who quoted or commented on these axioms and propositions. These include Martianus Capella, Philoponus in his commentary on Aristotle's Prior Analytics, Boethius, Eutocius in his commentary on Archimedes, Alexander of Aphrodisias, Themistius in his paraphrase of the Physics, Simplicius, and Proclus.

Common Notions 1 to 3: Martianus Capella VI, 723. 1: Philoponus in Prior Analytics II folio 5. Boethius page 378, 1. 2: Boethius page 378, 5. 3: Philoponus as cited above. Boethius page 378, 3. 4: Eutocius in Archimedes III page 254, 27. 7: Philoponus in Prior Analytics II folio 5. Boethius page 378, 7. Proposition 1: Alexander of Aphrodisias in Prior Analytics I folio 8 recto, in Topics page 11. Themistius, Paraphrase of the Physics folio 35 verso. Simplicius in Physics folio 119. Proclus page 102, 14; 223, 22. Philoponus in Prior Analytics II folio 4 verso. Martianus Capella VI, 724. Boethius page 380, 2 [pages 390, 6 to 25]. Proclus pages 208 to 210 repeats the entire proposition more freely.

1. Axioms: Proclus page 193. "These common notions" found in manuscripts B, F, and V. Numbers omitted in manuscripts P, B, and F. 3. "Equal things to equals" in Proclus. "Is equal" in Proclus. 4. "From equals, equals" is just "Equals" in Proclus. 5. "Is equal" in Proclus. Postulate 4 seems to have crept in from the commentary of Pappus: see Proclus.