In Book I of Euclid's Elements, the fifth common notion says "The whole is greater than the part".

For Euclid, magnitudes are objects that can be compared, added, and subtracted, provided they are of the “same kind.” In Book I, the kinds of magnitudes that Euclid considers are (lengths of) line segments, (measures of) angles, and (areas of) triangles and quadrilaterals.

  1. Is there a criticism in literature to this axiom?

  2. If Euclid didn't state "The whole is equal to the sum of its parts", then who did?


The debate on Common notions is an old one.

See Heath's commentary to his edition of Euclid's Element (vol.I, page 221-on) for an overview and some useful informations.

According to Paul Tannery, Sur l'authenticité des axiomes d'Euclide (1884) all common notions were not in Euclid's work but were interpolated later.

A more recent debate is : Abraham Seidenberg, Did Euclid's Elements, book I, develop geometry axiomatically ? (AHES, 1975).

According to Heath (page 232) :

"Christopher Clavius (1538 – 1612) added the axiom that the whole is the equal to the sum of its parts."

See Euclidis Elementorum libri 15 (1574), page 20 :

XVIII Omne totum aequale est omnibus suis partibus simul sumptis.

But we can find an earlier occurrence in Thomas Bradwaride (ca.1300 – 1349)'s Geometria speculativa (printed in Paris, 1530) :

totus est equalibus suis partibus simul sumptis.


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