Definition: 5.01: Magnitude is Aliquot Part

(Definition 1 from Book 5 of Euclid's “Elements”)

A magnitude is a part of a(nother) magnitude, the lesser of the greater, when it measures the greater.

Modern Formulation

A positive real number \(\alpha > 0\) is called an aliquot part of another positive real number¹ \(\beta\), if there exists a natural number \(k > 1\) such that² \[\beta=k\cdot \alpha.\]

Definitions: 1 2 3 4
Proofs: 5 6 7 8 9 10 11 12
Propositions: 13 14 15 16 17 18 19 20

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References

Bibliography

Health, T.L.: "The Thirteen Books of Euclid's Elements - With Introduction and Commentary by T. L. Health", Cambridge at the University Press, 1968, Vol 1, 2, 3

Adapted from (subject to copyright, with kind permission)

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

Footnotes

From a geometrical point of view, \(\alpha\) and \(\beta\) could measure the lengths of some segments, the areas of some plane figures or the volumes of some solids. ↩
Please note that we always have the relation \(0 < \alpha < \beta\), because we require \(k\ge 2\). ↩

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