# 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

Github: non-Github:
@Fitzpatrick

### References

#### Bibliography

1. 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

1. 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.
2. Please note that we always have the relation $$0 < \alpha < \beta$$, because we require $$k\ge 2$$.