Proof
By the Archimedean property (Definition V.4), some multiple of the
smaller magnitude exceeds the larger. Iterated halving (or more)
brings the remainder below the smaller in a finite number of steps.
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Full neighborhood
Required by (dependents) (6)
- X.2Proposition X.2If, when the lesser of two unequal magnitudes is continually subtracted in turn from the greater, that which is left…
- XII.2Proposition XII.2Circles are to one another as the squares on the diameters.
- XII.5Proposition XII.5Pyramids which are of the same height and have triangular bases are to one another as their bases.
- XII.10Proposition XII.10Any cone is a third part of the cylinder which has the same base with it and equal height.
- XII.16Proposition XII.16Given two circles about the same centre, to inscribe in the greater circle an equilateral polygon with an even number…
- XII.18Proposition XII.18Spheres are to one another in the triplicate ratio of their respective diameters.
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