...last is proved via a noted convergence principle (X,l): if from a given magnitude there is removed more than its half, and from the remainder more than its half, and so on, the remainder eventually becomes smaller than any preassigned finite magnitude. It is interesting to...

...to paraphrase Euclid, we can say that given two unequal quantities, from the greater we can subtract more than its half, and from the remainder more than its half, such that a quantity smaller than a given smaller quantity is always reached (Elements X. prop I)....

...greater of two proposed magnitudes be taken not less than its half, and from the remainder not less than its half, and so on ; there shall at length remain...magnitude less than the least of the proposed magnitudes. For if AB and с be the two proposed magnitudes, of which А в in tin greater, then because (by Cor....

...Euclid's Tenth Book: If fron the greater of two unequal magnitudes there be taken more than its half, arm from the remainder more than its half, and so on, there shall at length remain magnitude less than the smaller of the proposed magnitudes. Here the smaller of the proposed magnitudes...