Editing Euclidean geometry (section)

==Treatment of infinity==

===Infinite objects===
Euclid sometimes distinguished explicitly between "finite lines" (e.g., Postulate 2) and "[[infinity|infinite]] lines" (book I, proposition 12). However, he typically did not make such distinctions unless they were necessary. The postulates do not explicitly refer to infinite lines, although for example some commentators interpret postulate 3, existence of a circle with any radius, as implying that space is infinite.<ref name="Heath, p.&nbsp;200"/>

The notion of [[infinitesimals|infinitesimal quantities]] had previously been discussed extensively by the [[Eleatic School]], but nobody had been able to put them on a firm logical basis, with paradoxes such as [[Zeno's paradox]] occurring that had not been resolved to universal satisfaction. Euclid used the [[method of exhaustion]] rather than infinitesimals.<ref>Ball, p.&nbsp;31.</ref>

Later ancient commentators, such as [[Proclus]] (410–485 CE), treated many questions about infinity as issues demanding proof and, e.g., Proclus claimed to prove the infinite divisibility of a line, based on a proof by contradiction in which he considered the cases of even and odd numbers of points constituting it.<ref>Heath, p.&nbsp;268.</ref>

At the turn of the 20th century, [[Otto Stolz]], [[Paul du Bois-Reymond]], [[Giuseppe Veronese]], and others produced controversial work on [[Archimedean property|non-Archimedean]] models of Euclidean geometry, in which the distance between two points may be infinite or infinitesimal, in the [[Isaac Newton|Newton]]–[[Gottfried Leibniz|Leibniz]] sense.<ref>Giuseppe Veronese, On Non-Archimedean Geometry, 1908. English translation in Real Numbers, Generalizations of the Reals, and Theories of Continua, ed. Philip Ehrlich, Kluwer, 1994.</ref> Fifty years later, [[Abraham Robinson]] provided a rigorous logical foundation for Veronese's work.<ref>Robinson, Abraham (1966). Non-standard analysis.</ref>

===Infinite processes===
Ancient geometers may have considered the parallel postulate – that two parallel lines do not ever intersect – less certain than the others because it makes a statement about infinitely remote regions of space, and so cannot be physically verified.<ref>Nagel and Newman, 1958, p.&nbsp;9.</ref>

The modern formulation of [[proof by induction]] was not developed until the 17th century, but some later commentators consider it implicit in some of Euclid's proofs, e.g., the proof of the infinitude of primes.<ref>Cajori (1918), p.&nbsp;197.</ref>

Supposed paradoxes involving infinite series, such as [[Zeno's paradox]], predated Euclid. Euclid avoided such discussions, giving, for example, the expression for the partial sums of the [[geometric series]] in IX.35 without commenting on the possibility of letting the number of terms become infinite.