 (3) The Mapping Argument (burhān al-taṭābuq or al-taṭbīq): Consider an actually infinite line \(AC\) which starts from \(A\) and extends infinitely in the direction of \(C\). Remove a finite segment \(AB\) from the beginning of \(AC\). Suppose that \(B^{*}C^{*}\) is a copy of (and, accordingly, of the same length as) \(BC\). Compare the size of \(B^{*}C^{*}\) with \(AC\) by mapping the former upon the latter so that the two lines are parallel and \(B^{*}\) is right in front of \(A\). \(B^{*}C^{*}\) must extend infinitely in the direction of \(C^{*}\). Otherwise, \(B^{*}C^{*}\) would be finite. This means that \(BC\) would be finite too. As a result, \(AC\)—which is the sum of \(BC\) with the finite segment \(AB\)—would be finite. Since this contradicts the initial assumption that \(AC\) is actually infinite, \(B^{*}C^{*}\) must extend infinitely in the direction of \(C^{*}\). But if so, then \(B^{*}C^{*}\) and \(AC\) correspond to each other, in the sense that no part of one of them remains uncovered by the other. So, based on the fourth common notion of the first book of Euclid’s Elements—according to which things which correspond to one another are equal to one another ([1908: vol. 1, 155])—we can conclude that \(AC\) is equal to \(B^{*}C^{*}\). This indicates that \(AC\) would also be equal to \(BC\), which is a proper part of \(AB\). However, the fifth Euclidean common notion states that such a whole-part equality is absurd ([1908: vol. 1, 155]). Therefore, \(AC\) cannot be equal to \(BC\). Consequently, the initial assumption that \(AC\) can be an actually infinite line must be rejected. There can be no such actually infinite magnitude.