線喻
參考文獻

1 2  "divided line," The Cambridge Dictionary of Philosophy, 2nd edition, Cambridge University Press, 1999, ISBN 0-521-63722-8, p. 239.
↑  Older Greek texts do not differentiate unequal from an equal (ανίσα, αν ίσα)
1 2 3  Plato, The Republic, Book 6, translated by Benjamin Jowett, online 的存檔，存档日期18 April 2009.
1 2  Desmond Lee and Rachana Kamtekar, The Republic, Notes to Book 6, Penguin, 1987, ISBN 0-14-044914-0.
↑  Let the length of AE be equal to  \scriptstyle 1 and that of AC equal to  \scriptstyle x, where  \scriptstyle 0<x<1 (following Socrates, however,  \scriptstyle 0<x<1/2; insofar as the equality of the lengths of BC and CD is concerned, the latter restriction is of no significance). The length of CE is thus equal to  \scriptstyle 1-x. It follows that the length of BC must be equal to  \scriptstyle x-x\times x\equiv (1-x)\times x, which is seen to be equal to the length of CD.
↑  Gail Fine, Knowledge and Belief in Republic V-VII, in G. Fine (ed.) Plato I (1990), also in S. Everson (ed.) Cambridge Companions to Ancient Thought I: Epistemology (Cambridge University Press: New York, 1990), pp. 85–115.
↑  Nicholas Denyer, Sun and line: the role of the Good, in G. R. F. Ferrari (ed.) The Cambridge Companion to Plato's Republic (Cambridge University Press: New York, 2007), pp. 284–309.
↑  Republic 520c （页面存档备份，存于）: "For once habituated ... you will know what each of the 'idols' is and whereof it is a semblance, because you have seen the reality of the beautiful, the just and the good."
↑  Rachel G.K. Singpurwalla, Plato’s Defense of Justice in the Republic （页面存档备份，存于） "justice is a virtue appropriate to both cities and individuals, and the nature or form of justice is the same in both (Republic, 435a)."
1 2  James Danaher, The Laws of Thought （页面存档备份，存于） "The restrictions Plato places on the laws of thought (i.e., "in the same respect," and "at the same time,") are an attempt to isolate the object of thought by removing it from all other time but the present and all respects but one."
↑  Cratylus 439d-e （页面存档备份，存于） "For if it is ever in the same state, then obviously at that time it is not changing (Plato's realism); and if it is always in the same state and is always the same, how can it ever change or move without relinquishing its own form (Aristotle's realism)"
↑  Republic 520c （页面存档备份，存于） "For once habituated you will discern them infinitely better than the dwellers there, and you will know what each of the 'idols' is and whereof it is a semblance, because you have seen the reality of the beautiful, the just and the good."
↑  Cratylus 439d-e （页面存档备份，存于） "How, then, can that which is never in the same state be anything? ... nor can it be known by anyone."
↑  Republic 4.436b （页面存档备份，存于） "It is obvious that the same thing will never do or suffer opposites in the same respect in relation to the same thing and at the same time"
↑  Republic 4.437a （页面存档备份，存于） "let us proceed on the hypothesis that this is so, with the understanding that, if it ever appear otherwise, everything that results from the assumption shall be invalidated"
↑  Also see the Timaeus 29c （页面存档备份，存于） on empirical "likelihood"
↑  It is interesting to note that modern logical analysis claims to prove that the potentially falsifiable "empirical content of a theory is exactly captured by ... axiomatization ... that uses axioms which are universal negations of conjunctions of atomic formulas" (C. Chambers, The Axiomatic Structure of Empirical Content) 的存檔，存档日期27 June 2010.
↑  Plato's Principle of Non-Contradiction (Republic, 4.436b) for the objective, physical world is presented with three axiomatic restrictions: The same thing ... cannot act or be acted upon ... in contrary ways ... (1) in the same part (2) in relation to the same thing (3) at the same time.
線喻（ A至C ）代表可见世界，（ CE ）代表可理解的世界。