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MA232A - Euclidean and Non-Euclidean Geometry
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| Euclid, Book I, Proposition 27
(Heath's Edition) If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another. |
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In this statement of Proposition 27 of Book I of Euclid, the “alternate angles” are the angles AEF and EFD. The proof is a proof by contradiction (reductio ad absurdum) based on Proposition 16 of Book I of Euclid. For if the lines EB and FD were to meet at a point G, then EFG would be a triangle. Proposition 16 would ensure that the interior angle EFG of that triangle would be strictly less than the exterior angle AEF of that triangle, contradicting the requirement that these two angles be equal.
| Euclid, Book I, Proposition 28
(Heath's Edition) If a straight line falling on two straight lines make the exterior angle equal to the interior and opposite angle on the same side, or the interior angles on the same side equal to two right angles, the straight lines will be parallel to one another. |
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Suppose first that the “exterior angle” EGB is equal to the “interior and opposite angle” GHD. It follows from Proposition 15 of Book I of Euclid that the angle EGB equals the angle AGH. Therefore the “alternate angles” AGH and GHD are equal. Proposition 27 then ensures that the straight lines are parallel to one another.
| Euclid, Book I, Proposition 29
(Heath's Edition) A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior angles on the same side equal to two right angles. |
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To prove this, Euclid notes that if the alternate angles AGH and GHD were not equal then one would be greater than the other. Without loss of generality suppose that AGH were greater than GHD. Then the sum of AGH and BGH would be greater than the sum of GHD and BGH. But the sum of AGH and BGH is two right angles. Therefore the sum of GHD and BGH would be less than two right angles. The Parallel Postulate (Postulate 5, or Axiom 12) would then ensure that the lines GB and HD would intersect, contradicting the requirement that they be parallel. Thus the alternate angles AGH and GHD must be equal. But the angles AGH and EGB are opposite angles, and are therefore equal by Proposition 15. Thus the angles GHD, AGH and EGB are equal. In particular the exterior angle EGB is equal to the interior and opposite angle GHD. It also follows that the sum of GHD and BGH is equal to the sum of EGB and BGH and is therefore equal to two right angles.
In a note entitled Aristotle on Parallels, Heath presents some quotations from Aristotle that demonstrate that the results of Propositions and 27 and 28 were known and included in standard texts written by ancient Greek mathematicians prior to Euclid. Euclid then states a result, Proposition 29, which is essentially a converse of Propositions 27 and 28. However Euclid justifies Proposition 29 as an immediate consequence of the Parallel Postulate (which is Postulate 5 in editions of Euclid that derive from Heiberg's text, or Axiom 12 in earlier editions that derive from the Latin text of Commandinus). The remaining propositions of Book I of Euclid then depend on the Parallel Postulate itself or on the equivalent result expressed in Proposition 29. In a note on Postulate 5, Heath argues that this postulated originated with Euclid himself. Heath then continues by describing attempts by Ptolemy, Proclus, Nasir al-Din al-Tusi, John Wallis, Gerolamo Saccheri, Lambert and Legendre to prove the result of Postulate 5 from the other postulates and common notions. This is followed in Heath's edition by a listing of alternatives to Postulate 5.
In John Playfair's Edition of Euclid (1795), the postulates and axioms correspond to those in, for example, Todhunter's edition except that the axiom that Two straight lines cannot enclose a space has been deleted, the axiom that All right angles are equal to one another has been renumbered as Axiom 10, and old Axiom 12 has been replaced by the following (numbered as Axiom 11) which is today referred to as Playfair's Axiom:
Two straight lines cannot be drawn through the same point, parallel to the same straight line, without coinciding with one another.
Playfair deduces Proposition 29 of Book I of Euclid as a consequence of Propositions 13, 15, 27 of that book, together with Playfair's Axiom.
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Dr. David R. Wilkins, School of Mathematics, Trinity College Dublin.