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Euclid: Elementa
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Proposition 27. 
THEOR. 18. PROPOS. 27. 
第二十七題 
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. 
SI in duas rectas lineas recta incidens linea alternatim angulos æquales inter se fecerit: parallelæ erunt inter se illæ rectæ lineæ. 
兩直線。有他直線交加其上若內相對兩角等。卽兩直線必平行。 
For let the straight line EF falling on the two straight lines AB, CD make the alternate angles AEF, EFD equal to one another;  I say that AB is parallel to CD. 
IN duas rectas A B, C D, incidens recta E F, faciat angulos alternatim A G H, D H G, inter se æquales.  Dico lineas A B, C D, esse parallelas. 
.  . 
For, if not, AB, CD when produced will meet either in the direction of B, D or towards A, C.  Let them be produced and meet, in the direction of B, D, at G.  Then, in the triangle GEF, the exterior angle AEF is equal to the interior and opposite angle EFG: which is impossible. [I. 16]  Therefore AB, CD when produced will not meet in the direction of B, D.  Similarly it can be proved that neither will they meet towards A, C.  But straight lines which do not meet in either direction are parallel; [Def. 23]  therefore AB is parallel to CD. 
Si enim non sunt parallelæ, coibunt tandem, si producantur infinite. Si namque non coirent unquam, parallelæ essent, ex parallelarum definitione.  Conueniant ergo ad partes B, & D, in puncto I.  Quoniam igitur triangulum est G I H, (cum A B, recta continuata sit, item recta C D, usque ad punctum I,) & angulus A G H, positus est æqualis angulo D H G; erit externus angulus A G H, æqualis interno, & opposito D H G, quod est absurdum; quoniam externus interno maior est.    Quod si A B, C D, coire dicantur ad partes A, & C, in puncto K, erit rursus eadem ratione angulus externus D H G, æqualis interno, & opposito A G H, quod est absurdum. Non igitur coibunt lineæ A B, C D.    Quare parallelæ erunt. Eodem modo, si ponantur anguli alterni B G H, C H G, æquales, demonstrabitur, lineas A B, C D, esse parallelas. 
.  .  .  .  .  .  . 
Therefore etc.  Q. E. D. 
Si igitur in duas rectas lineas recta incidens, &c.  Quod erat ostendendum.

SCHOLION
NECESSE est, ut lineæ, quæ dicuntur parallelæ, in eodem existant plano, ut ex definitione constat: Quare non satis est, duos angulos alternos æquales inter se esse, ut duæ lineæ probentur esse parallelæ, nisi ponatur, eas in uno, eodemque existere plano. Fieri enim potest, ut lineæ rectæ incidens in duas rectas non in eodem plano existentes, faciat alternos angulos æquales. Sit enim C D, perpendicularis ad A B, rectam, quæ in subiecto plano existit; & ex C, in alio plano, ad C D, ducatur alia perpendicularis C E, ita ut punctum E, intelligatur in sublimi. Quo posito, perspicuum est, rectam C D, incidentem in rectas C E, A B, facere duos angulos E C D, A D C, alternos æquales, cum sint recti; & tamen C E, A B, non sunt parallelæ, quod non in eodem existant plano. Non apposuit autem Euclides in propositione hanc conditionem; in eodem plano existentes: sicut neque in subsequentibus; quoniam cum in prioribus sex libris agatur de planis duntaxat, ut supra diximus, omnia intelligenda sunt necessario in eodem plano exsistere. In undecimo vero libro, & aliis, qui ipsum sequuntur, monebit semper, lineas aliquas in eodem esse plano, vel in diuersis planis, quia in illis libris disseritur de solidis, in quibus diuersa plana confiderari possunt. Quod idem dicendum est de punctis exira lineas, & superficies, &c. 
.  . 
 
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