For, if the angle AGH is unequal to the angle GHD, one of them is greater.
Let the angle AGH be greater.
Let the angle BGH be added to each;
therefore the angles AGH, BGH are greater than the angles BGH, GHD.
But the angles AGH, BGH are equal to two right angles; [I. 13]
therefore the angles BGH, GHD are less than two right angles.
But straight lines produced indefinitely from angles less than two right angles meet; [Post. 5]
therefore AB, CD, if produced indefinitely, will meet;
but they do not meet, because they are by hypothesis parallel.
Therefore the angle AGH is not unequal to the angle GHD, and is therefore equal to it.
Again, the angle AGH is equal to the angle EGB; [I. 15]
therefore the angle EGB is also equal to the angle GHD. [C.N. 1]
Let the angle BGH be added to each;
therefore the angles EGB, BGH are equal to the angles BGH, GHD. [C.N. 2]
But the angles EGB, BGH are equal to two right angles; [I. 13]
therefore the angles BGH, GHD are also equal to two right angles.
Si enim non sunt æquales, sit alter, nempe A G H, maior.
Quoniam igitur angulus A G H, maior est angulo D H G,
si addatur communis angulus BGH;
erunt duo A G H, B G H, maiores duobus D H G, B G H:
At duo A G H, B G H, æquales sint duobus rectis.
duo D H G, B G H, minores sunt duobus rectis.
Quare cum sint interni, & ad easdem partes, B, & D, coibunt lineæ A B, C D, ad eas partes,
quod est absurdum, cum ponantur esse parallelæ.
Non est igitur angulus A G H, maior angulo D H G: Sed neque minor. Eadem enim ratione ostenderetur, rectas coire ad partes A, & C.
Igitur æquales erunt anguli alterni A G H, D H G. Eademque est ratio de angulis alternis B G H, C H G.
DICO secundo, angulum externum A G E, æqualem esse interno, & ad easdem partes opposito C H G. Quoniam enim angulo B G H, æqualis est alternus C H G, ut ostensum est; & eidem B G H, æqualis est angulus A G E. Erunt anguli A G E, C H G, inter se quoque æquales. Eodem modo demonstrabitur, angulum B G E, æqualem esse angulo D H G.
DICO tertio, angulos internos ad easdem partes, A G H, C H G, æquales esse duobus rectis. Quoniam enim ostensum fuit, angulum externum A G E, æqualem esse angulo C H E, interno; si addatur communis A G H,
erunt duo A G E, A G H, duobus C H G, A G H, æquales:
Sed duo A G E, A G H, æquales sunt duobus rectis.
Igitur & duo anguli C H G, A G H, æquales duobus rectis erunr. Eodem modo anguli B G H, D H G, duobus erunt rectis æquales.
論曰。如云不然、
而甲庚辛、大於丁辛庚。
則丁辛庚、加辛庚乙。
宜小於辛庚甲、加辛庚乙矣。公論四
夫辛庚甲、辛庚乙。元與兩直角等。本篇十三據如彼論。
則丁辛庚、辛庚乙、兩角、小於兩直角。
而甲乙、丙丁、兩直線。向乙丁行。必相遇也。公論十一
可謂平行線乎。
次解曰。戊庚甲外角、與同方相對之庚辛丙內角、等。論曰。乙庚辛、與相對之丙辛庚、兩內角等。本題 則乙庚辛交角相等之戊庚甲、本篇十五 與丙辛庚、必等。公論一。
後解曰。甲庚辛、丙辛庚、兩內角、與兩直角等。
論曰。戊庚甲、與庚辛丙、兩角旣等。本題 而每加一甲庚辛角。
則庚辛丙、甲庚辛、兩角、與甲庚辛、戊庚甲、兩角、必等。公論二
夫甲庚辛、戊庚甲、本與兩直角等。本篇十三
則甲庚辛、丙辛庚、兩內角、亦與兩直角等。
Complete text
Book I