The Elements (1482) - Page 16

Three geometric diagrams are placed in the left margin, corresponding to the propositions: 1\. (Prop. 29) Two horizontal lines, labeled 'b-a' and 'd-c', are intersected by a diagonal transversal line labeled 'e-f' passing through point 'g'. An additional point 'h' is marked on the bottom line to show intersection points. 2\. (Prop. 30) Three parallel horizontal lines are shown, labeled 'a-b' (top), 'e-f' (middle), and 'c-d' (bottom). A transversal line intersects all three at points labeled 'k' (on ab), 'l' (on ef), and 'm' (on cd). 3\. (Prop. 31) A point 'a' is situated above a horizontal line 'b-c'. A line 'a-d' is drawn to intersect the horizontal line, and another line 'a-e' is drawn through point 'a' parallel to 'b-c'.

...[right] angles: I say that the two lines c.d. and e.f. are parallel. First, let angle d.g.a be equal to angle f.b.g. Then, by Proposition 15 which states that vertical angles are equal, angle c.g.b will be equal to that same angle f.b.g. Therefore, by the preceding proposition, c.d. and e.f. are parallel. Again, let the two angles d.g.b and f.b.g. be equal to two right angles; and since by Proposition 13 two angles d.g.b and c.g.b are likewise equal to two right angles, angle c.g.b will be equal to angle f.b.g. Therefore, by the preceding proposition, c.d. and e.f. will be parallel; which is what was proposed.

Proposition 29.

If a line falls upon two parallel lines, the alternate angles will be equal to one another; and the exterior angle will be equal to the interior and opposite angle on the same side; and the two interior angles on the same side will be equal to two right angles.
¶ Let there be two parallel lines a.b. and c.d. upon which falls the line e.f., cutting them at points g. and h. The transcription reads 'b', but the geometric context and diagrams indicate the point 'h'. I say that the alternate angles g. and h. are equal; and that the exterior angle g. is equal to the interior and opposite angle h. taken on the same side; and that the interior angles g. and h. taken on the same side are equal to two right angles. This is the converse of the two preceding propositions. The first part is shown thus: If angle a.g.h. original: b.g.b - likely a transcription error for a.g.h is not equal to angle c.h.g., one of them will be greater. Let angle c.h.g. be greater. Since the two angles c.h.g. and g.h.d. are equal to two right angles, then by Proposition 13, the two angles a.g.h. and d.h.g. will be less than two right angles. Therefore, by the fourth postulate original: "quartam petitionem." This refers to Euclid's Fifth Postulate, often called the Parallel Postulate; in some medieval traditions, it was numbered as the fourth, the two lines a.b. and c.d., if extended, will meet on the side of b. and d. at some point, such as k. Therefore, they are not parallel by definition, which contradicts the hypothesis; and since this is impossible, the two alternate angles a.g.h. and c.h.g. will be equal, which is the first part of the proposition. From this, the second is clear: for by Proposition 15, angle a.g.h. is equal to angle e.g.b.; therefore, angle e.g.b. will be equal to angle c.h.g.—namely, the exterior to the interior—which is the second part of the proposition. From this again the third is clear: for by Proposition 13, the two angles e.g.b. and b.g.h. are equal to two right angles. Therefore, the two angles b.g.h. and d.h.g. original: c.b.g. will also be equal to two right angles, which are the two interior angles taken on the same side; which is the third part of the proposition.

Proposition 30.

If two lines are each parallel to the same line, they will also be parallel to each other.
¶ Let there be two lines a.b. and c.d., each of which is parallel to line e.f. I say that those two, namely a.b. and c.d., are parallel. This is universally true whether the two lines a.b. and c.d. are in the same plane original: "superficie", meaning a two-dimensional surface or plane with line e.f. or not; however, here it is understood only as if all are in one plane. For the fact that they are parallel when in different planes is proved in the ninth proposition of the eleventh book The author clarifies that while this proof focuses on 2D plane geometry, the principle holds true in 3D solid geometry as well. So, let them all be in one plane. Let a line be drawn cutting lines a.b., e.f., and c.d. at points k., l., and m. And because a.b. is parallel to e.f., angle b.k.l. will be equal to angle e.l.k. by the first part of the preceding proposition, since they are alternate; and because c.d. is parallel to e.f., the exterior angle k.l.e. will be equal to the interior angle c.m.l. by the second part of the preceding proposition. Therefore, angle b.k.l. is equal to angle c.m.l., and since these are alternate original: "coalterni", the lines a.b. and c.d. will be parallel by Proposition 27; which was the proposition.

Proposition 31.

To draw a line parallel to a given line through a given point outside it.
¶ A point given outside a line is understood to be such that when the line is extended in both directions, it does not pass through the point. Let the point a. be given outside the line b.c., from which it is required to draw a line parallel to b.c. I draw the line a.d. in any way it may happen, and upon point a., which is the extremity...